🧠:[<think>Okay, so I need to find the binomial coefficient of the maximum term in the expansion of ((2x + frac{1}{x^2})^9). The options are given as A to D, with different terms specified. Let me recall how binomial expansions work. First, the general term in the expansion of ((a + b)^n) is given by (T_{k+1} = binom{n}{k} a^{n - k} b^{k}). In this case, (a = 2x) and (b = frac{1}{x^2}), and (n = 9). So the general term should be (T_{k+1} = binom{9}{k} (2x)^{9 - k} left(frac{1}{x^2}right)^k).Let me write that out more clearly. Let's compute each part step by step. The binomial coefficient is (binom{9}{k}), then we have ( (2x)^{9 - k} ) which is (2^{9 - k} x^{9 - k}), and (left(frac{1}{x^2}right)^k) is (x^{-2k}). So multiplying those together, the term is (binom{9}{k} 2^{9 - k} x^{9 - k - 2k} = binom{9}{k} 2^{9 - k} x^{9 - 3k}).So each term's exponent of x is (9 - 3k). But the problem is asking for the maximum term. Wait, maximum term in terms of numerical value? Since the expansion could have positive and negative terms, but here the coefficients might be positive because all terms in ((2x + 1/x^2)^9) would have coefficients multiplied by powers of 2 and 1, which are positive. So perhaps all terms are positive? Let me check.Wait, 2x is positive, 1/x² is positive for x positive, but the question doesn't specify the value of x. Hmm. Wait, but the problem is about the binomial coefficient of the maximum term. Wait, hold on. The question says "the binomial coefficient of the maximum term". Wait, the binomial coefficient is just (binom{9}{k}), right? So even if the term has other factors like 2^{9 - k}, the binomial coefficient itself is just (binom{9}{k}). But wait, maybe I need to check whether the question is asking for the term with the maximum coefficient (numerical factor) or the maximum binomial coefficient.Wait, the problem states: "The binomial coefficient of the maximum term in the expansion...". So the binomial coefficient corresponding to the maximum term. So first, we need to find the maximum term in the expansion, then report its binomial coefficient. But the options are given as terms (like 5th term, etc.), so maybe they mean which term (position) has the maximum binomial coefficient. But that might not be accurate.Wait, perhaps the problem is ambiguous. Let me check the exact wording again: "The binomial coefficient of the maximum term in the expansion of ((2x + frac{1}{x^2})^9) is ( )". Then the options are terms: A: the 5th term, etc. Hmm. So maybe they are asking which term (by its position) has the maximum binomial coefficient. But in binomial expansions, the binomial coefficients are symmetric. For example, in ((a + b)^n), the coefficients (binom{n}{k}) are symmetric around k = n/2. But here, the expansion isn't symmetric because the two terms in the binomial have different coefficients and different exponents. So the maximum term in the expansion isn't necessarily the one with the maximum binomial coefficient.Wait, maybe the problem is mixing two things: the term's coefficient (including the constants from 2x and 1/x²) and the binomial coefficient. Let me parse the question again carefully: "The binomial coefficient of the maximum term in the expansion..." So, binomial coefficient here refers to (binom{9}{k}), which is part of the term. But the "maximum term" would be the term with the greatest numerical value. However, since the terms involve variables (x), unless x is specified, the maximum term would depend on x. But the question is given without specifying x, so perhaps they are referring to the term with the maximum coefficient when the expression is written in terms of x. But even then, terms have different exponents. For example, the general term is (binom{9}{k} 2^{9 - k} x^{9 - 3k}). Each term has a different power of x, so unless they are comparing coefficients for the same power of x, which is not the case here. Therefore, this seems confusing.Wait, maybe the problem is actually asking for the term with the maximum coefficient when considering the coefficients multiplied by the constants from the binomial terms (i.e., the numerical coefficient in front of x^{9 - 3k}). So the coefficient would be (binom{9}{k} 2^{9 - k}), and then the question is asking which term's binomial coefficient (i.e., (binom{9}{k})) corresponds to the maximum term (i.e., the term with the maximum numerical coefficient (binom{9}{k} 2^{9 - k})). But then the answer would be the k where this product is maximized. Then, the binomial coefficient itself is part of the term's coefficient. Alternatively, maybe the problem is misworded and actually asking for the term with the maximum coefficient (numerical factor), and the options are which term that is. Then the answer would be found by maximizing (binom{9}{k} 2^{9 - k}). Let's proceed under that assumption because otherwise, if it's about binomial coefficients alone, the maximum binomial coefficient in the expansion is (binom{9}{4}) or (binom{9}{5}) due to symmetry, but that might not be the case here due to the 2^{9 - k} factor.So let's think step by step.The general term is (T_{k+1} = binom{9}{k} 2^{9 - k} x^{9 - 3k}). The coefficient of each term is (binom{9}{k} 2^{9 - k}). To find the maximum term, we need to find the value of k (from 0 to 9) that maximizes this coefficient. Since x is a variable, the term with the highest coefficient would be the "maximum term" if we consider coefficients regardless of the x exponent. So assuming that's the case, we need to find k such that (binom{9}{k} 2^{9 - k}) is maximized.Let me denote (C(k) = binom{9}{k} 2^{9 - k}). We need to find the k that maximizes C(k).To find the maximum, we can compute the ratio (C(k+1)/C(k)) and find when this ratio crosses 1.Compute (C(k+1)/C(k) = [binom{9}{k+1} 2^{9 - (k+1)}] / [binom{9}{k} 2^{9 - k}]) = ([frac{9!}{(k+1)!(9 - (k+1))!} / frac{9!}{k! (9 - k)!}] times 2^{-1}) = ([frac{k! (9 - k)!}{(k+1)! (8 - k)!}] times frac{1}{2}) = ([frac{9 - k}{(k + 1)}] times frac{1}{2}).So the ratio is (frac{9 - k}{2(k + 1)}). We can set this ratio equal to 1 to find the critical point:(frac{9 - k}{2(k + 1)} = 1)Multiply both sides by (2(k + 1)):(9 - k = 2(k + 1))Simplify:9 - k = 2k + 29 - 2 = 2k + k7 = 3kk = 7/3 ≈ 2.333...Since k must be an integer between 0 and 9, the maximum occurs around k = 2.333. So we check k = 2 and k = 3 to see which gives a larger C(k).Compute the ratio at k = 2:(C(3)/C(2) = frac{9 - 2}{2(2 + 1)} = frac{7}{6} ≈ 1.1667), which is greater than 1. So C(k) increases from k=2 to k=3.Then compute the ratio at k = 3:(C(4)/C(3) = frac{9 - 3}{2(3 + 1)} = frac{6}{8} = 0.75), which is less than 1. So C(k) decreases from k=3 to k=4.Therefore, the maximum occurs at k = 3. So the term is T_{k+1} = T_{4}. Wait, term numbers start at 1, so k=0 is the first term. So k=3 corresponds to the 4th term. But the options given are A: 5th term, B: 10th term, C: 5th and 6th terms, D: 9th and 10th terms. None of these is the 4th term. That suggests my approach is wrong.Wait, maybe I miscalculated. Let me check again.Wait, the ratio (C(k+1)/C(k)) is (frac{9 - k}{2(k + 1)}). So for k=0: ratio is (9 - 0)/(2(0 +1))=9/2=4.5>1, so C(k) increases.k=1: ratio=(9-1)/2(2)=8/4=2>1, still increasing.k=2: ratio=(9-2)/2(3)=7/6≈1.166>1, increasing.k=3: ratio=(9-3)/2(4)=6/8=0.75<1, decreasing.Therefore, maximum at k=3. So the 4th term. But none of the options have the 4th term. So maybe the problem is not asking for the term with maximum coefficient, but something else.Wait, going back to the problem statement: "The binomial coefficient of the maximum term in the expansion of ((2x + frac{1}{x^2})^9) is ( )". The binomial coefficient is (binom{9}{k}), so maybe the question is saying that among all the terms, which term's binomial coefficient is the maximum. But in that case, binomial coefficients (binom{9}{k}) are maximum at k=4 and 5, since binomial coefficients are symmetric. For n=9, which is odd, the maximum occurs at k=4 and k=5 (since 9/2=4.5), so the middle terms. So the 5th term (k=4) and 6th term (k=5) have the maximum binomial coefficients. So that would be option C: the 5th and 6th terms.But wait, that's assuming the question is asking for the term with the maximum binomial coefficient, regardless of the other factors (2^{9 -k} and x exponents). If that's the case, then yes, binomial coefficients are max at k=4 and 5. But the problem says "the binomial coefficient of the maximum term". That wording is a bit confusing. If "maximum term" is the term with the largest coefficient (including the 2^{9 -k} factor), then the binomial coefficient of that term is different. However, if "maximum term" refers to the term with the maximum binomial coefficient, then it's the 5th and 6th terms. The problem is ambiguous.But given that the options include C: 5th and 6th terms, which correspond to the maximum binomial coefficients in a standard binomial expansion (without coefficients on the terms), and D: 9th and 10th terms which would be the last terms, which have binomial coefficients (binom{9}{8}=9) and (binom{9}{9}=1), which are small. So likely, the answer is C.But wait, in our previous calculation, when considering the coefficient (binom{9}{k} 2^{9 -k}), the maximum was at k=3 (4th term). But none of the options have 4th term. So this is conflicting. Therefore, perhaps there is a misinterpretation here.Wait, maybe the problem is asking for the term with the maximum value when x=1. Then, substituting x=1, the term's value is (binom{9}{k} 2^{9 -k}). Then, the maximum term would be as we found at k=3 (4th term). But again, 4th term is not among the options. So maybe the problem is not about the numerical coefficient, but about the term with the highest degree or something else.Alternatively, maybe the problem is referring to the term with the maximum absolute value when considering x. But since x is a variable, unless restricted, the term with the highest degree would dominate for large x, and the term with the lowest degree (most negative exponent) would dominate as x approaches zero. So there isn't a universal maximum term unless x is fixed.But since the problem is given without specifying x, maybe the question is referring to the middle term, which in some cases is the term with the maximum binomial coefficient. For n=9, there are 10 terms, so the 5th and 6th terms are the middle ones, which have the maximum binomial coefficients. That is, (binom{9}{4}) and (binom{9}{5}) (since (binom{9}{4} = binom{9}{5})), which are equal and are the maximum coefficients. Therefore, if the question is asking for the term with the maximum binomial coefficient, the answer is C: the 5th and 6th terms.But why then the given binomial is ((2x + 1/x^2)^9) instead of a simple ((a + b)^9)? Maybe the 2 and the 1/x^2 affect the term's coefficient. However, the problem specifically mentions "the binomial coefficient", which is (binom{9}{k}), not the entire coefficient (which would include 2^{9 -k} and x^{9 -3k}). So if it's strictly asking for the binomial coefficient of the maximum term, where "maximum term" is defined in some way, but the problem is unclear. Alternatively, maybe the question is translated incorrectly, and "maximum term" refers to the term with the highest exponent. Let's see. The exponent of x in each term is (9 - 3k). To maximize the exponent, we need to maximize (9 - 3k). Since k starts at 0, the first term (k=0) has exponent 9, the second term (k=1) has exponent 6, then 3, 0, -3, etc. So the first term has the highest exponent. But the binomial coefficient of the first term is (binom{9}{0} = 1). That would be the 1st term, but option A is 5th term, which is not the case.Alternatively, the term with the maximum coefficient when x is set to 1. Then, as earlier, the coefficient would be (binom{9}{k} 2^{9 -k}). Let's compute this for k=3 and k=4 to see if maybe the maximum is spread over two terms.Wait, previously, when k=3, the ratio C(k+1)/C(k) is 0.75, which is less than 1, so the maximum is at k=3. But let's compute the actual values.Compute C(k) for k=2,3,4:C(2) = (binom{9}{2} 2^{7} = 36 times 128 = 4608)C(3) = (binom{9}{3} 2^{6} = 84 times 64 = 5376)C(4) = (binom{9}{4} 2^{5} = 126 times 32 = 4032)So indeed, C(3) is larger than C(2) and C(4). So the maximum is at k=3, which is the 4th term. But 4th term is not an option. Therefore, there must be a misunderstanding here.Wait, maybe the problem is asking for the term with the maximum value when considering both the coefficient and the variable part. But without specifying x, that's impossible. Unless the question is using "maximum term" in a different sense, such as the term with the maximum exponent, which is the first term, but that's not in the options.Wait, another approach. Let's recall that in binomial expansions, the term with the maximum coefficient can be found by the formula. If the binomial is (a + b)^n, the term with the maximum coefficient is the integer part of (n+1) * |a| / (|a| + |b|). Wait, is that a valid formula?Wait, yes, there is a formula for the term with the greatest coefficient in the expansion of (a + b)^n. If a and b are positive, then the term with the maximum coefficient is at k = floor[(n + 1) * b / (a + b)]. Wait, maybe I need to check.Alternatively, the ratio of consecutive coefficients can help. For (a + b)^n, the ratio T_{k+1}/T_k = (n - k + 1)/k * b/a. So set this ratio to 1 to find the turning point. So:(n - k + 1)/k * b/a = 1(n - k + 1) * b = k * a(n + 1) * b = k (a + b)k = (n + 1) * b / (a + b)In our case, the binomial is (2x + 1/x²)^9. But if we treat the coefficients as '2' and '1', because the variables x and 1/x² are part of the terms. Wait, but the formula I mentioned is for the expansion (a + b)^n where a and b are constants. However, here, a and b have variable parts. But if we consider only the coefficients (the constants multiplied by the variables), then the coefficients are 2^{9 -k} and 1^k, so the coefficient of each term is (binom{9}{k} 2^{9 -k}). Therefore, applying the formula for the term with the maximum coefficient in (A + B)^n where A = 2 and B = 1. Then, the ratio would be:k = (n + 1) * B / (A + B) = (9 + 1) * 1 / (2 + 1) = 10 / 3 ≈ 3.333.So the maximum coefficient occurs around k=3.333, which is between k=3 and k=4. So we check the coefficients at k=3 and k=4.As calculated before, C(3)=5376 and C(4)=4032. Therefore, the maximum is at k=3, which is the 4th term. But again, the options do not include the 4th term.This suggests a conflict between the calculated maximum term and the provided answer options. Therefore, perhaps the problem is not asking about the coefficient's maximum but the binomial coefficient's maximum. If we ignore the constants 2^{9 -k} and just consider the binomial coefficients (binom{9}{k}), then the maximum is at k=4 and 5 (terms 5 and 6), which is option C. This must be the intended answer because the problem mentions "the binomial coefficient of the maximum term". If "maximum term" is defined as the term with the maximum binomial coefficient (rather than the maximum overall coefficient), then the answer is C. However, the wording is confusing because the binomial coefficient is part of the term's coefficient, but the term's "maximum" could be interpreted in different ways.Alternatively, maybe "maximum term" refers to the term with the highest exponent, but as discussed earlier, that's the first term. Alternatively, the term with the middle exponent, but in the expansion of ((2x + 1/x^2)^9), the exponents of x are 9 - 3k. Let's list them:For k=0: exponent 9k=1: 6k=2: 3k=3: 0k=4: -3k=5: -6k=6: -9k=7: -12k=8: -15k=9: -18So the exponents decrease as k increases. The exponents are positive for k=0,1,2,3 (i.e., first four terms), zero at k=3 (4th term), and negative for k=4 to 9. If the problem is considering only the terms with positive exponents, then the maximum exponent is 9 (first term). If considering absolute values, but exponents can't be negative if x is positive. But without x's value, it's unclear.Alternatively, maybe the problem is asking for the term with the maximum coefficient when the expression is written in descending powers of x. In that case, the first few terms (with positive exponents) would come first, then the constant term, then the negative exponents. But again, the maximum coefficient would still be at k=3 as before.Given that none of the options match k=3 (4th term), but option C is 5th and 6th terms. Let's see what k those correspond to. The 5th term is k=4 (since terms start at k=0), and the 6th term is k=5. So binomial coefficients for k=4 and k=5 are both (binom{9}{4} = binom{9}{5} = 126), which are the maximum binomial coefficients. Therefore, if the question is asking for the terms where the binomial coefficients are maximum, then the answer is C. But the wording "binomial coefficient of the maximum term" is ambiguous. However, given the options, and since the standard maximum binomial coefficients in a degree 9 expansion are at the middle terms (5th and 6th), the answer is likely C.But why does the original binomial have 2x and 1/x²? If those coefficients affect the term's coefficient, but if the question specifically refers to the binomial coefficient (the combinatorial part (binom{9}{k})), then those constants don't affect it. Therefore, if the question is strictly about (binom{9}{k}), then the maximum is at 5th and 6th terms. Hence, answer C.However, this contradicts the earlier calculation where considering the 2^{9 -k} factor leads to maximum at k=3. But if the problem mentions only the binomial coefficient (not the entire coefficient), then it's separate. The question says "the binomial coefficient of the maximum term". So, parse this as: which term is the maximum one, and then what is its binomial coefficient. But the options are about the term's position, not the binomial coefficient's value. Wait, the options are A: the 5th term, B: the 10th term, etc. So the answer is one of these options, which are term positions. Therefore, interpreting the question: "In the expansion, which term has the maximum binomial coefficient?" Then, the answer is the 5th and 6th terms, which have binomial coefficients of 126, the highest. Therefore, the answer is C.Alternatively, if the question is asking for the term with the maximum coefficient (numerical factor, including the 2^{9 -k}), then the answer isn't among the options, which is conflicting. Since the provided options include C and given that binomial coefficients peak at the middle, the answer expected is probably C. Therefore, I think the answer is C.Final Answerboxed{C}</think>To determine the binomial coefficient of the maximum term in the expansion of ((2x + frac{1}{x^2})^9), we start by considering the general term in the binomial expansion. The general term is given by:[ T_{k+1} = binom{9}{k} (2x)^{9-k} left(frac{1}{x^2}right)^k = binom{9}{k} 2^{9-k} x^{9-3k} ]The binomial coefficient here is (binom{9}{k}). To find the maximum binomial coefficient, we note that for a binomial expansion ((a + b)^n), the binomial coefficients (binom{n}{k}) are maximized at (k = leftlfloor frac{n}{2} rightrfloor) and (k = leftlceil frac{n}{2} rightrceil). For (n = 9), the maximum binomial coefficients occur at (k = 4) and (k = 5), which correspond to the 5th and 6th terms in the expansion (since terms are indexed starting from (k = 0)).Thus, the binomial coefficients (binom{9}{4}) and (binom{9}{5}) are both equal to 126, which are the maximum values. Therefore, the terms with the maximum binomial coefficients are the 5th and 6th terms.[boxed{C}],<think>Okay, so I need to figure out which term in the expansion of ((2x + frac{1}{x^2})^9) has the maximum binomial coefficient. The options are given as A: 5th term, B: 10th term, C: 5th and 6th terms, D: 9th and 10th terms. Hmm, let me recall how binomial expansions work.First, the general term in the expansion of ((a + b)^n) is given by (binom{n}{k} a^{n - k} b^{k}) where (k) ranges from 0 to (n). In this case, (a = 2x) and (b = frac{1}{x^2}), and (n = 9). So each term in the expansion will be (binom{9}{k} (2x)^{9 - k} left(frac{1}{x^2}right)^k). Simplifying that, each term becomes (binom{9}{k} 2^{9 - k} x^{9 - k} x^{-2k} = binom{9}{k} 2^{9 - k} x^{9 - 3k}). So the exponent of (x) in each term is (9 - 3k). But the question is about the binomial coefficient of the maximum term. Wait, does it mean the term with the maximum coefficient (numerical value) in the expansion, or the term with the maximum binomial coefficient (binom{9}{k})? The wording says "the binomial coefficient of the maximum term". Hmm. Let me parse that again. It might be ambiguous, but "binomial coefficient of the maximum term" probably refers to the binomial coefficient (i.e., (binom{9}{k})) that is the largest among all coefficients in the expansion. But wait, binomial coefficients in the expansion of ((a + b)^n) are symmetric. For (n=9), the coefficients (binom{9}{k}) are maximized at (k = 4) and (k=5) because when (n) is odd, the middle coefficients are the largest. Wait, no, for (n=9), the maximum binomial coefficient is at (k = lfloor frac{9}{2} rfloor = 4) and (k = lceil frac{9}{2} rceil = 5), so the 5th and 6th terms? Because the terms start counting from (k=0) as the first term. Wait, actually, in binomial expansions, the first term corresponds to (k=0), so the term number is (k + 1). So for (k=0), it's the 1st term; (k=1), 2nd term, etc. So if the maximum coefficients are at (k=4) and (k=5), that would correspond to the 5th and 6th terms. So then option C?But wait, hold on. Wait, in this case, the binomial coefficients are (binom{9}{k}), so they are symmetric around (k = 4.5), so indeed, (k=4) and (k=5) will have the same binomial coefficients, which are the maximum. Therefore, the 5th term (k=4) and the 6th term (k=5) both have the maximum binomial coefficient. So that would be answer C.But let me verify. For example, (binom{9}{4} = binom{9}{5} = 126), which is the maximum. So yeah, k=4 and k=5 (5th and 6th terms) have the maximum binomial coefficients. Therefore, answer C.Wait, but the question says "the binomial coefficient of the maximum term in the expansion". Wait, maybe I misinterpreted. Maybe it's asking for the binomial coefficient of the term that is the maximum term in the expansion (i.e., the term with the highest numerical value when expanded). That would be different. So maybe the problem is asking for which term is the maximum term in the expansion, and then what is its binomial coefficient, but the options are about which term(s) have the maximum binomial coefficient. Wait, the wording is confusing.Wait, let's read the problem again: "The binomial coefficient of the maximum term in the expansion of ((2x + frac{1}{x^2})^9) is ( )". The options are about which term has the maximum binomial coefficient. Wait, no. Wait, "the binomial coefficient of the maximum term". So maybe "the binomial coefficient corresponding to the maximum term in the expansion is which term number". Hmm. Hmm.Alternatively, maybe the problem is translated from Chinese, and the translation is a bit off. Alternatively, maybe it's asking for which term has the maximum coefficient (i.e., the largest numerical coefficient when expanded), but the answer options are term positions. But the term "binomial coefficient" specifically refers to the (binom{n}{k}) part, not multiplied by the coefficients from (a) or (b). So maybe the question is just asking which term has the largest binomial coefficient (binom{9}{k}), which as we saw is the 5th and 6th terms (k=4 and k=5), so answer C. That seems plausible.But let me check with the alternative interpretation. Suppose instead that the question is asking for the term with the maximum numerical coefficient in the expansion. Then we would need to compute the coefficients (binom{9}{k} 2^{9 - k}) and find where they are maximum. Because each term's coefficient is (binom{9}{k} 2^{9 - k}), since (b = frac{1}{x^2}) doesn't contribute a coefficient, only (2x) does. So the coefficient is (binom{9}{k} 2^{9 - k}). So in that case, to find the maximum coefficient, we need to compare (binom{9}{k} 2^{9 - k}) for different k.But the problem says "the binomial coefficient of the maximum term". If "binomial coefficient" refers to (binom{9}{k}), then it's separate from the rest. But if the "maximum term" is the term with the highest coefficient, then "the binomial coefficient of that maximum term" would be the (binom{9}{k}) part of that term. Then the answer would depend on which term is maximum. So perhaps the problem is mixing these two concepts. Hmm. So it's ambiguous.Wait, maybe the problem is asking for the term(s) where the binomial coefficient (i.e., (binom{9}{k})) is maximized. Since binomial coefficients are maximized at the middle terms, for n=9, which is odd, the middle terms are at k=4 and k=5, so the 5th and 6th terms, which is option C. So that's straightforward.But I need to make sure. Let me think again. The binomial coefficients themselves, (binom{9}{k}), are maximum at k=4 and 5, so the 5th and 6th terms. So the answer is C. Therefore, the correct answer is C.But wait, let me check again. The first term is k=0: (binom{9}{0}), second term k=1: (binom{9}{1}), ..., fifth term k=4: (binom{9}{4}), sixth term k=5: (binom{9}{5}), and so on. Since (binom{9}{4} = binom{9}{5} = 126), which is the maximum binomial coefficient. Therefore, the binomial coefficients of the 5th and 6th terms are the maximum. So answer C.Yes, that seems right. So I think the answer is C.Final Answerboxed{C}</think>To determine which term in the expansion of ((2x + frac{1}{x^2})^9) has the maximum binomial coefficient, we start by recalling the general term in the binomial expansion, which is given by:[binom{9}{k} (2x)^{9 - k} left(frac{1}{x^2}right)^k]Simplifying this term, we get:[binom{9}{k} 2^{9 - k} x^{9 - 3k}]The binomial coefficient for each term is (binom{9}{k}). For (n = 9), the binomial coefficients (binom{9}{k}) are maximized at (k = 4) and (k = 5) because the coefficients are symmetric around (k = frac{9}{2} = 4.5). These values of (k) correspond to the 5th term (when (k = 4)) and the 6th term (when (k = 5)).Thus, the binomial coefficients (binom{9}{4}) and (binom{9}{5}) are both 126, which is the maximum value. Therefore, the 5th and 6th terms have the maximum binomial coefficients.[boxed{C}]]

🔑:Solution: The expansion has a total of 9+1=10 terms. Therefore, the binomial coefficients are at their maximum in the middle two terms, specifically the 5th and 6th terms.Hence, the answer is: boxed{text{C}}.According to the properties of binomial coefficients, the maximum binomial coefficient in the expansion is in the middle two terms of the 10 terms. This problem primarily tests the application of the binomial theorem, and understanding the properties of binomial coefficients is the key to solving this problem.

🧠:[<think>Okay, so I need to solve this geometry problem where the locus of point P is contained in an equilateral hyperbola. Let me try to visualize the setup first. We have two circles, C₁ and C₂. The radius of C₁ is r₁, and the radius of C₂ is r₂. A variable tangent t to circle C₁ intersects circle C₂ at points A and B. Then, the tangents to C₂ from points A and B intersect at point P. The problem asks for the distance between the centers of C₁ and C₂ such that the locus of P, as the tangent t varies, is contained in an equilateral hyperbola. First, let me recall that an equilateral hyperbola (also known as a rectangular hyperbola) has perpendicular asymptotes, which means its axes are equal in length. For a hyperbola in standard form, this occurs when the coefficients of x² and y² are equal in magnitude but opposite in sign. So, for example, the hyperbola x²/a² - y²/a² = 1 is equilateral because the asymptotes y = ±x are perpendicular.Given that, the key is to find the condition on the distance between the centers (let's denote them as O₁ and O₂ for C₁ and C₂ respectively) such that all possible points P lie on such a hyperbola.Let me try to model the problem with coordinates. Let's place the centers of the circles in a coordinate system. Without loss of generality, I can place O₁ at the origin (0,0) and O₂ along the x-axis at (d, 0), where d is the distance between the centers that we need to find.So, circle C₁ is x² + y² = r₁², and circle C₂ is (x - d)² + y² = r₂².A variable tangent t to C₁. The tangent to a circle can be represented in various forms. For a circle centered at the origin, the equation of a tangent line at angle θ can be written as x cos θ + y sin θ = r₁. This is the standard equation for a tangent line to a circle centered at the origin with radius r₁. The line makes an angle θ with the x-axis, and the point of contact is (r₁ cos θ, r₁ sin θ). Alternatively, if I don't want to parameterize by θ, I can represent a general tangent line to C₁. Any tangent line to C₁ can be written as x cos θ + y sin θ = r₁ for some θ. This parametrization will be useful because as θ varies, the tangent line t varies, which in turn causes points A and B to vary, leading to different points P. The locus of P is what we need to analyze.So, the tangent line t has equation x cos θ + y sin θ = r₁. This line intersects circle C₂: (x - d)² + y² = r₂². Let's find the points A and B where the tangent t intersects C₂.To find the points of intersection, substitute y from the tangent line equation into the circle equation. Wait, but solving for y might be messy. Alternatively, solve for x or y in the tangent line equation and substitute into the circle equation.Let's solve the tangent equation for x:x cos θ + y sin θ = r₁ ⇒ x = (r₁ - y sin θ)/cos θSubstitute this into the equation of C₂:[( (r₁ - y sin θ)/cos θ - d )² + y²] = r₂²This seems complicated. Alternatively, maybe it's better to use parametric coordinates.Alternatively, since the line t is tangent to C₁, and intersects C₂ at A and B, perhaps we can find the coordinates of A and B in terms of θ.Alternatively, since line t is x cos θ + y sin θ = r₁, let's find its intersection with C₂.So, substituting x from the tangent line into C₂'s equation:(x - d)^2 + y^2 = r₂²But x = (r₁ - y sin θ)/cos θSo,[( (r₁ - y sin θ)/cos θ - d )² + y²] = r₂²Let me expand this:First, compute ( (r₁ - y sin θ)/cos θ - d ):= (r₁ - y sin θ - d cos θ)/cos θThus, squared:[(r₁ - y sin θ - d cos θ)/cos θ]^2 = [ (r₁ - d cos θ - y sin θ)^2 ] / cos² θTherefore, the equation becomes:[ (r₁ - d cos θ - y sin θ)^2 ] / cos² θ + y² = r₂²Multiply both sides by cos² θ to eliminate the denominator:(r₁ - d cos θ - y sin θ)^2 + y² cos² θ = r₂² cos² θLet me expand (r₁ - d cos θ - y sin θ)^2:= (r₁ - d cos θ)^2 - 2 (r₁ - d cos θ)(y sin θ) + y² sin² θTherefore, the equation becomes:(r₁ - d cos θ)^2 - 2 (r₁ - d cos θ) y sin θ + y² sin² θ + y² cos² θ = r₂² cos² θNotice that y² sin² θ + y² cos² θ = y² (sin² θ + cos² θ) = y². So, simplifying:(r₁ - d cos θ)^2 - 2 (r₁ - d cos θ) y sin θ + y² = r₂² cos² θBring all terms to one side:y² - 2 (r₁ - d cos θ) sin θ y + (r₁ - d cos θ)^2 - r₂² cos² θ = 0This is a quadratic equation in y. Let's write it as:y² - 2 (r₁ - d cos θ) sin θ y + [ (r₁ - d cos θ)^2 - r₂² cos² θ ] = 0Let me denote this as:y² + B y + C = 0where:B = -2 (r₁ - d cos θ) sin θC = (r₁ - d cos θ)^2 - r₂² cos² θThe solutions for y are given by the quadratic formula:y = [ -B ± sqrt(B² - 4 * 1 * C) ] / 2Compute discriminant D:D = B² - 4CCompute B²:B² = [ -2 (r₁ - d cos θ) sin θ ]² = 4 (r₁ - d cos θ)^2 sin² θCompute 4C:4C = 4 [ (r₁ - d cos θ)^2 - r₂² cos² θ ]Thus, D = 4 (r₁ - d cos θ)^2 sin² θ - 4 [ (r₁ - d cos θ)^2 - r₂² cos² θ ]Factor out the 4:D = 4 [ (r₁ - d cos θ)^2 sin² θ - (r₁ - d cos θ)^2 + r₂² cos² θ ]Simplify the expression inside the brackets:= (r₁ - d cos θ)^2 (sin² θ - 1) + r₂² cos² θSince sin² θ - 1 = -cos² θ,= - (r₁ - d cos θ)^2 cos² θ + r₂² cos² θFactor out cos² θ:= cos² θ [ - (r₁ - d cos θ)^2 + r₂² ]Thus, D = 4 cos² θ [ - (r₁ - d cos θ)^2 + r₂² ]For real intersections A and B, the discriminant D must be non-negative. So:4 cos² θ [ - (r₁ - d cos θ)^2 + r₂² ] ≥ 0Since 4 cos² θ is always non-negative (as it's a square), the condition reduces to:- (r₁ - d cos θ)^2 + r₂² ≥ 0Which implies:r₂² ≥ (r₁ - d cos θ)^2But this is a condition that must hold for the line t to intersect C₂. However, since the tangent t is variable, θ can vary such that this inequality is satisfied. However, since we need the locus of P to exist for varying t, we need that for some θ, the line t intersects C₂. But perhaps the exact condition on d will come from the requirement that the locus is a hyperbola.But maybe let's not get bogged down here. Let's proceed to find coordinates of points A and B.The solutions for y are:y = [ 2 (r₁ - d cos θ) sin θ ± sqrt(D) ] / 2= (r₁ - d cos θ) sin θ ± sqrt(D)/2But sqrt(D) = sqrt(4 cos² θ [ - (r₁ - d cos θ)^2 + r₂² ]) = 2 |cos θ| sqrt( r₂² - (r₁ - d cos θ)^2 )Assuming cos θ ≠ 0, otherwise the tangent line would be vertical (x = r₁ or x = -r₁, but since the circle C₁ is centered at the origin, the vertical tangents are x = r₁ and x = -r₁. However, if the center O₂ is at (d, 0), then a vertical tangent at x = r₁ would intersect C₂ if |d - r₁| ≤ r₂, but maybe this is a special case. For generality, let's assume cos θ ≠ 0.Therefore, sqrt(D) = 2 |cos θ| sqrt(r₂² - (r₁ - d cos θ)^2 )Therefore, the solutions for y are:y = (r₁ - d cos θ) sin θ ± |cos θ| sqrt(r₂² - (r₁ - d cos θ)^2 )But the ± and the absolute value complicate things. However, perhaps we can write it as:y = (r₁ - d cos θ) sin θ ± cos θ sqrt(r₂² - (r₁ - d cos θ)^2 )Assuming cos θ is positive (since θ can vary, we might need to consider both signs, but maybe we can parameterize θ such that cos θ is positive. Alternatively, express in terms of t = tan θ or something else. Hmm.Alternatively, maybe use a parameterization for the tangent line. Wait, perhaps there's a better approach. Let's recall that the points A and B lie on C₂, and the tangents from P to C₂ meet at P. Therefore, P is the external homothety center of the two tangents from A and B to C₂. Wait, actually, since PA and PB are both tangents to C₂, P lies on the polar of line AB with respect to C₂. Alternatively, since AB is the chord of contact of point P with respect to C₂.Yes, that's right. For any point P outside a circle, the polar line of P with respect to the circle is the chord of contact of the tangents from P to the circle. Therefore, AB is the chord of contact of P with respect to C₂. Therefore, if AB is the chord of contact, then the equation of AB is the polar of P with respect to C₂.Given that C₂ is (x - d)^2 + y^2 = r₂², the equation of the polar line of point P(h, k) is:(h - d)(x - d) + k y = r₂²But in our case, the chord AB is the given tangent line t to C₁, which is x cos θ + y sin θ = r₁. So, equating the two expressions for AB:(h - d)(x - d) + k y = r₂²andx cos θ + y sin θ = r₁Therefore, these two equations must represent the same line. Therefore, their coefficients must be proportional.First, let's rewrite the polar equation:(h - d)(x - d) + k y = r₂²Expanding:(h - d)x - d(h - d) + k y = r₂²Rearranged:(h - d)x + k y = r₂² + d(h - d)Therefore, the polar line is:(h - d)x + k y = r₂² + d h - d²This must be the same line as x cos θ + y sin θ = r₁. Therefore, the coefficients must be proportional:(h - d) / cos θ = k / sin θ = [r₂² + d h - d²] / r₁Therefore, we have the system:1. (h - d) / cos θ = k / sin θ ⇒ (h - d) sin θ = k cos θ2. (h - d) / cos θ = [r₂² + d h - d²] / r₁ ⇒ (h - d) r₁ = cos θ [r₂² + d h - d²]From the first equation, k = (h - d) tan θFrom the second equation, (h - d) r₁ = cos θ [r₂² + d h - d²]Let me solve for cos θ from the second equation.cos θ = (h - d) r₁ / [r₂² + d h - d²]But from the first equation, sin θ = k / (h - d) * cos θ, assuming h ≠ d.Wait, let's substitute cos θ from the second equation into the first equation.From equation 1: k = (h - d) tan θtan θ = sin θ / cos θ, so k = (h - d) (sin θ / cos θ )But sin θ can be expressed from equation 1 as sin θ = k cos θ / (h - d)Wait, maybe we can express sin θ and cos θ in terms of h and k.Alternatively, let's square and add the two equations derived from the proportional coefficients.Let me denote the ratios as a constant λ:(h - d) = λ cos θk = λ sin θr₂² + d h - d² = λ r₁Therefore, from the first two equations:(h - d) = λ cos θk = λ sin θTherefore, (h - d)^2 + k^2 = λ² (cos² θ + sin² θ) = λ²From the third equation: λ = (r₂² + d h - d²)/r₁Thus,(h - d)^2 + k^2 = [ (r₂² + d h - d²)/r₁ ]²Therefore, the locus of P(h, k) is given by:(h - d)^2 + k^2 = [ (r₂² + d h - d² ) / r₁ ]²This is the equation relating h and k for the point P. To find the condition that this locus is an equilateral hyperbola, we need to expand this equation and see under what conditions on d (the distance between centers) it represents a rectangular hyperbola.So, let's expand both sides.Left-hand side (LHS):(h - d)^2 + k^2 = h² - 2 d h + d² + k²Right-hand side (RHS):[ (r₂² + d h - d² ) / r₁ ]² = (d h + r₂² - d² )² / r₁²Therefore, the equation becomes:h² - 2 d h + d² + k² = (d h + r₂² - d² )² / r₁²Multiply both sides by r₁² to eliminate the denominator:r₁² (h² - 2 d h + d² + k²) = (d h + r₂² - d² )²Expand the RHS:(d h + (r₂² - d²))² = d² h² + 2 d h (r₂² - d²) + (r₂² - d²)^2Therefore, expanding the entire equation:r₁² h² - 2 d r₁² h + r₁² d² + r₁² k² = d² h² + 2 d (r₂² - d²) h + (r₂² - d²)^2Bring all terms to the left-hand side:r₁² h² - 2 d r₁² h + r₁² d² + r₁² k² - d² h² - 2 d (r₂² - d²) h - (r₂² - d²)^2 = 0Group like terms:h² (r₁² - d²) + k² r₁² + h [ -2 d r₁² - 2 d (r₂² - d²) ] + [ r₁² d² - (r₂² - d²)^2 ] = 0Simplify each coefficient:1. Coefficient of h²: (r₁² - d²)2. Coefficient of k²: r₁²3. Coefficient of h: -2 d r₁² - 2 d r₂² + 2 d³ = -2 d (r₁² + r₂² - d²)4. Constant term: r₁² d² - (r₂² - d²)^2So the equation becomes:(r₁² - d²) h² + r₁² k² - 2 d (r₁² + r₂² - d²) h + [ r₁² d² - (r₂^4 - 2 r₂² d² + d^4) ] = 0Simplify the constant term:r₁² d² - r₂^4 + 2 r₂² d² - d^4= (r₁² d² + 2 r₂² d² - d^4) - r₂^4= d² (r₁² + 2 r₂² - d²) - r₂^4Thus, the entire equation is:(r₁² - d²) h² + r₁² k² - 2 d (r₁² + r₂² - d²) h + d² (r₁² + 2 r₂² - d²) - r₂^4 = 0To analyze this conic section, let's write it in general form:A h² + B k² + C h + D = 0But in this case, there's a cross term? Wait, no. Wait, the equation is:(r₁² - d²) h² + r₁² k² - 2 d (r₁² + r₂² - d²) h + [ d² (r₁² + 2 r₂² - d²) - r₂^4 ] = 0So, variables h and k, with coefficients:A = r₁² - d²B = r₁²C = -2 d (r₁² + r₂² - d²)D = d² (r₁² + 2 r₂² - d²) - r₂^4So, this is a quadratic equation in h and k. To determine the type of conic, we can check the discriminant of the quadratic form. For the equation A h² + B k² + C h + D = 0, the type of conic can be determined by the coefficients A and B. If A and B are both non-zero and have opposite signs, it's a hyperbola. Additionally, for it to be a rectangular hyperbola, the asymptotes must be perpendicular, which in the case of a conic without cross terms (i.e., no h k term) requires that A = -B. Wait, if there are no cross terms, then the axes are aligned with the coordinate axes, so for the hyperbola to be rectangular (equilateral), we need the coefficients of h² and k² to be equal in magnitude and opposite in sign. That is, A = -B.Wait, let me recall. For a hyperbola of the form A h² + B k² + C h + D k + E = 0, if there are no cross terms (i.e., no h k term), then the axes are aligned with the coordinate axes. For it to be a rectangular hyperbola, the asymptotes are perpendicular, which requires that the slopes of the asymptotes are ±1. The asymptotes of the hyperbola A h² + B k² + C h + D k + E = 0 (when there are no cross terms) can be found by setting the linear terms and constant term to zero: A h² + B k² = 0. For this to represent asymptotes with slopes ±1, we need that A = -B. So, if A = -B, then the equation A h² + B k² = 0 becomes A(h² - k²) = 0, which factors as (h - k)(h + k) = 0, i.e., the lines y = h and y = -h, which are perpendicular. Hence, the hyperbola is rectangular.Therefore, in our case, for the equation (r₁² - d²) h² + r₁² k² + ... = 0 to represent a rectangular hyperbola, we require that the coefficients of h² and k² satisfy A = -B. That is:r₁² - d² = - r₁²Solving for d:r₁² - d² = - r₁² ⇒ -d² = -2 r₁² ⇒ d² = 2 r₁² ⇒ d = r₁ √2Wait, so if d² = 2 r₁², then the coefficient of h² is - r₁², and the coefficient of k² is r₁², hence A = -B, which would make the equation a rectangular hyperbola. But wait, let's check that. If d² = 2 r₁², then:A = r₁² - d² = r₁² - 2 r₁² = - r₁²B = r₁²So A = -B, which would imply that the quadratic form is -r₁² h² + r₁² k² + ... = 0. But let's check the rest of the equation. If we set d² = 2 r₁², then we can substitute d = r₁ √2 into the other coefficients and the constant term to see if the equation indeed represents a hyperbola. However, even if A = -B, we still need to ensure that the other terms don't cause the equation to degenerate or become another type of conic.But if A = -B, and assuming the equation is non-degenerate (i.e., determinant not zero), then it's a hyperbola. The key condition for it to be a rectangular hyperbola is A = -B. Therefore, if we set A = -B, that is, r₁² - d² = - r₁², leading to d² = 2 r₁², then the quadratic part is -r₁² h² + r₁² k², which factors as r₁²(k² - h²), leading to a rectangular hyperbola. But we need to check if the entire equation, including linear and constant terms, corresponds to a rectangular hyperbola. However, even with linear terms, the type of hyperbola is determined by the quadratic terms. The linear terms only affect the position (translation) of the hyperbola, not its type or the angles between its asymptotes. Hence, if the quadratic terms satisfy A = -B, then the hyperbola is rectangular, regardless of the linear terms. Therefore, setting A = -B is sufficient.Therefore, the condition is d² = 2 r₁², so d = r₁√2.Wait, but hold on. The problem says "the locus of P when t varies is contained in an equilateral hyperbola". So, even if after translating, the hyperbola remains equilateral. Since equilateral hyperbola's asymptotes are perpendicular regardless of translation, because translation doesn't affect the asymptotes' angles. Therefore, even if there are linear terms, the quadratic form's coefficients determine the asymptotes' angles. Therefore, the key condition is that the quadratic terms have A = -B, hence the distance d must satisfy d² = 2 r₁². So, d = r₁√2. But wait, the problem says "as a function of r₁ and r₂". Wait, but in our derivation, we didn't use r₂. That seems odd. The answer is only dependent on r₁? But in the problem statement, it's possible that r₂ affects the condition. Hmm. Maybe my mistake.Wait, let's go back. When we set A = -B, which is (r₁² - d²) = - r₁², which leads to d² = 2 r₁², but in this case, the equation of the locus is:- r₁² h² + r₁² k² - 2 d (r₁² + r₂² - d²) h + [ d² (r₁² + 2 r₂² - d²) - r₂^4 ] = 0But if d² = 2 r₁², let's substitute that into the coefficients:First, d² = 2 r₁², so d = r₁√2.Substitute into the coefficient of h:-2 d (r₁² + r₂² - d²) = -2 r₁√2 (r₁² + r₂² - 2 r₁² ) = -2 r₁√2 ( r₂² - r₁² )And the constant term:d² (r₁² + 2 r₂² - d²) - r₂^4 = 2 r₁² ( r₁² + 2 r₂² - 2 r₁² ) - r₂^4 = 2 r₁² ( - r₁² + 2 r₂² ) - r₂^4 = -2 r₁^4 + 4 r₁² r₂² - r₂^4So the equation becomes:- r₁² h² + r₁² k² - 2 r₁√2 ( r₂² - r₁² ) h + ( -2 r₁^4 + 4 r₁² r₂² - r₂^4 ) = 0Divide both sides by r₁² to simplify:- h² + k² - 2 √2 ( r₂² - r₁² ) / r₁ * h + ( -2 r₁² + 4 r₂² - ( r₂^4 ) / r₁² ) = 0But this seems complicated, but regardless, the quadratic terms are - h² + k², so the equation is of the form k² - h² + linear terms + constant = 0, which is a hyperbola, and since the coefficients of h² and k² are equal in magnitude and opposite in sign, the asymptotes will be perpendicular. Hence, even with the linear and constant terms, the asymptotes are determined by the quadratic terms, leading to slopes ±1. Therefore, the hyperbola is equilateral.But this conclusion seems to depend only on d = √2 r₁, regardless of r₂. However, the problem states that the answer should be a function of r₁ and r₂. This suggests that my previous approach might be missing something. Maybe the condition also involves r₂?Wait, let me think again. When we set A = -B, which is r₁² - d² = - r₁², leading to d² = 2 r₁², this was derived purely from the quadratic terms. However, perhaps there is another condition from the rest of the equation. For instance, if the linear terms or the constant term impose a relation involving r₂, maybe the distance d also relates to r₂.But in the previous steps, when we set A = -B, we obtained d² = 2 r₁², but the rest of the equation (linear and constant terms) involve both r₁ and r₂. However, the type of conic (rectangular hyperbola) is determined solely by the quadratic terms. Therefore, even if there are linear terms, the hyperbola remains rectangular as long as the quadratic terms satisfy A = -B. Hence, the only condition needed is d = √2 r₁, regardless of r₂. But the problem states "as a function of r₁ and r₂", which suggests that the answer might depend on both. Therefore, I must have made a mistake in my reasoning.Alternatively, perhaps there's a different approach where the answer does depend on both r₁ and r₂.Wait, let's consider inversion. Maybe not. Alternatively, use the theory of poles and polars. Alternatively, consider some geometric relations.Let me think differently. The locus of P is the set of all points from which the chord of contact with respect to C₂ is a tangent to C₁. Wait, because AB is the chord of contact of P with respect to C₂, and AB is tangent to C₁. Therefore, the locus of P is the set of points whose chord of contact with respect to C₂ is tangent to C₁. This is called the conic called the " reciprocal conic" or something similar.Alternatively, the locus of P such that their polar lines with respect to C₂ are tangent to C₁. In that case, the locus is known to be a conic, specifically an ellipse, hyperbola, etc., depending on the configuration.In general, the locus of points whose polars with respect to a circle touch another circle is a conic. The type of conic depends on the position of the circles. For it to be a hyperbola, the circles must be separated in some way.However, to get a rectangular hyperbola, certain conditions on the circles must be satisfied.Alternatively, recall that the equation we derived earlier was:(h - d)^2 + k^2 = [ (r₂² + d h - d² ) / r₁ ]²If we expand this, we get:h² - 2 d h + d² + k² = (d h + r₂² - d² )² / r₁²Multiplying both sides by r₁²:r₁² h² - 2 d r₁² h + r₁² d² + r₁² k² = d² h² + 2 d (r₂² - d²) h + (r₂² - d²)^2Bring all terms to the left:(r₁² - d²) h² + r₁² k² - 2 d (r₁² + r₂² - d²) h + [ r₁² d² - (r₂² - d²)^2 ] = 0As before. So, as mentioned, for this to be a rectangular hyperbola, we require that the quadratic terms in h² and k² are negatives of each other. So:r₁² - d² = - r₁² ⇒ d² = 2 r₁²But this gives d = √2 r₁, independent of r₂. However, the problem states "as a function of r₁ and r₂", which suggests that the answer might involve both r₁ and r₂. So perhaps my mistake was in assuming that the locus is a hyperbola only when A = -B, but maybe there is another condition that also involves r₂. Let me check.Wait, suppose d² ≠ 2 r₁². Then, the quadratic terms would be (r₁² - d²) h² + r₁² k². For this to represent a hyperbola, we need that the coefficients of h² and k² have opposite signs. So either:1. (r₁² - d²) > 0 and r₁² < 0 (impossible, since r₁² is positive)2. (r₁² - d²) < 0 and r₁² > 0. So, if (r₁² - d²) < 0, which means d² > r₁². In this case, the equation is a hyperbola if also the other terms don't make it degenerate.But the problem states that the locus is contained in an equilateral hyperbola, not just any hyperbola. Therefore, even if it is a hyperbola, we need it to be equilateral. For that, as before, the quadratic terms must satisfy A = -B. Therefore, regardless of other terms, we must have (r₁² - d²) = - r₁² ⇒ d² = 2 r₁². Therefore, the distance between centers is d = √2 r₁. But why does the problem mention r₂ then? Maybe there is an error in my approach. Let me verify with specific cases.Suppose r₂ = r₁. If d = √2 r₁, then the centers are separated by √2 r₁, and the locus of P is a rectangular hyperbola. If I take r₂ to be different, say r₂ = 2 r₁, then does the same condition hold? According to the previous analysis, yes. The condition d = √2 r₁ is independent of r₂, which seems counter-intuitive. Maybe I need to consider the discriminant or another condition.Wait, in the equation of the locus:(r₁² - d²) h² + r₁² k² - 2 d (r₁² + r₂² - d²) h + [ r₁² d² - (r₂² - d²)^2 ] = 0If we set d² = 2 r₁², then substituting back into the equation:(r₁² - 2 r₁²) h² + r₁² k² - 2 * sqrt(2) r₁ ( r₁² + r₂² - 2 r₁² ) h + [ r₁² * 2 r₁² - (r₂² - 2 r₁²)^2 ] = 0Simplify each term:- r₁² h² + r₁² k² - 2 sqrt(2) r₁ ( - r₁² + r₂² ) h + ( 2 r₁^4 - ( r₂^4 - 4 r₂² r₁² + 4 r₁^4 ) ) = 0Simplify the constant term:2 r₁^4 - r₂^4 + 4 r₂² r₁² - 4 r₁^4 = -2 r₁^4 + 4 r₁² r₂² - r₂^4So, the equation becomes:- r₁² h² + r₁² k² + 2 sqrt(2) r₁ ( r₁² - r₂² ) h -2 r₁^4 + 4 r₁² r₂² - r₂^4 = 0Divide the entire equation by r₁²:- h² + k² + 2 sqrt(2) ( r₁² - r₂² ) / r₁ * h -2 r₁² + 4 r₂² - ( r₂^4 ) / r₁² = 0This is a hyperbola with quadratic terms -h² + k², so indeed a rectangular hyperbola. The presence of linear terms means the hyperbola is shifted, but it's still rectangular. The coefficients involving r₂ affect the position and shape (like the axes lengths), but not the angle between the asymptotes, which is determined solely by the quadratic terms. Therefore, regardless of r₂, as long as d = sqrt(2) r₁, the locus is a rectangular hyperbola.But the problem says "as a function of r₁ and r₂", so maybe the answer is d = sqrt(2) r₁, regardless of r₂. However, the problem might be expecting a different answer. Alternatively, perhaps there's an error in my analysis.Alternatively, maybe the condition also requires that the linear terms vanish, which would center the hyperbola at the origin. But the problem doesn't state that the hyperbola is centered at the origin, just that it's equilateral (rectangular). Since a hyperbola can be translated and still be rectangular, the linear terms don't affect the rectangular nature.Alternatively, maybe there is a different condition that involves r₂. For example, in the equation above, even if d = sqrt(2) r₁, the constant term and the linear term still involve r₂. For the equation to represent a non-degenerate hyperbola, the coefficients shouldn't contradict. For example, if for some values of r₂, the equation becomes a degenerate hyperbola, but as long as it's non-degenerate, it's a rectangular hyperbola. Alternatively, maybe I need to ensure that the original geometric configuration is possible. For instance, the line t must intersect circle C₂, so the distance between the line t and the center of C₂ must be less than or equal to r₂. The tangent line t to C₁ is at distance r₁ from O₁, and the distance from O₂ to the line t must be ≤ r₂. The distance from O₂(d, 0) to the line x cos θ + y sin θ = r₁ is |d cos θ - r₁| / sqrt(cos² θ + sin² θ) = |d cos θ - r₁|. For the line t to intersect C₂, this distance must be ≤ r₂. So |d cos θ - r₁| ≤ r₂. For all θ such that this holds, the line t intersects C₂. But since t is a tangent to C₁, which is fixed, and we need the locus of P to be contained in a hyperbola. But in our previous analysis, we found that if d = sqrt(2) r₁, then regardless of r₂, the locus is a rectangular hyperbola. However, if r₂ is too small, maybe there are no such points P, but the problem states "the locus of P when t varies is contained in an equilateral hyperbola". Therefore, as long as there are points P, the hyperbola exists, and the condition is only on d being sqrt(2) r₁. Alternatively, maybe the problem requires the locus to be exactly a hyperbola, not a degenerate conic or empty set. So, as long as the line t can intersect C₂ for some θ, which would require that the maximum distance from O₂ to line t is ≥ r₂. The maximum value of |d cos θ - r₁| occurs when cos θ = ±1, so |d - r₁| and |-d - r₁|. Since d is positive (distance between centers), the maximum is d + r₁, and the minimum is |d - r₁|. For the line t to intersect C₂, we need |d cos θ - r₁| ≤ r₂. Therefore, for some θ, this must hold. The minimum value of |d cos θ - r₁| is 0 if d cos θ = r₁, which is possible if d ≥ r₁ (since cos θ ≤ 1). If d < r₁, then the minimum is r₁ - d. Therefore, the condition for the existence of intersections is r₂ ≥ r₁ - d if d < r₁, or always possible if d ≥ r₁. But this is a separate condition for the existence of points A and B. However, the problem doesn't specify that the locus must be non-degenerate or non-empty, just that when it exists, it's contained in a rectangular hyperbola. Therefore, even if for some d and r₂ the locus is empty, but when it's non-empty, it's a rectangular hyperbola. Therefore, the main condition is d = sqrt(2) r₁. But the problem says "the locus of P when t varies is contained in an equilateral hyperbola". Therefore, if even part of the hyperbola is traced, but it's contained within an equilateral hyperbola, then the answer is still d = sqrt(2) r₁. Therefore, regardless of r₂, the required distance is sqrt(2) r₁. However, the problem states "as a function of r₁ and r₂", implying that the answer depends on both. This discrepancy suggests that my initial approach might be incorrect.Let me try another method. Let's use geometric transformations or properties.Since P is the intersection of tangents from A and B to C₂, and AB is the chord of contact from P to C₂, which is also tangent to C₁. So, for each P, its polar line with respect to C₂ is tangent to C₁. The set of all such P is called the reciprocal curve of C₁ with respect to C₂. In general, the reciprocal of a circle with respect to another circle is a conic. The type of conic depends on the relative positions of the two circles.A reciprocal curve can be an ellipse, hyperbola, or parabola depending on whether the original curve is inside, intersecting, or outside the circle with respect to which we take the reciprocal. In this case, since we're taking the reciprocal of C₁ with respect to C₂, and C₁ and C₂ are two circles with centers separated by distance d, the reciprocal curve will be a conic. For it to be a hyperbola, C₁ must lie outside C₂'s reciprocal base. But perhaps I need to recall the exact conditions.Alternatively, the reciprocal of a circle with respect to another circle is a conic whose type is determined by the position of the original circle relative to the base circle. If the original circle (C₁) is outside the base circle (C₂), the reciprocal is an ellipse; if it intersects, it's a hyperbola; if it's inside, it might be a hyperbola as well. Wait, not sure. Alternatively, use inversion. Inversion in the circle C₂. The inverse of C₁ with respect to C₂ would be another circle (if C₁ doesn't pass through the center of inversion). The reciprocal (which is the set of poles of tangents to C₁ with respect to C₂) is the inverse of C₁. So, if the inverse of C₁ with respect to C₂ is a circle, then the reciprocal curve is a circle, but the problem states it's a hyperbola. Therefore, the inverse must be a line, which happens when C₁ passes through the center of inversion. But since we are inverting with respect to C₂, if C₁ passes through O₂, then its inverse would be a line. But in that case, the reciprocal curve (which is the inverse) would be a line, which is a degenerate conic. But the problem states it's a hyperbola, so it must be non-degenerate. Therefore, maybe this approach isn't helpful.Alternatively, recall that the locus of P is the set of points for which the polar with respect to C₂ is tangent to C₁. This locus is known as the conic called the "polar conic" or "reciprocal curve". The equation of this locus is what we derived earlier. To determine when this conic is a rectangular hyperbola, we need to impose conditions on its equation. As we did before, the quadratic terms must satisfy A = -B. This gives d² = 2 r₁². Therefore, the answer is d = sqrt(2) r₁. But since the problem mentions "as a function of r₁ and r₂", and my answer doesn't involve r₂, I'm concerned. Maybe the condition also requires the linear term to disappear, but in our equation, the linear term is:-2 d (r₁² + r₂² - d²) hIf we set this coefficient to zero, then:-2 d (r₁² + r₂² - d²) = 0Which implies either d = 0 (which is impossible, since the circles would be concentric, and the problem likely assumes distinct centers), or:r₁² + r₂² - d² = 0 ⇒ d² = r₁² + r₂²But this is another condition. So if we set both:1. A = -B ⇒ d² = 2 r₁²2. Linear term coefficient = 0 ⇒ d² = r₁² + r₂²Then, combining these:From 1: d² = 2 r₁²From 2: d² = r₁² + r₂²Equating the two:2 r₁² = r₁² + r₂² ⇒ r₁² = r₂² ⇒ r₁ = r₂Therefore, this combined condition would require r₁ = r₂ and d² = 2 r₁². But this is more restrictive and depends on r₂. However, the problem doesn't state that r₁ = r₂, so this approach might not be valid.Alternatively, maybe we need only set the linear term to zero to get a standard hyperbola centered at the origin, but the problem doesn't specify the hyperbola's center. It just needs to be rectangular. Since a hyperbola can be translated and still be rectangular, the linear term doesn't affect the rectangular nature, only the position. Therefore, the only essential condition is A = -B, leading to d = sqrt(2) r₁. Alternatively, maybe the problem is from a source that assumes a specific relation or there's a different approach. Let me try to think differently.Consider the point P such that PA and PB are tangents to C₂, and AB is a tangent to C₁. Then, the locus of P is the set of points from which two tangents can be drawn to C₂, such that the chord of contact is tangent to C₁. This is similar to the definition of the director circle of a circle, but in this case, instead of the director circle (which is the locus of points from which the angle between the tangents is 90 degrees), it's the locus where the chord of contact is tangent to another circle.In general, the locus of points P such that the chord of contact of P with respect to a circle is tangent to another circle is a conic. For it to be a rectangular hyperbola, certain conditions on the circles must hold.In our case, the locus is given by the equation derived earlier, which is a conic. To make this conic a rectangular hyperbola, the quadratic terms must satisfy A = -B, leading to d² = 2 r₁², independent of r₂. Hence, my conclusion remains that the distance between centers must be sqrt(2) r₁.Alternatively, let's verify this with a specific example. Let me take r₁ = 1, r₂ = 1, and d = sqrt(2). Then, according to our condition, the locus should be a rectangular hyperbola.The equation becomes:-1² h² + 1² k² - 2 * sqrt(2) * (1² + 1² - (sqrt(2))²) h + [ (sqrt(2))² (1² + 2*1² - (sqrt(2))² ) - 1^4 ] = 0Simplify:- h² + k² - 2 sqrt(2) (1 + 1 - 2) h + (2 (1 + 2 - 2) - 1) = 0Simplify each part:- h² + k² - 2 sqrt(2) (0) h + (2 (1) - 1 ) = 0 ⇒ -h² + k² + 1 = 0 ⇒ k² - h² = -1 ⇒ h² - k² = 1Which is a standard rectangular hyperbola. So, yes, in this case, the locus is a rectangular hyperbola.If we take r₁ = 1, r₂ = 2, d = sqrt(2):Equation:-1 h² +1 k² - 2*sqrt(2)*(1 + 4 - 2)*h + [2*(1 + 8 - 2) - 16] = 0Calculates as:- h² + k² - 2*sqrt(2)*3 h + [2*7 - 16] = -h² + k² - 6 sqrt(2) h + (14 - 16) = -h² + k² - 6 sqrt(2) h - 2 = 0Which is a rectangular hyperbola, since the quadratic terms are -h² + k². It's just shifted and has a constant term, but the asymptotes are still y = ±x, so it's equilateral.Therefore, even with different r₂, the hyperbola remains equilateral. Therefore, the condition is indeed d = sqrt(2) r₁, independent of r₂. Thus, the distance between centers is sqrt(2) times the radius of C₁.But the problem says "as a function of r₁ and r₂", but according to this, it's only a function of r₁. Maybe there is an error in the problem statement, or perhaps my analysis is missing something. Alternatively, perhaps the problem assumes that the hyperbola is non-degenerate and requires a relation between r₁ and r₂ to ensure that. However, in our earlier example with r₁ = 1, r₂ = 1, d = sqrt(2), the hyperbola is non-degenerate, and similarly for r₂ = 2. So, perhaps the answer is indeed sqrt(2) r₁, regardless of r₂. Alternatively, maybe there's a different interpretation. Suppose the variable tangent t to C₁ intersects C₂ at A and B, and the tangents from A and B to C₂ intersect at P. The locus of P is the set of points P such that P lies on the polar of A and B with respect to C₂, but A and B lie on t, which is tangent to C₁. This seems similar to the earlier reasoning.Given that multiple approaches lead to the condition d = sqrt(2) r₁, and the problem's answer format requires a function of r₁ and r₂, but the derived condition doesn't involve r₂, I think the correct answer is d = sqrt(2) r₁, expressed as a function d = sqrt(2) r₁. But since the problem says "as a function of r₁ and r₂", maybe there is a misunderstanding. Perhaps the original problem had a different setup, but according to the analysis, the distance is sqrt(2) r₁.Wait, another idea: Maybe the hyperbola is equilateral only if the two circles are orthogonal. The condition for two circles to be orthogonal is that the square of the distance between centers is equal to the sum of the squares of the radii: d² = r₁² + r₂². But this is the condition for orthogonality, which results in the tangents from any point on one circle to the other being equal, etc. However, in our problem, the condition derived was d² = 2 r₁², which is different. But if orthogonality is d² = r₁² + r₂², and in our case, it's d² = 2 r₁², which would require r₂ = r₁. So if r₂ = r₁, then orthogonal circles would have d = sqrt(2) r₁. But in our case, the condition is d = sqrt(2) r₁ regardless of r₂. So unless r₂ = r₁, the orthogonality condition isn't met. Therefore, these are different conditions. In our problem, the requirement is that the locus is a rectangular hyperbola, which happens when d = sqrt(2) r₁, independent of r₂. Therefore, despite the problem mentioning r₂, the answer does not depend on it. Alternatively, maybe there's a miscalculation when I derived the equation. Let me check again.Starting from the beginning: We have two circles. A tangent t to C₁ intersects C₂ at A and B. Tangents from A and B to C₂ meet at P. We need the locus of P to be a rectangular hyperbola. We placed C₁ at (0,0), C₂ at (d,0). The tangent t to C₁ is x cos θ + y sin θ = r₁. This line intersects C₂ at points A and B. The polar of P with respect to C₂ is the line AB, which is the tangent t to C₁. Therefore, P lies on the polar line of t with respect to C₂. Wait, but no, AB is the polar of P, and AB is tangent to C₁. Therefore, the polar of P with respect to C₂ is tangent to C₁. Therefore, the locus of P is the set of points whose polars with respect to C₂ are tangent to C₁. This locus is known as the reciprocal curve of C₁ with respect to C₂, which is a conic. For this conic to be a rectangular hyperbola, certain metric conditions must hold. The equation of this conic was derived as:(r₁² - d²) h² + r₁² k² - 2 d (r₁² + r₂² - d²) h + [ r₁² d² - (r₂² - d²)^2 ] = 0For this to be a rectangular hyperbola, the coefficients of h² and k² must be negatives of each other. Hence:r₁² - d² = - r₁² ⇒ d² = 2 r₁²Thus, the distance between centers is sqrt(2) r₁.Since the problem asks for the distance as a function of r₁ and r₂, and this result doesn't involve r₂, there must be an error. However, multiple derivations confirm that d = sqrt(2) r₁ is the required condition. Possibly, the problem intended for the answer to involve both radii, but due to the specific geometric conditions, it cancels out. Alternatively, there's a different interpretation. For example, if we consider the hyperbola to be equilateral with a specific orientation or position, but since the hyperbola can be rotated or translated, the condition remains the same.Given the analysis, I will conclude that the distance between the centers is sqrt(2) times r₁, regardless of r₂. Therefore, the answer is d = sqrt(2) r₁, expressed as a function of r₁ and r₂ as sqrt(2) r₁. However, since the problem asks for a function of r₁ and r₂, but the answer doesn't depend on r₂, there might be a typo or my misunderstanding. But based on the derivations, the answer is:Final AnswerThe distance between the centers is boxed{sqrt{2} r_1}.</think>Given two circles ( C_1 ) and ( C_2 ) with radii ( r_1 ) and ( r_2 ) respectively, we need to find the distance between their centers such that the locus of point ( P ), formed by the intersection of tangents from points ( A ) and ( B ) (where a variable tangent to ( C_1 ) intersects ( C_2 )), is contained in an equilateral hyperbola.1. Setup and Equations: - Place ( C_1 ) at the origin ((0, 0)) and ( C_2 ) at ((d, 0)). - The tangent line to ( C_1 ) at angle ( theta ) is ( x cos theta + y sin theta = r_1 ). - This line intersects ( C_2 ) at points ( A ) and ( B ), which are used to find the tangents from ( P ).2. Intersection of Tangent Line with ( C_2 ): - Substitute the equation of the tangent line into ( C_2 )'s equation ((x - d)^2 + y^2 = r_2^2 ). - Solve for ( x ) and ( y ) to find the coordinates of ( A ) and ( B ).3. Locus of Point ( P ): - ( P ) is the external homothety center of the tangents from ( A ) and ( B ) to ( C_2 ). - The equation of the polar line of ( P ) with respect to ( C_2 ) must be the same as the tangent line to ( C_1 ).4. Quadratic Form and Hyperbola Condition: - Derive the quadratic equation for the locus of ( P ). - For the hyperbola to be equilateral (rectangular), the coefficients of ( h^2 ) and ( k^2 ) must be equal in magnitude and opposite in sign.5. Key Condition: - The quadratic terms in the derived equation must satisfy ( r_1^2 - d^2 = -r_1^2 ), leading to ( d^2 = 2r_1^2 ).6. Conclusion: - The distance between the centers of ( C_1 ) and ( C_2 ) must be ( sqrt{2} r_1 ).Thus, the distance between the centers of ( C_1 ) and ( C_2 ) such that the locus of ( P ) is contained in an equilateral hyperbola is (boxed{sqrt{2} r_1}).,<think>Okay, so I have this geometry problem here. Let me try to parse it step by step. The problem says:"A variable tangent t to the circle C₁, of radius r₁, intersects the circle C₂, of radius r₂ in points A and B. The tangents to C₂ through A and B intersect in P. Find, as a function of r₁ and r₂, the distance between the centers of C₁ and C₂ such that the locus of P when t varies is contained in an equilateral hyperbola. Note: A hyperbola is said to be equilateral if its asymptotes are perpendicular."Alright, so we have two circles, C₁ and C₂. The tangent to C₁ (radius r₁) is variable, meaning it can change. This tangent line t intersects circle C₂ (radius r₂) at points A and B. From A and B, we draw tangents to C₂, and these tangents intersect at point P. As the tangent t varies, the point P moves, and we need to find the distance between the centers of C₁ and C₂ such that the path (locus) of P is an equilateral hyperbola.First, let me recall that an equilateral hyperbola (also known as rectangular hyperbola) is a hyperbola whose asymptotes are perpendicular. This implies that the axes of the hyperbola are equal in length. In other words, the standard form of such a hyperbola is x² - y² = a² or xy = c², depending on the orientation.The problem is to find the distance d between the centers of C₁ and C₂ so that the locus of P is such a hyperbola.Let me start by setting up a coordinate system. Let's assume that the center of C₁ is at the origin (0,0) for simplicity. Then, the center of C₂ will be at some point (d, 0), since the distance between the centers is d along the x-axis. This choice should simplify calculations, as it aligns the centers along the x-axis.Given that, circle C₁ has equation x² + y² = r₁², and circle C₂ has equation (x - d)² + y² = r₂².Now, the tangent t to C₁ is variable. A tangent to a circle can be represented in various ways. For a circle centered at the origin, the tangent line at a point (x₀, y₀) on the circle is xx₀ + yy₀ = r₁². However, since the tangent is variable, it's better to parametrize it using a parameter, say θ, the angle made with the x-axis.Alternatively, we can use a parametric form of the tangent line. For the circle C₁ at the origin, a general tangent line can be written as x cos θ + y sin θ = r₁. This is derived from the point-slope form of the tangent line. The angle θ here represents the angle between the radius to the point of tangency and the x-axis. So, for each θ, we have a different tangent line t.Alternatively, we can parametrize the tangent lines using slope. Let me see. If we let the tangent line have slope m, then its equation would be y = mx ± r₁√(1 + m²). But maybe using θ is better because it gives a direct parameterization without square roots.So, using the angle θ, the tangent line t to C₁ is x cos θ + y sin θ = r₁.This line t intersects circle C₂ at points A and B. So, we need to find the intersection points between the line x cos θ + y sin θ = r₁ and the circle (x - d)² + y² = r₂².Once we have points A and B, we need to find the tangents from A and B to C₂. Wait, but A and B are already points on C₂, right? Because the line t intersects C₂ at A and B. So, if A and B are on C₂, then the tangent to C₂ at A is the line that touches C₂ only at A, and similarly for B. However, the problem says "the tangents to C₂ through A and B intersect in P". Wait, but if A and B are points on C₂, then the tangent at A is unique, and the tangent at B is unique. So, for each A and B, the tangent lines at those points will intersect at some point P.But then, if A and B are variable (since t is variable), then P will trace out a locus. The problem states that this locus must be contained in an equilateral hyperbola, and we need to find the distance d between the centers of C₁ and C₂ such that this condition holds.So, first, let's find the coordinates of points A and B by solving the intersection of the tangent line t to C₁ with circle C₂.Given the tangent line to C₁: x cos θ + y sin θ = r₁.And circle C₂: (x - d)² + y² = r₂².To find the points of intersection, we can solve these two equations simultaneously.Let me express y from the tangent line equation:From x cos θ + y sin θ = r₁, we get y = (r₁ - x cos θ)/sin θ.Substitute this into the equation of C₂:(x - d)² + [(r₁ - x cos θ)/sin θ]^2 = r₂².Let me expand this:(x - d)² + (r₁ - x cos θ)² / sin² θ = r₂².Let me compute each term:First term: (x - d)^2 = x² - 2 d x + d².Second term: (r₁ - x cos θ)^2 / sin² θ = (r₁² - 2 r₁ x cos θ + x² cos² θ)/sin² θ.Adding them together:x² - 2 d x + d² + (r₁² - 2 r₁ x cos θ + x² cos² θ)/sin² θ = r₂².Multiply through by sin² θ to eliminate the denominator:(x² - 2 d x + d²) sin² θ + r₁² - 2 r₁ x cos θ + x² cos² θ = r₂² sin² θ.Combine like terms:x² sin² θ + x² cos² θ - 2 d x sin² θ + d² sin² θ + r₁² - 2 r₁ x cos θ = r₂² sin² θ.Note that sin² θ + cos² θ = 1, so x² (sin² θ + cos² θ) = x².So, simplifying:x² - 2 d x sin² θ + d² sin² θ + r₁² - 2 r₁ x cos θ = r₂² sin² θ.Bring all terms to the left:x² - 2 d x sin² θ + d² sin² θ + r₁² - 2 r₁ x cos θ - r₂² sin² θ = 0.Let me collect like terms:x² terms: x².x terms: -2 d sin² θ x - 2 r₁ cos θ x.Constants: d² sin² θ + r₁² - r₂² sin² θ.So the equation becomes:x² - 2 x (d sin² θ + r₁ cos θ) + (d² sin² θ + r₁² - r₂² sin² θ) = 0.This is a quadratic in x. Let me write it as:x² - 2 x [d sin² θ + r₁ cos θ] + [sin² θ (d² - r₂²) + r₁²] = 0.Let me denote coefficients:A = 1,B = -2 [d sin² θ + r₁ cos θ],C = sin² θ (d² - r₂²) + r₁².The solutions are x = [ -B ± √(B² - 4AC) ] / 2A.But perhaps instead of solving for x, which might be complicated, we can note that points A and B lie on both the tangent line and circle C₂. However, since we need to find the tangents from A and B to C₂, but A and B are points on C₂, so the tangent at A is the line perpendicular to the radius of C₂ at A, and similarly for B.Therefore, given point A on C₂, the tangent at A has the equation (for circle C₂ centered at (d,0)):(Ax - d)(x - d) + Ay y = r₂²,Wait, no. The tangent line to circle C₂ at point (x₁, y₁) is (x₁ - d)(x - d) + y₁ y = r₂².Yes, because the general equation for the tangent to a circle at point (x₁, y₁) on the circle is (x₁ - h)(x - h) + (y₁ - k)(y - k) = r² for a circle centered at (h, k). So in our case, center is (d, 0), so it becomes (x₁ - d)(x - d) + (y₁ - 0)(y - 0) = r₂², which simplifies to (x₁ - d)(x - d) + y₁ y = r₂².Similarly for point B.Therefore, if we have points A = (x_A, y_A) and B = (x_B, y_B) on C₂, then the tangent at A is (x_A - d)(x - d) + y_A y = r₂², and the tangent at B is (x_B - d)(x - d) + y_B y = r₂².The intersection point P of these two tangents is what we need to find.Given that points A and B lie on the tangent line t to C₁, which is x cos θ + y sin θ = r₁, we can parametrize A and B in terms of θ. But since solving for A and B might be complex, maybe there's a better way to find the coordinates of P directly in terms of θ.Alternatively, perhaps we can use some properties of poles and polars. The point P is the intersection of the tangents at A and B to C₂, which means that P is the pole of the line AB with respect to C₂. Since AB is the chord of contact of P with respect to C₂.Given that, if line AB is the chord of contact of point P with respect to C₂, then the equation of line AB is (P_x - d)(x - d) + P_y y = r₂². Wait, the polar line of P with respect to C₂ is given by (X - d)(x - d) + Y y = r₂², where (X, Y) are the coordinates of P.But line AB is the polar of P, so AB is (X - d)(x - d) + Y y = r₂².But we also know that line AB is the tangent line t to C₁, which is x cos θ + y sin θ = r₁. Therefore, the two equations must represent the same line:(X - d)(x - d) + Y y = r₂²,andx cos θ + y sin θ = r₁.Therefore, these two equations must be proportional. That is, their coefficients must be scalar multiples of each other.Let me write both equations in standard form:First equation (polar line):(X - d)(x - d) + Y y = r₂².Expanding this:(X - d)x - d(X - d) + Y y = r₂²,which simplifies to:(X - d)x + Y y - d(X - d) - r₂² = 0,or:(X - d)x + Y y - [d(X - d) + r₂²] = 0.Second equation (tangent line t):x cos θ + y sin θ - r₁ = 0.Since these represent the same line, their coefficients must be proportional. Therefore:(X - d) / cos θ = Y / sin θ = - [d(X - d) + r₂²] / (-r₁).Note that the constants terms must also be proportional. Let me check:The first equation's constant term is - [d(X - d) + r₂²], and the second equation's constant term is -r₁. Therefore, the ratio is:- [d(X - d) + r₂²] / (-r₁) = [d(X - d) + r₂²] / r₁.Therefore, the ratios are:(X - d)/cos θ = Y/sin θ = [d(X - d) + r₂²]/r₁.Let me denote the common ratio as k. Then,(X - d) = k cos θ,Y = k sin θ,andd(X - d) + r₂² = k r₁.From the first two equations, we have:k = (X - d)/cos θ = Y/sin θ,so (X - d)/cos θ = Y/sin θ => Y = (X - d) tan θ.From the third equation:d(X - d) + r₂² = k r₁.But k = (X - d)/cos θ, so substituting:d(X - d) + r₂² = (X - d)/cos θ * r₁.Let me rearrange:d(X - d) + r₂² = r₁ (X - d)/cos θ.Multiply both sides by cos θ:d(X - d) cos θ + r₂² cos θ = r₁ (X - d).Bring all terms to one side:d(X - d) cos θ + r₂² cos θ - r₁ (X - d) = 0.Factor out (X - d):(X - d)(d cos θ - r₁) + r₂² cos θ = 0.Solve for (X - d):(X - d) = - r₂² cos θ / (d cos θ - r₁).Therefore,X = d - [r₂² cos θ / (d cos θ - r₁)].Similarly, from Y = (X - d) tan θ,Y = [ - r₂² cos θ / (d cos θ - r₁) ] tan θ= - r₂² cos θ * (sin θ / cos θ) / (d cos θ - r₁)= - r₂² sin θ / (d cos θ - r₁).Therefore, coordinates of P are:X = d - [r₂² cos θ / (d cos θ - r₁)]= [d (d cos θ - r₁) - r₂² cos θ] / (d cos θ - r₁)= [d² cos θ - d r₁ - r₂² cos θ] / (d cos θ - r₁)= [ (d² - r₂²) cos θ - d r₁ ] / (d cos θ - r₁).Similarly,Y = - r₂² sin θ / (d cos θ - r₁).So, X and Y are expressed in terms of θ. To find the locus of P, we need to eliminate θ from these equations and find the relation between X and Y.Let me write the parametric equations:X = [ (d² - r₂²) cos θ - d r₁ ] / (d cos θ - r₁),Y = - r₂² sin θ / (d cos θ - r₁).Let me try to express cos θ and sin θ in terms of X and Y, and then use the identity cos² θ + sin² θ = 1.First, let's denote denominator as D = d cos θ - r₁.Then, from the Y equation:Y = - r₂² sin θ / D => sin θ = - Y D / r₂².From the X equation:X = [ (d² - r₂²) cos θ - d r₁ ] / D.Multiply both sides by D:X D = (d² - r₂²) cos θ - d r₁.But D = d cos θ - r₁, so substitute D:X (d cos θ - r₁) = (d² - r₂²) cos θ - d r₁.Let me expand the left side:X d cos θ - X r₁ = (d² - r₂²) cos θ - d r₁.Bring all terms to the left:X d cos θ - X r₁ - (d² - r₂²) cos θ + d r₁ = 0.Factor cos θ terms:[ X d - (d² - r₂²) ] cos θ + ( - X r₁ + d r₁ ) = 0.Factor r₁ in the constant term:[ X d - d² + r₂² ] cos θ + r₁ ( - X + d ) = 0.Let me write this as:[ d(X - d) + r₂² ] cos θ + r₁ ( d - X ) = 0.Hmm, perhaps solving for cos θ:[ d(X - d) + r₂² ] cos θ = r₁ ( X - d ).Thus,cos θ = [ r₁ ( X - d ) ] / [ d(X - d) + r₂² ].Similarly, from the expression for sin θ:sin θ = - Y D / r₂² = - Y ( d cos θ - r₁ ) / r₂².But we have an expression for cos θ, so substitute:sin θ = - Y [ d ( r₁ ( X - d ) / [ d(X - d) + r₂² ]) - r₁ ] / r₂².Let me compute the term inside the brackets:d ( r₁ ( X - d ) / [ d(X - d) + r₂² ]) - r₁= [ d r₁ ( X - d ) - r₁ ( d(X - d) + r₂² ) ] / [ d(X - d) + r₂² ]= [ d r₁ ( X - d ) - d r₁ ( X - d ) - r₁ r₂² ] / [ d(X - d) + r₂² ]= [ 0 - r₁ r₂² ] / [ d(X - d) + r₂² ]= - r₁ r₂² / [ d(X - d) + r₂² ].Therefore, sin θ = - Y [ - r₁ r₂² / ( d(X - d) + r₂² ) ] / r₂²= - Y [ - r₁ r₂² / ( denominator ) ] / r₂²= Y r₁ / ( d(X - d ) + r₂² ).Thus,sin θ = Y r₁ / ( d(X - d ) + r₂² ).Now, we have expressions for cos θ and sin θ in terms of X and Y:cos θ = [ r₁ ( X - d ) ] / [ d(X - d ) + r₂² ],sin θ = [ Y r₁ ] / [ d(X - d ) + r₂² ].Since cos² θ + sin² θ = 1, we can substitute these into that identity:[ r₁² ( X - d )² / ( d(X - d ) + r₂² )² ] + [ Y² r₁² / ( d(X - d ) + r₂² )² ] = 1.Factor out r₁² / ( denominator )²:[ r₁² / ( d(X - d ) + r₂² )² ] [ ( X - d )² + Y² ] = 1.Multiply both sides by ( denominator )² / r₁²:( X - d )² + Y² = [ d(X - d ) + r₂² ]² / r₁².So, the equation of the locus of P is:( X - d )² + Y² = [ d(X - d ) + r₂² ]² / r₁².Let me write this as:( ( X - d )² + Y² ) r₁² = ( d(X - d ) + r₂² )².Expanding both sides:Left side: r₁² (X - d)^2 + r₁² Y².Right side: [ d(X - d ) + r₂² ]² = d² (X - d )² + 2 d r₂² (X - d ) + r₂⁴.Therefore, bringing all terms to the left:r₁² (X - d)^2 + r₁² Y² - d² (X - d )² - 2 d r₂² (X - d ) - r₂⁴ = 0.Factor (X - d )²:( r₁² - d² ) (X - d )² + r₁² Y² - 2 d r₂² (X - d ) - r₂⁴ = 0.Let me write this as:( r₁² - d² ) (X - d )² - 2 d r₂² (X - d ) + r₁² Y² - r₂⁴ = 0.This is the equation of the locus. We need this to be an equilateral hyperbola. Let's recall that an equilateral hyperbola can be written in the form:A X² + B Y² + C X + D Y + E = 0,with certain conditions on the coefficients to ensure it's a hyperbola with perpendicular asymptotes.Alternatively, we can rotate the coordinate system to eliminate cross terms if necessary. But since the problem states that the hyperbola is equilateral (rectangular), it suffices that after simplification, the equation can be transformed into the standard form of a rectangular hyperbola, possibly rotated.But perhaps we can analyze the given equation to see under what conditions it represents a rectangular hyperbola.First, let's expand the equation in terms of X and Y.Let me denote x = X and y = Y for simplicity. Then the equation becomes:( r₁² - d² ) (x - d )² - 2 d r₂² (x - d ) + r₁² y² - r₂⁴ = 0.Expanding (x - d )²:( r₁² - d² )(x² - 2 d x + d² ) - 2 d r₂² (x - d ) + r₁² y² - r₂⁴ = 0.Multiply out the terms:( r₁² - d² ) x² - 2 d ( r₁² - d² ) x + ( r₁² - d² ) d² - 2 d r₂² x + 2 d² r₂² + r₁² y² - r₂⁴ = 0.Combine like terms:The x² term: ( r₁² - d² ) x².The x terms: -2 d ( r₁² - d² ) x - 2 d r₂² x = -2 d [ r₁² - d² + r₂² ] x.The constant terms: ( r₁² - d² ) d² + 2 d² r₂² - r₂⁴.The y² term: r₁² y².So, putting it all together:( r₁² - d² ) x² - 2 d ( r₁² - d² + r₂² ) x + r₁² y² + [ ( r₁² - d² ) d² + 2 d² r₂² - r₂⁴ ] = 0.Now, to check if this is a hyperbola, we need the coefficients of x² and y² to be of opposite signs (for a hyperbola), and for it to be rectangular, the asymptotes should be perpendicular. For a hyperbola in the form A x² + B y² + C x + D y + E = 0, the condition for it to be rectangular is that A + B = 0. Wait, let me recall.Wait, if we have a hyperbola of the form (x²/a²) - (y²/b²) = 1, the asymptotes are y = ± (b/a)x, which are perpendicular if b/a = 1, i.e., a = b, making it a rectangular hyperbola. But in general, for a hyperbola rotated by an angle θ, the condition for being rectangular is that the angle between asymptotes is 90°, which translates to certain relationships between the coefficients. However, in the general quadratic equation:A x² + B y² + C x + D y + E + F xy = 0,the condition for being a rectangular hyperbola is that the eccentricity is √2, which can be checked via the discriminant, but perhaps more straightforwardly, if we can rotate the axes to eliminate the cross term (F xy), then the coefficients of x² and y² should be equal in magnitude but opposite in sign. Wait, no. Wait, in the rotated coordinate system, for a rectangular hyperbola, the equation becomes X²/a² - Y²/a² = 1, so the coefficients are equal in magnitude but opposite in sign. Therefore, the condition after rotation would be A' = -B'.But in our case, the equation is quadratic in x and y, but with no cross term (xy term). Therefore, the axes of the hyperbola (if it is one) are aligned with the coordinate axes. Therefore, the hyperbola would be of the form A x² + B y² + C x + D y + E = 0. For this to be a hyperbola, A and B must have opposite signs. For it to be a rectangular hyperbola, the asymptotes must be perpendicular, which would require that the slopes of the asymptotes are ±1. However, in the case where there is no xy term, the asymptotes are along the lines y = ± sqrt(-A/B) x. For them to be perpendicular, sqrt(-A/B) must be 1, so -A/B = 1 => A = -B. Therefore, the condition for a rectangular hyperbola in this case is that the coefficients of x² and y² sum to zero: A + B = 0.Wait, let me check this. If the hyperbola is of the form A x² + B y² + ... = 0, with A and B of opposite signs. Let's say A > 0 and B < 0. The asymptotes can be found by ignoring the lower-degree terms. So, setting A x² + B y² = 0 => y = ± sqrt(-A/B) x. For these asymptotes to be perpendicular, the product of their slopes should be -1. But the slopes are ± sqrt(-A/B), so the product is (-A/B). For perpendicularity, we need (-A/B) = -1 => A/B = 1 => A = B. But wait, since A and B have opposite signs, this would require A = -B. Wait, let me clarify.Suppose A is positive and B is negative. Let B = -C where C is positive. Then the equation becomes A x² - C y² + ... = 0. The asymptotes are A x² - C y² = 0 => y = ± sqrt(A/C) x. For these lines to be perpendicular, the slopes must satisfy (sqrt(A/C))*(-sqrt(A/C)) = -A/C = -1. Therefore, A/C = 1 => A = C. Since B = -C, this implies A = -B. Therefore, the condition is A + B = 0.Therefore, in our quadratic equation, for it to represent a rectangular hyperbola, the coefficients of x² and y² must satisfy A + B = 0.Looking back at our equation:( r₁² - d² ) x² - 2 d ( r₁² - d² + r₂² ) x + r₁² y² + [ ( r₁² - d² ) d² + 2 d² r₂² - r₂⁴ ] = 0.Here, the coefficients of x² and y² are ( r₁² - d² ) and r₁², respectively. For the hyperbola to be rectangular, we need:( r₁² - d² ) + r₁² = 0.That is:2 r₁² - d² = 0 => d² = 2 r₁² => d = r₁ sqrt(2).Wait, but let me check. Wait, according to the condition above, A + B = 0, where A is the coefficient of x² and B is the coefficient of y². But in our case, A is ( r₁² - d² ) and B is r₁². So:( r₁² - d² ) + r₁² = 0 => 2 r₁² - d² = 0 => d² = 2 r₁² => d = r₁ sqrt(2).But wait, is that all? Wait, but maybe there are other conditions for this quadratic equation to represent a hyperbola. For example, besides A + B = 0 (for rectangular hyperbola), we need that the equation actually represents a hyperbola, which requires that the discriminant of the quadratic form is positive. The discriminant for a general quadratic equation A x² + B y² + C x + D y + E = 0 (without xy term) is B² - 4 A E, but actually, the discriminant for conic sections is computed differently. Wait, the general discriminant for conic sections is B² - 4 A C, but in our case, since there is no xy term, B=0 (the coefficient of xy), so the discriminant is -4 A C, where A and C are the coefficients of x² and y². Wait, no. Let me recall.The general conic equation is:A x² + B y² + C x + D y + E + F x y = 0.The discriminant is calculated as follows:Δ = B² - 4 A C if we consider it as a quadratic in one variable, but actually, the discriminant for the type of conic is determined by the matrix:| A F/2 C/2 || F/2 B D/2 || C/2 D/2 E |But perhaps more straightforward is to use the invariants. For a quadratic equation without the xy term (F = 0), the discriminant simplifies. The type of conic is determined by the discriminant of the quadratic terms. Specifically, if the coefficients of x² and y² are non-zero and have opposite signs, it's a hyperbola. If they have the same sign, it's an ellipse or a circle, and if one is zero, it's a parabola.In our case, the coefficients of x² and y² are ( r₁² - d² ) and r₁². So, for it to be a hyperbola, these coefficients must have opposite signs. So, either ( r₁² - d² ) > 0 and r₁² < 0, which is impossible since r₁² is positive, or ( r₁² - d² ) < 0 and r₁² > 0, which is possible. So, we need ( r₁² - d² ) < 0, i.e., d² > r₁². However, if we set the condition for the hyperbola to be rectangular, which is ( r₁² - d² ) + r₁² = 0 => d² = 2 r₁², then d² = 2 r₁² which is greater than r₁², so this satisfies the hyperbola condition. Therefore, under the condition d² = 2 r₁², the equation becomes a rectangular hyperbola.But wait, let's check this. If d² = 2 r₁², then:The coefficient of x² is ( r₁² - d² ) = ( r₁² - 2 r₁² ) = - r₁²,and the coefficient of y² is r₁².So, the equation becomes:- r₁² x² - 2 d ( r₁² - d² + r₂² ) x + r₁² y² + [ ( r₁² - d² ) d² + 2 d² r₂² - r₂⁴ ] = 0.But if d² = 2 r₁², then let's substitute d² = 2 r₁²:First, coefficient of x²: - r₁²,Coefficient of y²: r₁²,Therefore, A = - r₁², B = r₁², so A + B = 0, satisfying the rectangular hyperbola condition.Now, let's compute the other coefficients:Term with x: -2 d ( r₁² - d² + r₂² ) x.Substitute d² = 2 r₁²:-2 d ( r₁² - 2 r₁² + r₂² ) x = -2 d ( - r₁² + r₂² ) x = 2 d ( r₁² - r₂² ) x.Similarly, the constant term:( r₁² - d² ) d² + 2 d² r₂² - r₂⁴.Substitute d² = 2 r₁²:( r₁² - 2 r₁² ) * 2 r₁² + 2 * 2 r₁² * r₂² - r₂⁴ =( - r₁² ) * 2 r₁² + 4 r₁² r₂² - r₂⁴ =- 2 r₁⁴ + 4 r₁² r₂² - r₂⁴.Therefore, the equation becomes:- r₁² x² + r₁² y² + 2 d ( r₁² - r₂² ) x + ( - 2 r₁⁴ + 4 r₁² r₂² - r₂⁴ ) = 0.Divide the entire equation by r₁² to simplify:- x² + y² + 2 d (1 - ( r₂² / r₁² )) x + ( - 2 r₁² + 4 r₂² - ( r₂⁴ / r₁² ) ) = 0.But this might not be necessary. Let me factor the equation:- r₁² x² + r₁² y² + 2 d ( r₁² - r₂² ) x + ( - 2 r₁⁴ + 4 r₁² r₂² - r₂⁴ ) = 0.Let me rearrange terms:r₁² y² - r₁² x² + 2 d ( r₁² - r₂² ) x - 2 r₁⁴ + 4 r₁² r₂² - r₂⁴ = 0.Factor out -r₁² from the x² term:r₁² y² - r₁² x² + ... = r₁² ( y² - x² ) + ... So, perhaps write it as:r₁² ( y² - x² ) + 2 d ( r₁² - r₂² ) x - 2 r₁⁴ + 4 r₁² r₂² - r₂⁴ = 0.Let me see if this can be expressed in the standard form of a rectangular hyperbola. A rectangular hyperbola can be written as (x - h)²/a² - (y - k)²/a² = 1 or similar, with equal denominators. Alternatively, after translating the coordinates, it can be written as X² - Y² = c².So, perhaps completing the square for x and y.First, group the x terms and y terms:r₁² ( -x² + y² ) + 2 d ( r₁² - r₂² ) x + ( -2 r₁⁴ + 4 r₁² r₂² - r₂⁴ ) = 0.Let me write this as:- r₁² x² + 2 d ( r₁² - r₂² ) x + r₁² y² + ( -2 r₁⁴ + 4 r₁² r₂² - r₂⁴ ) = 0.Let me focus on the x terms:- r₁² x² + 2 d ( r₁² - r₂² ) x.Factor out - r₁²:- r₁² ( x² - [ 2 d ( r₁² - r₂² ) / r₁² ] x ).Complete the square inside the parentheses:x² - [ 2 d ( r₁² - r₂² ) / r₁² ] x = x² - 2 [ d ( r₁² - r₂² ) / r₁² ] x.The square completion would require adding and subtracting [ d ( r₁² - r₂² ) / r₁² ]².Therefore:= [ x - d ( r₁² - r₂² ) / r₁² ]² - [ d² ( r₁² - r₂² )² / r₁⁴ ].Therefore, the x terms become:- r₁² [ (x - d ( r₁² - r₂² ) / r₁² )² - d² ( r₁² - r₂² )² / r₁⁴ ]= - r₁² (x - d ( r₁² - r₂² ) / r₁² )² + d² ( r₁² - r₂² )² / r₁².Putting this back into the equation:- r₁² (x - d ( r₁² - r₂² ) / r₁² )² + d² ( r₁² - r₂² )² / r₁² + r₁² y² + ( -2 r₁⁴ + 4 r₁² r₂² - r₂⁴ ) = 0.Rearrange terms:- r₁² (x - h )² + r₁² y² + [ d² ( r₁² - r₂² )² / r₁² - 2 r₁⁴ + 4 r₁² r₂² - r₂⁴ ] = 0,where h = d ( r₁² - r₂² ) / r₁².Divide the entire equation by r₁² to simplify:- (x - h )² + y² + [ d² ( r₁² - r₂² )² / r₁⁴ - 2 r₁² + 4 r₂² - r₂⁴ / r₁² ] = 0.Let me compute the constant term:Term1 = d² ( r₁² - r₂² )² / r₁⁴,Term2 = -2 r₁²,Term3 = +4 r₂²,Term4 = - r₂⁴ / r₁².So, the total constant term is Term1 + Term2 + Term3 + Term4.But this is getting complicated. Maybe there's a better way.Alternatively, since we know that under the condition d² = 2 r₁², the quadratic equation reduces to a rectangular hyperbola. Therefore, setting d² = 2 r₁² would satisfy the problem's condition. However, we need to confirm if this is the only condition required.Wait, but in the problem statement, the distance d is to be found as a function of r₁ and r₂ such that the locus is contained in an equilateral hyperbola. Our analysis suggests that setting d² = 2 r₁² would make the quadratic equation a rectangular hyperbola. However, this seems independent of r₂. That seems odd. Why would the answer not depend on r₂? Perhaps I made a mistake in the analysis.Wait, let's recap. The condition A + B = 0 (coefficients of x² and y² sum to zero) gives d² = 2 r₁², which is independent of r₂. However, in the quadratic equation, there are other terms involving r₂² and r₂⁴. Therefore, even if d² = 2 r₁², the constants and linear terms depend on r₂. Therefore, the equation would represent a rectangular hyperbola, but shifted and scaled. However, the problem states that the locus is "contained in" an equilateral hyperbola, which might mean that it is a subset of such a hyperbola. Therefore, even if there are linear terms, completing the square would show that it's a rectangular hyperbola, possibly translated.Therefore, maybe the only condition needed is d² = 2 r₁², regardless of r₂. However, this seems counterintuitive because the position of C₂ (distance d) should depend on both radii r₁ and r₂. Perhaps there is an error in assuming that A + B = 0 is sufficient.Wait, another approach: suppose we let d = sqrt(2) r₁. Then, regardless of r₂, the locus is a rectangular hyperbola. But this seems unlikely because the presence of r₂ in the equation would affect the shape. But according to our previous steps, when d² = 2 r₁², the quadratic equation reduces to a rectangular hyperbola's equation, regardless of r₂. Therefore, maybe the answer is d = sqrt(2) r₁. But I need to verify this.Alternatively, perhaps there's another condition to ensure that the quadratic equation actually represents a hyperbola. Wait, but when d² = 2 r₁², the coefficient of x² is -r₁² and coefficient of y² is r₁², so they have opposite signs, which is the condition for a hyperbola. Therefore, regardless of other terms, this would be a hyperbola. Then, due to the condition A + B = 0, it's a rectangular hyperbola. The other terms (linear and constant) would translate the hyperbola but not affect its rectangularity.Therefore, the key condition is d² = 2 r₁². Therefore, the distance between centers is d = r₁ sqrt(2).But the problem states "as a function of r₁ and r₂", so if the answer is independent of r₂, that's possible, but seems strange. Maybe my mistake lies in the assumption that A + B = 0 is the only condition.Wait, let's check with specific values. Suppose r₂ = 0. Then circle C₂ is a point at (d, 0). Then, the tangent line t to C₁ intersects the point C₂, so points A and B coincide at (d, 0). Then, the tangents from A and B to C₂ are undefined, as C₂ is a point. So, this edge case is problematic. Similarly, if r₂ is very small, the locus might not be a hyperbola. Therefore, the answer must depend on r₂. Therefore, my previous conclusion that d = sqrt(2) r₁ is incorrect.Alternatively, perhaps I missed a step where r₂ comes into play. Let me revisit the equations.We had the condition A + B = 0, where A = ( r₁² - d² ) and B = r₁². So, ( r₁² - d² ) + r₁² = 0 => d² = 2 r₁². But in the process, we might have ignored the other terms. Wait, no. The condition for the hyperbola to be rectangular is on the quadratic terms only. The linear and constant terms affect the position and scaling but not the rectangularity. Therefore, even with those terms, the hyperbola would still be rectangular as long as A + B = 0.Therefore, the answer is d = sqrt(2) r₁, regardless of r₂.But that seems odd. Let me consider specific values for r₁ and r₂. Let's take r₁ = 1, r₂ = 1. Then d = sqrt(2). Let's see what the equation becomes.The equation:( (x - sqrt(2) )² + y² ) * 1² = ( sqrt(2) (x - sqrt(2) ) + 1² )².Wait, this might not be helpful. Alternatively, substitute into the equation:( (X - d )² + Y² ) r₁² = ( d(X - d ) + r₂² )².With d = sqrt(2) r₁, let me set r₁ = 1, d = sqrt(2), r₂ = 1.Equation becomes:( (X - sqrt(2))² + Y² ) * 1 = ( sqrt(2)(X - sqrt(2)) + 1 )².Left side: (X - sqrt(2))² + Y².Right side: [ sqrt(2) X - 2 + 1 ]² = [ sqrt(2) X - 1 ]².Therefore, equation is:(X - sqrt(2))² + Y² = ( sqrt(2) X - 1 )².Expanding both sides:Left: X² - 2 sqrt(2) X + 2 + Y².Right: 2 X² - 2 sqrt(2) X + 1.Subtracting right from left:X² - 2 sqrt(2) X + 2 + Y² - 2 X² + 2 sqrt(2) X - 1 = 0.Simplify:- X² + Y² + 1 = 0 => Y² - X² + 1 = 0 => Y² - X² = -1 => X² - Y² = 1.Which is indeed a rectangular hyperbola. So, in this case, even with r₂ = 1, the equation reduces to X² - Y² = 1, a rectangular hyperbola.Therefore, the answer d = sqrt(2) r₁ is correct, independent of r₂. Wait, but in the problem statement, it says "as a function of r₁ and r₂", which suggests that the answer should depend on both r₁ and r₂. However, according to this analysis, the answer is d = sqrt(2) r₁, regardless of r₂. Is this possible?But in the example above, with r₂ = 1, the hyperbola equation didn't depend on r₂. Wait, but in that case, r₂ was equal to r₁. Wait, no, in the example, I set r₁ = 1, r₂ = 1, d = sqrt(2). But when I did the substitution, the equation reduced to X² - Y² = 1, which does not involve r₂. However, in the general case, when we set d = sqrt(2) r₁, then the equation becomes:( (X - d )² + Y² ) r₁² = ( d(X - d ) + r₂² )².But with d = sqrt(2) r₁, substituting:Left side: r₁² ( (X - sqrt(2) r₁ )² + Y² ).Right side: ( sqrt(2) r₁ (X - sqrt(2) r₁ ) + r₂² )².Let me expand both sides.Left side:r₁² [ X² - 2 sqrt(2) r₁ X + 2 r₁² + Y² ].Right side:[ sqrt(2) r₁ X - 2 r₁² + r₂² ]².= ( sqrt(2) r₁ X + ( r₂² - 2 r₁² ) )².= 2 r₁² X² + 2 sqrt(2) r₁ X ( r₂² - 2 r₁² ) + ( r₂² - 2 r₁² )².Therefore, the equation becomes:r₁² X² - 2 sqrt(2) r₁³ X + 2 r₁⁴ + r₁² Y² = 2 r₁² X² + 2 sqrt(2) r₁ X ( r₂² - 2 r₁² ) + ( r₂² - 2 r₁² )².Bring all terms to the left:r₁² X² - 2 sqrt(2) r₁³ X + 2 r₁⁴ + r₁² Y² - 2 r₁² X² - 2 sqrt(2) r₁ X ( r₂² - 2 r₁² ) - ( r₂² - 2 r₁² )² = 0.Simplify term by term:X² terms: r₁² - 2 r₁² = - r₁².X terms: -2 sqrt(2) r₁³ X - 2 sqrt(2) r₁ X ( r₂² - 2 r₁² )= -2 sqrt(2) r₁ X [ r₁² + r₂² - 2 r₁² ]= -2 sqrt(2) r₁ X ( - r₁² + r₂² )= 2 sqrt(2) r₁ X ( r₁² - r₂² ).Constant terms: 2 r₁⁴ - ( r₂² - 2 r₁² )²= 2 r₁⁴ - ( r₂⁴ - 4 r₁² r₂² + 4 r₁⁴ )= 2 r₁⁴ - r₂⁴ + 4 r₁² r₂² - 4 r₁⁴= -2 r₁⁴ + 4 r₁² r₂² - r₂⁴.Y terms: + r₁² Y².Therefore, the equation becomes:- r₁² X² + 2 sqrt(2) r₁ ( r₁² - r₂² ) X + r₁² Y² -2 r₁⁴ + 4 r₁² r₂² - r₂⁴ = 0.Divide all terms by r₁²:- X² + 2 sqrt(2) ( r₁² - r₂² ) X / r₁ + Y² - 2 r₁² + 4 r₂² - ( r₂⁴ / r₁² ) = 0.This equation is still a rectangular hyperbola because the coefficients of X² and Y² are -1 and 1, summing to zero. Therefore, regardless of the value of r₂, as long as d = sqrt(2) r₁, the equation represents a rectangular hyperbola. The terms involving r₂ affect the position and the shape's scaling but not the rectangularity.Therefore, the answer is that the distance between centers must be d = sqrt(2) r₁, regardless of r₂. However, the problem states "as a function of r₁ and r₂", which suggests that the intended answer might involve both r₁ and r₂. Therefore, there must be a mistake in my reasoning.Wait, let me go back to the problem statement. It says:"A variable tangent t to the circle C₁, of radius r₁, intersects the circle C₂, of radius r₂ in A and B. The tangents to C₂ through A and B intersect in P. Find, as a function of r₁ and r₂, the distance between the centers of C₁ and C₂ such that the locus of P when t varies is contained in an equilateral hyperbola."In the previous analysis, we found that if d² = 2 r₁², the locus is a rectangular hyperbola, regardless of r₂. However, in the example where r₁ = 1, r₂ = 1, d = sqrt(2), we saw that the equation reduces to X² - Y² = 1, which is a rectangular hyperbola. However, if we take r₂ different from r₁, say r₁ = 1, r₂ = 2, d = sqrt(2), then the equation becomes:- X² + Y² + 2 sqrt(2) (1 - 4) X + ( -2 + 16 - 16 ) = 0=> - X² + Y² - 6 sqrt(2) X - 2 = 0Completing the square for X:- (X² + 6 sqrt(2) X ) + Y² - 2 = 0=> - [ (X + 3 sqrt(2))² - 18 ] + Y² - 2 = 0=> - (X + 3 sqrt(2))² + 18 + Y² - 2 = 0=> - (X + 3 sqrt(2))² + Y² + 16 = 0=> Y² - (X + 3 sqrt(2))² = -16Which is a rectangular hyperbola (since coefficients are 1 and -1) centered at (-3 sqrt(2), 0), opening upwards and downwards. So yes, even with different r₂, the hyperbola is rectangular. Hence, the condition d = sqrt(2) r₁ is sufficient regardless of r₂. Therefore, the answer is d = sqrt(2) r₁.But the problem says "as a function of r₁ and r₂", implying the answer should depend on both. This contradiction suggests that my initial analysis missed something.Wait, another thought: maybe the initial parametrization assumes that the center of C₁ is at the origin and C₂ is at (d, 0). However, the problem doesn't specify the relative position of the circles other than the distance between centers. Therefore, by choosing coordinates where C₁ is at the origin and C₂ is on the x-axis, we might have imposed some symmetry which makes the result independent of r₂. However, in reality, the position of C₂ relative to C₁ might need to satisfy a different condition that involves r₂.Alternatively, perhaps there's an error in assuming that the condition A + B = 0 is sufficient for the hyperbola to be equilateral. Wait, but in our example, it worked. However, let me think.An equilateral hyperbola is a hyperbola with perpendicular asymptotes, which is equivalent to it being a rectangular hyperbola. The condition A + B = 0 (for the quadratic equation without cross terms) ensures that the asymptotes are perpendicular. Therefore, this condition is both necessary and sufficient.Therefore, according to the mathematics, the distance d must satisfy d² = 2 r₁², independent of r₂. Therefore, the answer is d = sqrt(2) r₁.But why does the problem mention r₂? Is there a misunderstanding in the problem setup?Wait, let's re-examine the problem statement:"A variable tangent t to the circle C₁, of radius r₁, intersects the circle C₂, of radius r₂ in A and B. The tangents to C₂ through A and B intersect in P. Find, as a function of r₁ and r₂, the distance between the centers of C₁ and C₂ such that the locus of P when t varies is contained in an equilateral hyperbola."The key here is that the tangent line t to C₁ intersects C₂ at A and B. The positions of A and B depend on both the distance d and the radii r₁, r₂. Then, the point P is the intersection of the tangents at A and B to C₂. The locus of P must be a rectangular hyperbola. Our analysis shows that this occurs when d = sqrt(2) r₁, regardless of r₂.But maybe there's a different approach. Let me think of the locus of P.Another approach: The set of points P such that P is the intersection of tangents from A and B, which are points where a tangent to C₁ meets C₂. This locus P is called the conic inscribed or related to both circles. Maybe there's a reciprocal relation or something in bipolar coordinates.Alternatively, consider inversion. Inversion in a circle might simplify the problem, but that could be complex.Alternatively, note that P lies on the polar of line AB with respect to C₂. Since AB is the tangent to C₁, maybe there's a relation between the polars of the two circles.Alternatively, recall that the locus of the poles of a fixed line with respect to a circle is a conic. But in this case, the line AB is variable, being the tangent to C₁.Alternatively, think of P as the pole of line AB with respect to C₂. Since AB is tangent to C₁, which is another circle. So, maybe the locus of P is the set of poles of tangents to C₁ with respect to C₂. The set of such poles is known to be a conic, specifically an ellipse, hyperbola, etc., depending on the relative positions.In general, the locus of poles of a fixed line with respect to a conic is another conic. But here, the line is variable (tangent to C₁), and we take its pole with respect to C₂. The locus should be a conic, and we need it to be a rectangular hyperbola.The condition for this conic to be a rectangular hyperbola would then depend on the relative positions of C₁ and C₂.Alternatively, using La Hire's theorem: If a point lies on the polar of another point with respect to a circle, then the latter lies on the polar of the former.But I'm not sure if this helps directly.Alternatively, think of the problem in terms of the director circle or the radical axis. The director circle of a circle is the locus of points where the tangents are perpendicular. For a circle of radius r, the director circle has radius sqrt(2) r. But I'm not sure.Wait, but if we consider that the locus of P is a rectangular hyperbola, which is the director circle of another conic, but maybe this is a stretch.Alternatively, recall that in our coordinate system, the equation of the locus is a rectangular hyperbola when d = sqrt(2) r₁, regardless of r₂, as shown earlier. Therefore, the answer is d = sqrt(2) r₁.But the problem says "as a function of r₁ and r₂". If the answer does not depend on r₂, then it's simply d = sqrt(2) r₁. However, since the problem mentions both radii, maybe the intended solution requires a different approach where r₂ plays a role.Wait, another possibility: perhaps in my analysis, I assumed that line AB is the tangent to C₁, but in reality, line AB is the variable tangent to C₁, which intersects C₂. The position of C₂ relative to C₁ would affect how line AB cuts C₂, and thus the location of points A and B, which in turn affects the location of P. Therefore, the condition might involve both r₁ and r₂.But according to our previous detailed calculation, the condition d² = 2 r₁² makes the quadratic equation a rectangular hyperbola regardless of r₂, which seems to contradict this intuition.Wait, let's consider another example. Let me take r₁ = 1, r₂ = 2, and d = sqrt(2). Then, following the equation derived earlier:The equation of the locus is:( (X - sqrt(2))² + Y² ) = ( sqrt(2)(X - sqrt(2)) + 4 )².Left side: X² - 2 sqrt(2) X + 2 + Y².Right side: [ sqrt(2) X - 2 + 4 ]² = [ sqrt(2) X + 2 ]² = 2 X² + 4 sqrt(2) X + 4.Equation: X² - 2 sqrt(2) X + 2 + Y² = 2 X² + 4 sqrt(2) X + 4.Bring all terms to left:X² - 2 sqrt(2) X + 2 + Y² - 2 X² - 4 sqrt(2) X - 4 = 0.Simplify:- X² - 6 sqrt(2) X - 2 + Y² = 0.Rearranged:Y² - X² - 6 sqrt(2) X - 2 = 0.Complete the square for X:Y² - (X² + 6 sqrt(2) X ) - 2 = 0.Complete the square inside the parentheses:X² + 6 sqrt(2) X = (X + 3 sqrt(2))² - (3 sqrt(2))² = (X + 3 sqrt(2))² - 18.Substitute back:Y² - [ (X + 3 sqrt(2))² - 18 ] - 2 = 0 => Y² - (X + 3 sqrt(2))² + 16 = 0 => Y² - (X + 3 sqrt(2))² = -16.Which is equivalent to:(X + 3 sqrt(2))² - Y² = 16.Dividing both sides by 16:( (X + 3 sqrt(2))² ) / 16 - Y² / 16 = 1.This is a standard hyperbola, but since the coefficients are 1 and -1 (before dividing by 16), it's a rectangular hyperbola. Therefore, even with r₂ = 2, d = sqrt(2) r₁ gives a rectangular hyperbola. Therefore, my initial conclusion stands.Therefore, despite the problem stating "as a function of r₁ and r₂", the distance d is independent of r₂ and is simply sqrt(2) times r₁. Therefore, the answer is d = sqrt(2) r₁.But why does the problem mention r₂? Maybe there's a misunderstanding in the problem translation or in the original problem statement. Alternatively, perhaps the solution depends on r₂, and my analysis is flawed.Wait, another angle: Maybe the variable tangent t to C₁ is not just any tangent, but specifically the external tangent, but the problem states "variable tangent t to C₁", which can be any tangent. However, depending on the relative positions of C₁ and C₂, the line t might not always intersect C₂. Therefore, perhaps the distance d must be chosen such that the tangent t to C₁ always intersects C₂, but the problem doesn't specify this. It just says "intersects the circle C₂ in A and B". If the tangent t can be such that it doesn't intersect C₂, then the locus of P wouldn't be defined. Therefore, perhaps d must be chosen such that all tangents to C₁ intersect C₂, i.e., C₂ is enclosed within the tangent lines of C₁. But this would require that the distance d is less than or equal to r₁ + r₂, but I'm not sure.But given that the problem states "intersects the circle C₂ in A and B", it is implicit that the tangent t to C₁ intersects C₂ at two points A and B, so for this to happen for all tangents t to C₁, C₂ must be positioned such that all tangents to C₁ intersect C₂. This would require that C₂ is inside the dual circle (the director circle) of C₁, but the director circle of C₁ has radius sqrt(2) r₁. Wait, the director circle (locus of points from which the tangents to C₁ are perpendicular) has radius sqrt(2) r₁. If C₂ is inside this director circle, then all lines t (which are tangents to C₁) would intersect C₂. But this is speculative.Alternatively, if d > r₁ + r₂, then some tangents to C₁ would not intersect C₂. If d < r₁ + r₂, then some tangents would intersect, others not. However, the problem states that t is a tangent to C₁ that intersects C₂, so perhaps we only consider those tangents t that do intersect C₂, and among those, the locus of P is to be a hyperbola. Therefore, even if d is such that only some tangents t intersect C₂, we consider the locus for those t that do intersect, and require that this locus is a rectangular hyperbola.However, according to our previous analysis, the condition d = sqrt(2) r₁ is sufficient, regardless of r₂. Therefore, I think the answer is d = sqrt(2) r₁, and the mention of r₂ in the problem might be a red herring, or perhaps a translation error.Alternatively, there's another approach where the answer does depend on r₂. Let me consider the following:The locus of P is the set of points from which the tangents to C₂ meet at points A and B which lie on a tangent to C₁. This is similar to the definition of a conic section: the locus of points P such that the tangents from P to C₂ meet a fixed line (but here, the line is variable, being any tangent to C₁).This might relate to the concept of a conic hull or reciprocal curve.Alternatively, the locus of P such that the polar of P with respect to C₂ is a tangent to C₁. Because the polar of P with respect to C₂ is the line AB, which is a tangent to C₁. Therefore, the locus of P is the set of points whose polar lines with respect to C₂ are tangents to C₁. This is known to be a conic called the reciprocal curve or the conic polar of C₁ with respect to C₂. The condition for this conic to be a rectangular hyperbola would impose a condition on the relative positions of C₁ and C₂.The reciprocal curve (conic) of C₁ with respect to C₂ is given by the equation:The polar line of P with respect to C₂ is tangent to C₁. For P = (X, Y), the polar line with respect to C₂ is (X - d)(x - d) + Y y = r₂². This line must be tangent to C₁: x² + y² = r₁². The condition for a line to be tangent to a circle is that the distance from the center to the line equals the radius.The distance from the center of C₁ (origin) to the polar line (X - d)(x - d) + Y y = r₂² is | (X - d)(0 - d) + Y * 0 - r₂² | / sqrt( (X - d)^2 + Y^2 ). This distance must equal r₁.Therefore:| -d(X - d) - r₂² | / sqrt( (X - d)^2 + Y^2 ) = r₁.Squaring both sides:[ d(X - d) + r₂² ]² / ( (X - d)^2 + Y^2 ) = r₁².Which rearranges to:[ d(X - d) + r₂² ]² = r₁² [ (X - d)^2 + Y² ].This is exactly the same equation we derived earlier. Therefore, the locus of P is indeed defined by this equation, and the condition for it to be a rectangular hyperbola is that the quadratic terms satisfy A + B = 0, leading to d² = 2 r₁².Therefore, regardless of r₂, the distance between centers must be d = sqrt(2) r₁. Thus, the answer is boxed{d = r_1 sqrt{2}}.But since the problem specifies "as a function of r₁ and r₂", and my conclusion is that it's independent of r₂, perhaps there's an error. However, according to the mathematical derivation, this is the case. Therefore, the answer is d = sqrt(2) r₁.Final AnswerThe distance between the centers is boxed{r_1 sqrt{2}}.</think>Given two circles ( C_1 ) and ( C_2 ) with radii ( r_1 ) and ( r_2 ) respectively, we need to find the distance ( d ) between their centers such that the locus of point ( P ), formed by the intersection of tangents from points ( A ) and ( B ) (where a variable tangent to ( C_1 ) intersects ( C_2 )), is contained in an equilateral hyperbola.1. Setup and Equations: - Circle ( C_1 ) is centered at the origin ((0, 0)) with radius ( r_1 ). - Circle ( C_2 ) is centered at ((d, 0)) with radius ( r_2 ). - A variable tangent to ( C_1 ) can be written as ( x cos theta + y sin theta = r_1 ).2. Intersection Points: - The tangent line intersects ( C_2 ) at points ( A ) and ( B ). Solving the equations of the tangent line and ( C_2 ) gives the coordinates of ( A ) and ( B ).3. Tangents from ( A ) and ( B ): - The tangents to ( C_2 ) at ( A ) and ( B ) intersect at point ( P ). Using the polar line properties, ( P ) is the pole of the line ( AB ) with respect to ( C_2 ).4. Locus Equation: - The coordinates of ( P ) are derived in terms of ( theta ) and simplified. The resulting equation of the locus of ( P ) is found to be: [ ( (X - d)^2 + Y^2 ) r_1^2 = ( d(X - d) + r_2^2 )^2 ]5. Condition for Equilateral Hyperbola: - For the locus to be an equilateral hyperbola, the quadratic terms must satisfy the condition that their coefficients sum to zero. This leads to the equation: [ ( r_1^2 - d^2 ) + r_1^2 = 0 ] - Solving this gives ( d^2 = 2 r_1^2 ), hence ( d = r_1 sqrt{2} ).6. Verification: - The derived condition is verified through examples and geometric reasoning, confirming that the distance ( d ) is indeed independent of ( r_2 ) and solely depends on ( r_1 ).Thus, the distance between the centers of ( C_1 ) and ( C_2 ) such that the locus of ( P ) is an equilateral hyperbola is (boxed{r_1 sqrt{2}}).]

🔑:1. Identify the given elements and their relationships: - Circle ( C_1 ) with radius ( r_1 ). - Circle ( C_2 ) with radius ( r_2 ). - A variable tangent ( t ) to ( C_1 ) intersects ( C_2 ) at points ( A ) and ( B ). - Tangents to ( C_2 ) at ( A ) and ( B ) intersect at point ( P ).2. Determine the locus of midpoints ( M ) of the chords ( AB ): - The locus of midpoints ( M ) of the chords ( AB ) is the pedal curve of ( C_1 ) with respect to ( O_2 ) (the center of ( C_2 )). - This pedal curve is a Pascal's limaçon ( mathcal{L} ) with cusp at ( O_2 ) and embracing ( C_1 ).3. Relate the distances involving ( M ) and ( P ): - Since ( overline{O_2M} cdot overline{O_2P} = r_2^2 ), it follows that the locus of points ( P ) is the inverse of ( mathcal{L} ) under the inversion with respect to ( C_2 ).4. Determine the nature of the conic ( mathcal{H} ): - The locus of ( P ) is a conic ( mathcal{H} ) with focus at ( O_2 ) and focal axis along ( O_1O_2 ) (where ( O_1 ) is the center of ( C_1 )). - For ( mathcal{H} ) to be an equilateral hyperbola, its eccentricity ( e ) must equal ( sqrt{2} ).5. Set up the polar coordinate system: - Take ( O_2 ) as the pole of the polar coordinate system ((r, theta)) and ( O_2O_1 ) as the polar axis ((theta = 0^circ)). - Thus, ( O_1 equiv (d, 0) ), where ( d ) is the distance between the centers of ( C_1 ) and ( C_2 ).6. Write the equations of ( mathcal{L} ) and ( mathcal{H} ): - The equation of the Pascal's limaçon ( mathcal{L} ) is: [ r = r_1 + d cos theta ] - The equation of the conic ( mathcal{H} ) (inverse of ( mathcal{L} ) under inversion with respect to ( C_2 )) is: [ r = frac{r_2^2}{r_1 + d cos theta} ]7. Determine the condition for ( mathcal{H} ) to be an equilateral hyperbola: - For ( mathcal{H} ) to be an equilateral hyperbola, its eccentricity ( e ) must be ( sqrt{2} ). - The eccentricity ( e ) of the hyperbola is given by: [ e = frac{d}{r_1} ] - Setting ( e = sqrt{2} ), we get: [ sqrt{2} = frac{d}{r_1} ] - Solving for ( d ), we find: [ d = r_1 sqrt{2} ]The final answer is (boxed{d = r_1 sqrt{2}})

🧠:[<think>Okay, so I need to figure out the sum of angles x and y in this circle divided into 12 equal arcs labeled from A to L. The answer choices are 75, 80, 90, 120, and 150 degrees. Hmm, let me start by visualizing the problem. Since the circumference is divided into 12 equal arcs, each arc must correspond to 360/12 = 30 degrees. So each arc is 30 degrees. That means the central angle for each arc is 30 degrees. Now, the question mentions angles x and y. But the problem is, I don't have the diagram here. Hmm, so I need to recall or figure out where angles x and y are located. Since the answer choices are all multiples of 15 (except 80, but maybe it's derived from something else), maybe the angles x and y are inscribed angles subtended by some of these arcs. In a circle, the measure of an inscribed angle is half the measure of its subtended arc. So if x and y are inscribed angles, their measures would be half of the arcs they intercept. Let's assume that the problem refers to angles formed by connecting points A through L. For example, maybe x is an angle formed by three points, say A, B, and C, and similarly for y. But without the exact diagram, I need to make some assumptions.Alternatively, maybe the problem is referring to angles at the center, but if that's the case, each angle would be 30 degrees, but adding them would give 60, which isn't among the choices. So likely, x and y are inscribed angles somewhere.Alternatively, perhaps x and y are angles formed by two chords intersecting at a point on the circumference or inside the circle. Wait, if two chords intersect inside the circle, the angle formed is equal to half the sum of the intercepted arcs. If they intersect outside, it's half the difference. But since the answer is a sum, maybe x and y are angles formed by intersecting chords, each angle being half the sum of their intercepted arcs.But the problem says "the sum of the angles x and y". So maybe x and y are two different angles, each formed by some chords, and we need to add their measures. Let me think of a possible configuration.Since the circle is divided into 12 equal parts, each of 30 degrees. Let me consider points labeled A to L around the circle. Maybe angles x and y are at the center? But then each angle would be 30 degrees, sum 60. Not an option. So probably inscribed angles.Suppose that angle x intercepts arc AB and angle y intercepts arc BC. Then each angle would be half of 30 degrees, so 15 degrees, sum 30. Not in the options. Hmm.Alternatively, maybe the angles intercept more arcs. For example, if angle x intercepts arc AC, that's two arcs, so 60 degrees, so angle x would be 30 degrees. Similarly, if y intercepts another arc of two intervals, angle y would be 30 degrees, sum 60. Still not matching the answer choices.Wait, answer choices include 75, 80, 90, 120, 150. Let me think of another approach. Maybe the angles are formed by three points each. For example, angle x is at point B, formed by points A, B, and C. Then that angle is an inscribed angle intercepting arc AC, which is 60 degrees, so angle x is 30 degrees. Similarly, angle y at another point. Hmm, but sum would still be 60.Alternatively, maybe angles x and y are formed by more distant points. For example, angle x intercepts an arc of three intervals (90 degrees), so angle x is 45 degrees. Similarly, angle y intercepts another arc of three intervals, angle y is 45, sum 90. That's answer choice C. 90. Hmm. Maybe that's the case.But how can I confirm this? Let me try to reconstruct a possible figure. If the circle is divided into 12 equal arcs with letters A to L, each arc 30 degrees. Let's imagine that angle x is formed by points A, B, and D. Then arc AD is two arcs from A to D, which is 2*30=60 degrees. So inscribed angle ABD would be half of 60, which is 30 degrees. Not helpful.Wait, maybe if points are further apart. For example, angle at point B, formed by points A, B, and E. The arc AE is 4 arcs away, which is 120 degrees. So angle ABE would be half of 120, which is 60 degrees. Similarly, another angle somewhere else. If the sum is x + y = 60 + 60 = 120, which is answer D. But answer C is 90. Hmm.Alternatively, maybe the angles are vertical angles formed by intersecting chords. For example, if two chords intersect inside the circle, then the measure of the angle is half the sum of the intercepted arcs. Suppose chord AC intersects chord BD at some point inside the circle. Then angle x is formed there, and its measure is half the sum of arcs AB and CD. If each arc is 30 degrees, maybe arc AB is 30 and arc CD is 30, so angle x would be (30+30)/2 = 30. If another angle y is formed similarly, maybe (arc BC + arc DE)/2 = (30 +30)/2=30, so sum 60. Still not matching.Wait, maybe the intercepted arcs are longer. For example, if angle x is formed by chords that intercept arcs of 3 intervals each. Let's say angle x intercepts arcs of 90 degrees each. Then the angle would be half the sum, so (90 + 90)/2 = 90. But that's the angle itself. But if two such angles, sum 180. Not in the options.Alternatively, maybe angle x is half the sum of two arcs. Let's say the intercepted arcs are 60 and 30. Then angle x is (60 +30)/2=45. Similarly, angle y is another 45. Sum 90. That's possible. But how?Alternatively, maybe angles x and y are angles in a triangle. For example, if you have a triangle inscribed in the circle with vertices at three of the points, then the angles of the triangle would be half the measure of their intercepted arcs. Suppose the triangle is such that each angle intercepts an arc of 60 degrees, so each angle is 30, sum 90. But that's the triangle's angles summing to 180. Hmm.Alternatively, maybe the problem is referring to angles formed by tangents and secants. For example, if there's a tangent at point A and a secant from A to some other point, forming an angle. But without the diagram, this is challenging.Alternatively, perhaps the problem is similar to a clock face. Since there are 12 divisions, like hours on a clock. Each hour mark is 30 degrees apart. Maybe angles x and y are the angles between the hands at certain times? But the problem mentions letters A to L, not numbers. Hmm.Alternatively, think of a regular 12-gon inscribed in the circle. Each central angle is 30 degrees. Then, maybe x and y are interior angles of some polygon formed by connecting the points. But in a regular 12-gon, each interior angle is (180*(12-2))/12 = 150 degrees. But we are summing two angles, so 300. Not matching.Alternatively, maybe it's about the angles between adjacent points as seen from another point. For example, from point A, the angle subtended by points B and L. The arc from B to L is let's say 11 arcs, 330 degrees, but the inscribed angle would be half that, 165, which is not an option.Wait, maybe the problem is referring to two specific angles. Let me think of possible configurations. If the circle is divided into 12 equal parts, each 30 degrees. If x and y are inscribed angles intercepting arcs of 3 and 3 arcs, then each angle would be 45 degrees, sum 90. If intercepting arcs of 4 and 2 arcs, angles would be 60 and 30, sum 90. Alternatively, arcs of 5 and 1, angles 75 and 15, sum 90. Wait, 75 is an answer choice. But 75 +15=90. Hmm.Alternatively, maybe each angle intercepts 5 arcs (150 degrees), so inscribed angles would be 75 each, sum 150. But 150 is an answer choice. Alternatively, one intercepts 2 arcs (60) and another intercepts 4 arcs (120), so angles 30 and 60, sum 90. Hmm.Alternatively, think of the points labeled A to L. If we connect points that are spaced apart. For example, if angle x is formed by points A, K, and B, and angle y is formed by points D, E, and F. Wait, but again, without the diagram, this is hard.Wait, perhaps the answer is 150. Because 150 is a large angle. If an inscribed angle intercepts 5 arcs (150 degrees), the angle is 75. If two such angles, 75 +75=150. But answer E is 150. Alternatively, maybe each angle is 75, sum 150. But answer E is 150.Alternatively, if the angles are formed by three points each. For example, if angle x is at point A, formed by points L, A, B. Then the arc LB is 2 arcs, 60 degrees, so inscribed angle is 30. If another angle y is at point D, formed by points C, D, E. Similarly 30, sum 60. Not matching.Alternatively, maybe the angles are external angles formed by two tangents. The angle between two tangents can be calculated by half the difference of the intercepted arcs. For example, if two tangents are drawn from a point outside the circle to points A and C, the angle between the tangents would be half the difference of the intercepted arcs. The arc from A to C is 60 degrees, the opposite arc is 300, so the angle is (300 -60)/2=120. If two such angles, sum 240. Not matching.Alternatively, maybe the problem is combining two angles from different configurations. For instance, one angle is an inscribed angle intercepting 120 degrees (60 degrees), and another angle is a central angle of 30 degrees, sum 90. But answer options are 75, 80, 90, 120, 150.Wait, if x is an inscribed angle intercepting 150 degrees (75 degrees) and y is an inscribed angle intercepting 90 degrees (45 degrees), sum is 120. But answer D is 120. Hmm. But how to know?Alternatively, maybe both angles intercept 60 degrees each, sum 30+30=60, not an answer. Alternatively, both intercept 90, sum 45+45=90. Answer C. Maybe.But the problem is, without seeing the diagram, it's challenging. However, since the answer is among the options, and the most common answer for such problems is 90 degrees, maybe that's the case. But let me think of another way.If the circle is divided into 12 equal parts, each 30 degrees. If points are labeled A to L sequentially, then the chord between A and, say, E is 4 arcs apart, which is 120 degrees. If there's a triangle formed by A, E, and another point, the angles in the triangle would be inscribed angles intercepting various arcs.Alternatively, perhaps x and y are vertical angles formed by intersecting chords. For example, if chord AC intersects chord BD at point O (the center). Then, the vertical angles at O would each be equal to the sum of arcs AB and CD. Wait, no. The angle at the center is equal to the arc. So if two chords intersect at the center, the angle is equal to the arc. But if they intersect elsewhere, the angle is half the sum of intercepted arcs.Wait, let's recall the formula. If two chords intersect at a point inside the circle, then the measure of the angle is half the sum of the measures of the intercepted arcs. So, suppose two chords intersect, creating angles x and y. Then, x + y would be equal to half the sum of all four intercepted arcs. But since the total circumference is 360, the sum of all arcs is 360. Therefore, x + y = 1/2 * (sum of arcs). Wait, but if two angles are formed by intersecting chords, they are vertical angles, so they are equal. Wait, no. If two chords intersect, they form two pairs of vertical angles. Each pair is equal. So if x and y are the two different vertical angles, their sum would be... Hmm.Wait, suppose two chords intersect, forming angles x and y. Then, x is half the sum of two arcs, and the other vertical angle is also x. Then, the adjacent angles would be y, which is half the sum of the other two arcs. But since the total sum of arcs around the circle is 360, the sum of all four arcs is 360. Therefore, x = 1/2(a + b), y = 1/2(c + d), where a + b + c + d = 360. Therefore, x + y = 1/2(a + b + c + d) = 1/2*360 = 180. But 180 isn't an answer choice. Hmm, so maybe x and y are not vertical angles, but rather two angles at different intersection points.Alternatively, maybe x and y are angles at different locations. For example, angle x is formed by one pair of intersecting chords, and angle y by another pair. Then, x and y could each be half the sum of their respective intercepted arcs. If each intercepted arcs sum to, say, 90 degrees, then x + y = 45 + 45 = 90. If the intercepted arcs sum to 150 each, then 75 +75=150. So depending on how the chords are arranged, the sum could vary.Alternatively, maybe x and y are inscribed angles intercepting arcs that are three each. So each arc is 3*30=90, so each angle is 45, sum 90. That would be answer C. Alternatively, intercepting arcs of 5 each, 150, angles 75 each, sum 150. But since 90 and 150 are both answer choices, need to figure out which is more plausible.Alternatively, if the problem is from a past AMC 8 or similar competition, maybe it's problem number something with a known answer. But I don't have that context.Alternatively, think of the letters A to L. If the circle is divided into 12 equal arcs, each 30 degrees, the letters go around the circle every 30 degrees. Maybe x is the angle at A between two chords, say from A to C and from A to D. The arc from C to D is 30 degrees, so the angle at A would be half the difference of the intercepted arcs. Wait, if it's formed by two chords from A, then it's an inscribed angle. Wait, if two chords go from A to C and A to D, then angle CAD is an inscribed angle intercepting arc CD, which is 30 degrees. So angle x is 15 degrees. Similarly, angle y somewhere else. Hmm, but 15 + something is not among the answer choices.Alternatively, maybe angle x is formed by points A, B, and D. The arc from A to D is 3 arcs (90 degrees), so inscribed angle at B would be half of 90, which is 45. Similarly, angle y at another point. If angle y is also 45, sum 90. That's possible.Alternatively, if angle x is formed by points A, C, and E, intercepting arc AE which is 4 arcs (120 degrees), so angle x would be 60. Then angle y intercepting arc BF (30 degrees), angle y is 15. Sum 75, which is answer A. But 75 is also an option.Alternatively, maybe angle x intercepts arc AE (120 degrees) and angle y intercepts arc BF (30 degrees). Then x = 60 and y =15, sum 75. Answer A. Hmm. But not sure.Wait, maybe this is a standard problem. Let me think. If a circle is divided into 12 equal parts, each 30 degrees. If you inscribe a square, the central angles would be 90 degrees, but 90/30=3, so sides would be 3 arcs apart. But inscribed angles for the square would be 90 degrees, which is half of 180. Hmm.Alternatively, maybe the problem is similar to a 12-hour clock where each hour mark is 30 degrees. If the problem involves the hour and minute hands, but no, the problem mentions letters A-L.Alternatively, think of the circle as a dodecagon. In a regular dodecagon, each central angle is 30 degrees. The internal angles are 150 degrees each. But again, sum of two internal angles would be 300, which is not an option.Wait, the problem states "the number of degrees in the sum of the angles x and y". So, x and y are two specific angles in the diagram. Since it's multiple choice, maybe the answer is 90. If I have to guess, but I need to think more carefully.Alternatively, maybe angles x and y are angles in a triangle formed by three non-adjacent points. For example, if you connect every other point, forming a hexagon. But no, in that case, internal angles would be 120 each.Wait, perhaps the angles x and y are the angles of a quadrilateral inscribed in the circle. Then, the sum of opposite angles would be 180. But the problem says sum of x and y. If they are opposite angles, their sum is 180. But 180 isn't an option. Wait, answer E is 150. Hmm.Alternatively, maybe the quadrilateral is not convex. Hmm.Alternatively, think of two overlapping triangles. But this is getting too vague.Wait, another approach: since each arc is 30 degrees, any inscribed angle will be a multiple of 15 degrees (since 30/2=15). The answer choices are 75, 80, 90, 120, 150. Except 80, all are multiples of 15. So 80 is probably not the answer. So eliminate B. Then, the answer is among 75,90,120,150.If x and y are inscribed angles each intercepting 3 arcs (90 degrees), then each angle is 45, sum 90. If each intercepting 5 arcs (150), each angle 75, sum 150. If one intercepts 4 arcs (120), angle 60, another intercepts 2 arcs (60), angle 30, sum 90. Alternatively, intercepting 6 arcs (180), angle 90, but a straight line.But how to know which configuration is correct. Maybe the most straightforward is that angles x and y each intercept 3 arcs (90 degrees), so each is 45, sum 90. But why 3 arcs?Alternatively, if the diagram has angles x and y each formed by three arcs apart, then yes. Alternatively, if it's a rectangle inscribed in the circle, with angles 90 each. But inscribed rectangle would have sides separated by 6 arcs (180 degrees), so the angles would be 90. But sum would be 180. Not helpful.Alternatively, if the problem refers to two angles formed by the intersection of two chords each. For example, if two chords intersect such that each angle intercepts arcs adding up to 180 degrees. Then, each angle is 90, sum 180. But answer choices don't have 180.Alternatively, maybe angles x and y are angles at the center, each 30 degrees, sum 60. Not an answer. Hmm.Wait, another thought: if the circle is divided into 12 equal arcs, creating 12 points. If you connect every other point, you get a hexagon. Each central angle is 60 degrees. The inscribed angles would be 30 degrees. But again, sum 60.Alternatively, connect every three points: you get a square. Wait, 12 divided by 3 is 4. So connecting every three points would give a square, each central angle 90 degrees. Then inscribed angles would be 45 degrees. So if angles x and y are two such angles, sum 90. That's possible.Alternatively, if the problem has to do with an equilateral triangle inscribed in the circle. Each central angle would be 120 degrees, inscribed angles 60 degrees. Sum 120. Answer D. But need three points spaced 4 arcs apart. Since 12 divided by 3 is 4. So each vertex is 4 arcs apart, central angle 120. Inscribed angle 60. If two such angles, sum 120. Answer D. Hmm, but 120 is an option.Wait, but an equilateral triangle inscribed would have each angle 60, sum 180. But the problem is only summing two angles.Alternatively, if x and y are two angles of the equilateral triangle, each 60, sum 120. Answer D. But does that make sense?Alternatively, if the angles are external to the circle? For example, angles formed by tangents. If two tangents are drawn from a point outside the circle to two points separated by arc length. The angle between the tangents would be half the difference of the intercepted arcs. For example, if the arc between the points is 60 degrees, the angle outside is (360 - 60)/2 = 150, which is answer E. If two such angles, sum 300. Not an option.Alternatively, one tangent angle and one inscribed angle. For example, angle x is a tangent-secant angle, and angle y is an inscribed angle. But without the diagram, hard to tell.Wait, the answer choices include 75, which is half of 150. Maybe angle x intercepts 5 arcs (150), so 75, and angle y intercepts another 5 arcs, 75, sum 150. Answer E. But also, maybe angle x is 75 and angle y is something else. Hmm.Alternatively, maybe the problem is referring to the angles formed by the hands of a clock at certain times, as letters correspond to numbers. For example, A is 12, B is 1, ..., L is 11. Then, if x and y are angles between the hour and minute hands at two different times, but the problem states "sum of the angles x and y". However, without specific letters, this is speculative.Alternatively, maybe letters A-L are placed like a clock, but the problem is not about time. For example, if points A and G are opposite each other (180 degrees apart). Then, the angle between A, O, G is 180. But it's a straight line.Alternatively, if angles x and y are angles in two different triangles. For example, each triangle has angles that sum to 180, so sum of x and y would depend on their context.Wait, this is getting too convoluted. Let me think of the most probable answer. Since the circle is divided into 12 parts, 30 degrees each. If angles x and y are inscribed angles intercepting 3 arcs each (90 degrees), then each is 45, sum 90. Answer C. Alternatively, intercepting 5 arcs each (150), each angle 75, sum 150. Answer E. These are both possible. If the problem is from AMC 8, answer is likely 90 or 150. Alternatively, think of the circle divided into 12 parts. If you inscribe a square, each side skips 3 arcs (90 degrees). The central angle is 90, inscribed angle is 45. If the problem involves a square, then angles x and y could be 45 each, sum 90. Answer C. Alternatively, if inscribing an equilateral triangle, central angles 120, inscribed angles 60, sum 120. Answer D. But how do I know?Alternatively, if the problem is similar to the 2013 AMC 10A problem 12, but not sure.Wait, let's try to check the answer choices. If the answer is 75, that would be half of 150. If one angle is intercepting 5 arcs (150), angle is 75. If another angle is intercepting 1 arc (30), angle is 15. Sum 90. No. Alternatively, both intercepting 150. Sum 150. So answer E. Alternatively, angles formed by different arcs. Maybe one angle intercepts 4 arcs (120) so 60, another intercepts 2 arcs (60) so 30, sum 90. So 90 is possible. Hmm.Alternatively, if angle x is an inscribed angle intercepting 2 arcs (60 degrees) so angle x=30, and angle y intercepts 4 arcs (120 degrees), angle y=60, sum 90. Answer C. This is plausible. Maybe the problem has such a configuration.Alternatively, if angle x is formed by chords connecting points A, B, and D (arcs of 2 intervals, 60 degrees), angle x=30, and angle y is formed by points D, E, G (arcs of 2 intervals, 60 degrees), angle y=30, sum 60. Not an option. Alternatively, intercepting more arcs.Alternatively, if angle x is formed by points A, C, E (intercepts arc AE, 4 arcs, 120 degrees), angle x=60. Angle y is formed by points B, D, F (arc BF=4 arcs, 120 degrees), angle y=60. Sum 120. Answer D. That could be.Alternatively, if angle x is formed by points A, D, G (intercepts arc AG=6 arcs, 180 degrees), angle x=90. Similarly, angle y=90, sum 180. Not an option.Wait, but answer choices do not include 180, so that's out.Alternatively, angle x intercepts arc AE (120 degrees) and angle y intercepts arc BF (30 degrees). So x=60, y=15, sum=75. Answer A.Alternatively, angle x intercepts arc AJ (6 arcs, 180), angle is 90. Angle y intercepts arc something else. If intercepts 60, angle 30, sum 120. Hmm.This is really challenging without the diagram. Maybe the answer is 90 degrees, which is a common right angle, but I need to verify.Alternatively, think about the possible answer choices. 75, 80, 90, 120, 150. 90 is the only right angle, which is a common angle in circles, like inscribed angles intercepting semicircles. If both angles x and y are right angles, sum 180, but that's not an option. Wait, unless each is intercepting a semicircle, but that would make them 90 each, sum 180. Not an option. So not that.Alternatively, if one angle intercepts a semicircle (180) and another intercepts a 60-degree arc. Then, angles are 90 and 30, sum 120. Answer D. Hmm.Alternatively, think of the answer choices: 75,80,90,120,150. 80 is the only non-multiple of 15 or 30. Since each arc is 30 degrees, and inscribed angles would be multiples of 15 (half of 30). So 80 is likely a trap. So maybe eliminate 80. Then, 75,90,120,150.If it's an AMC problem, the answer is often C or D. But I can't rely on that.Alternatively, think of the answer 150. If each angle is 75, sum 150. 75 is half of 150. So if each angle intercepts 150 degrees, which is 5 arcs. If two such angles, sum 150. Answer E.Alternatively, answer 90. If each angle intercepts 90 degrees, which is 3 arcs. 3*30=90. Inscribed angle is 45. Two angles sum to 90.Alternatively, maybe it's a combination. For example, angle x intercepts arc of 120 degrees (60), angle y intercepts 60 (30), sum 90.But without the diagram, it's really difficult. However, given that the answer choices include 90 and 150, which are both common angles, and 75 and 120 as well. Wait, let me think of another approach. If the circle is divided into 12 equal arcs, each 30 degrees. Then, the points are every 30 degrees. If we connect points to form angles x and y, maybe x and y are angles in a star formed by connecting every fifth point or something. But that might create angles of different measures.Alternatively, think of the angles formed by the hands of a clock at 3:00 and 9:00, which are 90 degrees each. Sum 180. Not an option. Alternatively, at 2:00 and 10:00, which are 60 and 150, sum 210. Not helpful.Alternatively, think of the letters A to L as corresponding to numbers 1 to 12. Then, if angle x is between A(1) and C(3), that's 2 hours, 60 degrees. Similarly, angle y between D(4) and F(6), another 60 degrees. Sum 120. Answer D. But in this case, angles at the center. But the problem says "the angles x and y", which might not be central angles.Alternatively, if they are inscribed angles, then 60 degrees at the center becomes 30 degrees inscribed. So two such angles sum 60. Not an answer.Wait, if angle x is the angle between the tangents from point A to the circle. Wait, but all points are on the circle. So tangents at A would just touch at A. Not helpful.Alternatively, perhaps x and y are angles in a triangle inscribed in the circle. If the triangle is equilateral, each angle is 60, sum 120. Answer D. If it's a right triangle, angles 90, sum with another angle. But a right triangle inscribed has a right angle (90), sum with another angle would be more than 90. But answer options are single sums. Wait, the problem says "the sum of the angles x and y". So two angles, not three. So maybe two angles from different triangles.Alternatively, think of the problem in terms of the number of degrees between the letters. For example, if moving from A to B is 30 degrees. If x is the angle between A and C, which is 60 degrees, and y is between D and F, which is 60 degrees, sum 120. But if these are central angles, sum 120. If they are inscribed angles, sum 60. But 120 and 60 are possible.Wait, the problem says "the number of degrees in the sum of the angles x and y". So depending on whether they are central or inscribed angles. If they are central angles, each 30 degrees, sum 60. But 60 isn't an option. If they are inscribed angles intercepting two arcs each (60 degrees), angles 30, sum 60. Not an option. If intercepting three arcs (90), angles 45, sum 90. Answer C.Alternatively, if the problem is referring to the angles formed by the intersection of two chords, where each angle is calculated as half the sum of the intercepted arcs. For example, if two chords intersect, and the intercepted arcs are 90 and 90, then each angle is 90. But that would make the angles 90 each, sum 180. Not an option.Alternatively, if two pairs of arcs are 60 and 120. Then, angles would be (60 +120)/2=90 each. So sum 180. Still not helpful.Alternatively, if two different intersections: one angle x is formed by arcs summing to 150, angle x=75, and another angle y is formed by arcs summing to 150, angle y=75. Sum 150. Answer E.Alternatively, if angle x is formed by arcs summing to 90 (angle 45) and angle y is formed by arcs summing to 150 (angle 75), total sum 120. Answer D.Hmm. There are multiple ways to get to the answer choices, but without the diagram, it's impossible to be certain. However, given the answer choices, and the fact that 90 and 150 are both feasible, I need to think of the most probable.Wait, let me recall that in a regular dodecagon (12-gon), the measure of each internal angle is 150 degrees. But that's the internal angle. The central angle is 30. But inscribed angles would vary.Alternatively, if the problem has two angles formed by three points each, such that each angle is an inscribed angle intercepting 3 arcs (90 degrees), then each is 45, sum 90. This is a common problem setup. So I think the answer is 90 degrees, option C.But wait, another way: if you inscribe a regular hexagon in the circle, each side subtends 60 degrees. The inscribed angles would be 30 degrees. But sum of two would be 60. Not an option.Alternatively, a square inscribed in the circle. Each central angle 90, inscribed angle 45. Sum of two angles 90. Answer C. This seems plausible.Alternatively, if two right angles (90 each), but sum 180. Not an option.Alternatively, think of the answer 150. If each angle intercepts 150 degrees, which is 5 arcs. Each angle is 75. Sum 150. Answer E.Given that both 90 and 150 are possible, but since the problem says "sum of the angles x and y", maybe the answer is 150. If each angle is 75, then sum is 150. But why would each angle intercept 150 degrees?Alternatively, if the angles are exterior angles formed by two tangents. The formula for the angle between two tangents from an external point is half the difference of the intercepted arcs. For example, if two tangents are drawn from a point outside the circle to points A and B, the angle between them is (arc AB - arc BA)/2. But since arc AB is 30 degrees, arc BA is 330 degrees. So angle is (330 -30)/2=150. If two such angles, sum 300. Not an option.Alternatively, if one tangent angle is 150 (intercepting 30 and 330), and another angle is formed by two secants with intercepted arcs. For example, angle formed by two secants intercepting arcs of 60 and 120. The angle would be (120 -60)/2=30. Sum 150 +30=180. Not an option.This is really tough. Considering that the answer choices are 75,80,90,120,150. Given that the circle is divided into 12 parts, 30 each. The most probable answer is 90 degrees. Because 90 is a right angle, often comes up in geometry problems. For example, if angles x and y are each inscribed angles intercepting 180 degrees (though that's a semicircle), but 180 is not an arc here. Wait, but intercepting three arcs (90 degrees) each. Then each angle is 45, sum 90. Maybe the answer is 90. But I also see that 150 is possible. However, given the options, I think 90 is more likely. Alternatively, the sum is 90.Wait, let me make a decision. Given that without the diagram, but the arcs are 30 each. If x and y are angles each intercepting 3 arcs (90 degrees), sum is 45+45=90. Answer C. Alternatively, intercepting 5 arcs (150 degrees), sum 75+75=150. Answer E. Since both are possible, but in typical problems, intercepting 3 arcs is more common for inscribed angles. So I'll go with 90, answer C.But wait, the fact that 150 is an option makes me hesitate. If the intercepted arcs are five each, sum 150. But why would the problem choose five arcs? Maybe if the chords are connecting points that are five apart. For example, from A to F is five arcs, which is 150 degrees. Then the inscribed angle would be 75. If two such angles, sum 150. Answer E.Alternatively, if the problem is about a five-pointed star or something. But in a 12-division circle, a five-pointed star would not align.Alternatively, the fact that 12 arcs, 30 each. If you go five arcs from a point, it's equivalent to going back seven arcs the other way (since 5 +7=12). So the smaller arc is five arcs (150 degrees), the larger arc is seven arcs (210 degrees). But inscribed angle would consider the smaller arc. So angle is 75. Sum two angles, 150. Answer E.Alternatively, if the problem has two angles, each intercepting a 150-degree arc, sum 150. Given that 150 is a possible answer.Given the ambiguity, but considering that the answer choices include both 90 and 150, which are both feasible, but 150 is a larger angle, perhaps intercepting more arcs. But given that the problem is from an exam like AMC 8, where the answer is likely to be a more common angle. 90 is a right angle, more commonly encountered. Hence, I think the answer is 90 degrees, option C.But I'm not 100% sure. Another way: if each angle is formed by connecting three points spaced 3 arcs apart, forming a square. Each central angle 90, inscribed angle 45. Sum 90. That makes sense. Hence, I'll go with C, 90.Final Answerboxed{C}</think>The circumference of the circle with center ( O ) is divided into 12 equal arcs, each corresponding to 30 degrees. The problem requires finding the sum of the angles ( x ) and ( y ).1. Understanding the Circle Division: - The circle is divided into 12 equal arcs, each measuring ( frac{360}{12} = 30 ) degrees.2. Considering Inscribed Angles: - An inscribed angle is half the measure of its subtended arc. - If angles ( x ) and ( y ) are inscribed angles intercepting arcs of 90 degrees each (3 arcs of 30 degrees each), each angle would be ( frac{90}{2} = 45 ) degrees. - Summing these angles: ( 45^circ + 45^circ = 90^circ ).3. Verification with Answer Choices: - The answer choices include 75, 80, 90, 120, and 150 degrees. - Given the problem's context and common geometric configurations, the most plausible answer is 90 degrees, as it corresponds to a right angle formed by inscribed angles intercepting arcs of 90 degrees each.Thus, the number of degrees in the sum of the angles ( x ) and ( y ) is (boxed{C}).,<think>Okay, so I have this problem here where a circle with center O is divided into 12 equal arcs, labeled from A to L. The question is asking for the number of degrees in the sum of angles x and y. The answer choices are 75, 80, 90, 120, and 150. Hmm, let's try to visualize this. Since it's divided into 12 equal arcs, each arc must correspond to 360/12 = 30 degrees. So each arc is 30 degrees. Wait, but the problem mentions angles x and y. Since it's a circle divided into 12 equal parts, maybe x and y are angles formed by some chords or something? The problem probably comes with a diagram, but since I don't have that, I need to figure out based on standard positions. Let me recall that in such problems, often the angles x and y could be inscribed angles or central angles subtended by some of these arcs. If they are inscribed angles, their measure would be half the measure of the central angle subtending the same arc. If they are central angles, then their measure would be equal to the arc's measure.But since the answer choices are all multiples of 5 and 15, and the arcs are 30 degrees each, maybe the angles x and y each are formed by connecting points A to L or something. Without the diagram, this is a bit challenging, but perhaps there's a standard way these points are labeled. Let me assume that the points are labeled clockwise from A to L, each 30 degrees apart. So starting from point A at 0 degrees, then every 30 degrees we have B, C, ..., up to L at 330 degrees.But how are angles x and y defined? Since the problem is about the sum of angles x and y, maybe they are angles at the center or at the circumference. If they are at the center, then they would correspond to some arcs. If they are inscribed angles, then their measure would be half of the arcs they subtend.Wait, the problem just mentions "the angles x and y" but doesn't specify where they are. Since it's divided into 12 equal arcs, maybe it's like a clock face, with each hour mark being a letter. So A is at 12 o'clock, B at 1 o'clock (30 degrees), C at 2 o'clock (60 degrees), etc., up to L at 11 o'clock (330 degrees). Then, maybe x and y are angles formed by connecting certain points.Alternatively, maybe x and y are vertical angles or something else. Wait, maybe it's a triangle or quadrilateral inscribed in the circle, and x and y are two angles of that shape. The sum of those angles would then depend on the arcs they subtend.Alternatively, maybe x and y are angles at the center, each corresponding to a certain number of arcs. If they are central angles, then each angle would be a multiple of 30 degrees, so their sum would also be a multiple of 30. Looking at the answer choices, 75, 80, 90, 120, 150. Among these, 90, 120, 150 are multiples of 30, while 75 and 80 are not. Hmm, so maybe x and y are not central angles but inscribed angles. If that's the case, their measures would be half the measure of the central angles.For example, if angle x subtends an arc of 60 degrees, then angle x would be 30 degrees. Similarly, angle y might subtend another arc, and their sum would be the sum of half those arcs. So if x and y are inscribed angles subtending arcs of, say, 60 and 120 degrees, their sum would be 30 + 60 = 90 degrees, which is option C. Alternatively, if they subtend arcs of 90 and 150 degrees, their sum would be 45 + 75 = 120. But 120 is an option. Hmm.Alternatively, maybe the arcs that x and y subtend add up to a certain number. If the total circumference is 360, divided into 12 parts, so each 30 degrees. Maybe the arcs subtended by x and y are each multiple of 30 degrees. Suppose angle x subtends 3 arcs (90 degrees), so inscribed angle x would be 45 degrees. Similarly, angle y subtends 5 arcs (150 degrees), inscribed angle y would be 75 degrees. Then their sum is 45 + 75 = 120, which is option D. Alternatively, maybe both subtend 2 arcs each (60 degrees), so each angle is 30 degrees, sum 60, which isn't an option. Hmm.Alternatively, maybe x and y are angles at the center. If each is a central angle over, say, 2 arcs (60 degrees) and 3 arcs (90 degrees), then their sum is 60 + 90 = 150, which is an option. But 150 is E. But the answer choices are given, so maybe.But since the answer choices include 90, 120, 150, which are multiples of 30, maybe it's a combination. Wait, but the problem says "the number of degrees in the sum of the angles x and y". So depending on where these angles are located, the answer could vary.Wait, maybe the problem is similar to one where x and y are angles formed by intersecting chords. For example, when two chords intersect inside a circle, the measure of the angle formed is equal to half the sum of the measures of the intercepted arcs. Similarly, if two chords intersect outside the circle, the angle is half the difference of the intercepted arcs.But without the diagram, it's hard to tell. However, given the answer choices, and the fact that each arc is 30 degrees, maybe the intercepted arcs are such that the sum or difference leads to x and y.Alternatively, maybe x and y are angles in different triangles, but given the problem is about a circle divided into 12 equal parts, it's likely related to inscribed angles or central angles.Wait, let me think. If the circle is divided into 12 equal arcs, each 30 degrees. If we have points labeled A to L, then the arc between each consecutive letter is 30 degrees. Suppose that angles x and y are inscribed angles subtended by arcs of certain lengths.For example, if angle x is subtended by an arc of 2 intervals (60 degrees), then x is 30 degrees. If angle y is subtended by an arc of 4 intervals (120 degrees), then y is 60 degrees. Then x + y would be 90 degrees. Alternatively, if x is subtended by 3 intervals (90 degrees), then x is 45 degrees, and y is subtended by 5 intervals (150 degrees), so 75 degrees, sum is 120. Alternatively, if both are subtended by 3 intervals, sum is 90. Hmm.Alternatively, if the angles x and y are vertical angles formed by the intersection of two chords. Then each angle is half the sum of the intercepted arcs. For example, if two chords intersect, creating angles x and y, which are vertical angles, so they are equal. Wait, but the problem says sum of angles x and y. If they are vertical angles, they are equal, so sum would be 2x. But the answer choices are 75, 80, 90, etc., so perhaps not.Alternatively, if x and y are angles in different parts of the diagram. For example, maybe x is an inscribed angle subtended by arc AB (30 degrees), so x is 15 degrees, and y is subtended by arc CD (30 degrees), so 15 degrees. Then sum is 30, which is not an answer. So that's not.Alternatively, maybe x is an angle formed by three points, say A, B, C, forming triangle ABC, with angle at B. Then angle at B would be an inscribed angle subtended by arc AC. Since arc AC is 2 intervals (60 degrees), so angle at B is 30 degrees. Similarly, angle y could be at another point, say E, subtended by arc FG (2 intervals, 60 degrees), so 30 degrees. Then sum is 60, which is not an option.Alternatively, perhaps x and y are angles in a quadrilateral inscribed in the circle. Then opposite angles in a cyclic quadrilateral sum to 180 degrees. But if the problem is asking for the sum of two angles, perhaps they are opposite angles, but then the answer would be 180, which isn't an option. So maybe not.Alternatively, maybe it's two angles that are not in the same figure. For example, angle x is an inscribed angle subtended by arc A to C (60 degrees), so 30 degrees, and angle y is subtended by arc D to G (90 degrees), so 45 degrees. Then the sum is 75, which is option A. But 75 is an answer choice. Alternatively, if angle x is subtended by arc A to D (90 degrees), so 45 degrees, and angle y is subtended by arc E to H (90 degrees), sum is 90. Then the total sum is 90, which is option C.But without knowing the exact positions, this is speculative. Let's try to approach systematically.Given the circle is divided into 12 equal arcs, each 30 degrees. The letters are A through L, so 12 letters. Let me assign numbers to each point for easier reference. Let’s say point A is at 0 degrees, then each subsequent letter is 30 degrees more. So:A: 0 degreesB: 30 degreesC: 60 degreesD: 90 degreesE: 120 degreesF: 150 degreesG: 180 degreesH: 210 degreesI: 240 degreesJ: 270 degreesK: 300 degreesL: 330 degreesSo, points are every 30 degrees around the circle.Now, angles x and y could be angles at the center or inscribed angles. Let's consider possibilities.Case 1: Both x and y are central angles. Then each is a multiple of 30 degrees. Their sum would be a multiple of 30. Looking at answer choices, possible sums are 90, 120, 150. So maybe x is 60 and y is 60, sum 120. Or x=30, y=60, sum=90. Or x=90, y=60, sum=150.But which one? If the problem is from a diagram, maybe there's a standard configuration. For example, in a 12-part circle, common angles are 30, 60, 90, etc. If x and y are angles at the center, say between points A-B-C, so angle AOB is 30 degrees, angle BOC is 30 degrees. But unless they are combining angles.Alternatively, maybe x and y are angles formed by three points each, such as angle ABC, which is an inscribed angle. For example, if ABC is a triangle with points A, B, C. The angle at B would be an inscribed angle subtended by arc AC. Since arc AC is 60 degrees (from A to C is two arcs), then angle ABC is 30 degrees. Similarly, if there's another angle, say, at point D, subtended by arc EF (60 degrees), then angle DEF is 30 degrees, sum is 60. Not an answer choice. So maybe not.Alternatively, if angle x is subtended by a larger arc. For example, angle at point B subtended by arc A to D. Arc A to D is 90 degrees (three arcs), so inscribed angle is 45 degrees. Then angle y is subtended by another arc, say E to H (90 degrees), so 45 degrees. Sum is 90. So that would be answer C.Alternatively, maybe angle x is subtended by arc A to G (180 degrees), so inscribed angle is 90 degrees, and angle y is something else. But 90 is already an answer choice.Alternatively, maybe angle x is subtended by arc A to E, which is 120 degrees (four arcs), so inscribed angle is 60 degrees. Then angle y is subtended by arc G to K, 120 degrees, inscribed angle 60 degrees. Sum is 120.Alternatively, perhaps angles x and y are formed by intersecting chords. For example, if two chords intersect inside the circle, then the measure of the angle is half the sum of the intercepted arcs. Suppose chord AB intersects chord CD at some point inside the circle. Then angle x would be 1/2*(arc AC + arc BD). If arcs AC and BD are each, say, 60 degrees, then angle x is 1/2*(60+60)=60 degrees. If another pair of chords intersect, forming angle y similarly. Then depending on the arcs, the sum could be different.But again, without the diagram, it's tricky. Maybe looking at answer choices, 90 is a likely candidate because it's a right angle, which often comes up in circle theorems. If angles x and y are each 45 degrees, which is half of 90, but 45 isn't a multiple of 30. Wait, 45 is half of 90, which is three arcs (90 degrees). So if an inscribed angle is subtended by three arcs (90 degrees), it would be 45 degrees.Alternatively, maybe x and y are angles in a triangle inscribed in the circle. For example, if you have a triangle with vertices at A, E, and I (each 120 degrees apart), then each angle of the triangle would be half the arc opposite. So angle at A would be half the arc EI, which is 240 degrees, so angle is 120 degrees. Similarly, angles at E and I would each be 120 degrees, sum is 360, which is not possible. Wait, triangles have 180 degrees. Hmm, maybe not.Alternatively, maybe it's a square inscribed in the circle. If the circle is divided into 12 parts, but a square would take every 3 arcs (90 degrees). So points A, D, G, J. Then each angle of the square is 90 degrees. But sum of two angles would be 180. Not an option.Wait, perhaps the problem is similar to the following: If points are labeled every 30 degrees, and angles x and y are inscribed angles subtended by arcs of 3 and 5 units, each. Then 3*30=90, so inscribed angle is 45; 5*30=150, inscribed angle 75. Sum is 120. That's answer D. Alternatively, 2 and 4 arcs: 60 and 120, inscribed angles 30 and 60. Sum 90.Alternatively, maybe the arcs are 1 and 5: 30 and 150. Inscribed angles 15 and 75. Sum 90. Wait, 15+75=90. Hmm. That's possible. But 15 isn't a multiple of 30. Wait, but inscribed angles can be 15 if the arc is 30.Wait, inscribed angle is half the measure of the arc. So if an inscribed angle subtends a 30-degree arc, the angle is 15 degrees. If another inscribed angle subtends a 150-degree arc, it's 75 degrees. 15+75=90. So that's possible. So maybe x is 15 and y is 75. But how would that happen? If x is subtended by a single arc (30 degrees) and y is subtended by five arcs (150 degrees). If the diagram has, for example, angle x between two adjacent points (arc of 30) and angle y between five points apart (arc of 150). Then their sum would be 15+75=90. That's answer C. Alternatively, if both angles are subtended by three arcs (90 degrees), each angle is 45, sum 90. So that's another way.Alternatively, if one angle is subtended by two arcs (60 degrees) and another by four arcs (120 degrees), each would be 30 and 60, sum 90. So maybe that's it. So 90 is the answer. But why is 120 also an option? Hmm. Maybe if the angles are subtended by different arcs. For example, if one angle is subtended by three arcs (90) and another by five arcs (150). Then 45+75=120. So sum is 120. So depending on the arcs they subtend.But how to figure out which one is correct? Since I don't have the diagram, maybe looking for standard configurations. In a 12-part circle, common inscribed angles might correspond to 3 arcs (90 degrees), giving 45, and 5 arcs (150 degrees), giving 75. So 45+75=120. Alternatively, 2 arcs (60) and 4 arcs (120), 30+60=90.Alternatively, maybe the problem is combining angles from different positions. Wait, another approach: if the circle is divided into 12 equal parts, it's akin to a clock face. So perhaps the letters correspond to the numbers on a clock, with A at 12, B at 1, ..., L at 11. Then, angles x and y could be the angles between the hands of the clock or something similar. For example, the angle between 12 and 3 is 90 degrees. But the problem is about the sum of two angles. If, say, x is the angle between A and D (3 hours apart, 90 degrees), and y is the angle between D and G (another 90 degrees), then sum is 180. Not an option. Alternatively, maybe smaller angles.But this is too speculative. Wait, maybe the problem is referring to the angles formed by the intersecting chords at the center. If x and y are angles at the center, then each is a multiple of 30. For example, if x spans 2 arcs (60) and y spans 3 arcs (90), sum is 150. Answer E. But answer E is 150, which is possible. Alternatively, if x and y are both 60, sum is 120.Alternatively, maybe the angles are external angles formed by tangents. Wait, but the problem doesn't mention tangents. Hmm.Alternatively, maybe x and y are angles of a triangle inscribed in the circle. For instance, if you have a triangle with vertices at points A, D, and G (each 90 degrees apart). Then each angle of the triangle would be half the measure of the arc opposite. The arc opposite angle A is DG, which is 180 degrees, so angle A is 90 degrees. Similarly, angles D and G would also be 90 degrees, forming a rectangle. But a triangle can't have three right angles. Wait, that's impossible. So maybe not.Alternatively, a triangle with vertices at A, B, and D. Arc AB is 30 degrees, arc BD is 60 degrees. The angle at B would be half the arc AD, which is 90 degrees, so angle B is 45 degrees. The angle at A is half the arc BD, 60 degrees, so 30 degrees. The angle at D is half the arc AB, 30 degrees, so 15 degrees. Sum of angles is 45+30+15=90. So the sum of all three angles is 90, but the problem is asking for sum of x and y. If x and y are two angles of the triangle, say 45 and 30, sum is 75 (option A). 45 and 15, sum is 60. Not an option.Alternatively, maybe the problem is referring to two angles in different triangles. But this is getting too convoluted.Wait, another approach: the problem is from an old contest, perhaps AMC 10, where the diagram is a dodecagon (12-gon) inscribed in a circle with points labeled A to L. In such problems, often angles x and y are formed by connecting non-adjacent points. For example, in a regular dodecagon, each central angle is 30 degrees. The inscribed angles would be half the central angles for the same arcs.If angles x and y are two inscribed angles each subtended by 3 arcs (90 degrees), so each angle is 45 degrees, sum 90. If they are subtended by 2 arcs (60 degrees), each is 30, sum 60. If subtended by 4 arcs (120 degrees), each is 60, sum 120.But given the answer choices, 90, 120, 150 are possible. Maybe the intercepted arcs are 3 and 5, leading to 45 and 75, sum 120. Alternatively, 4 and 4 arcs, sum 120. Wait, if each angle is subtended by 4 arcs (120 degrees), then each angle is 60, sum 120. Alternatively, 1 and 5 arcs: 15 and 75, sum 90. Hmm.Alternatively, maybe the angles are vertical angles formed by intersecting chords. For example, if two chords intersect, and each angle is half the sum of the intercepted arcs. Suppose one pair of intercepted arcs is 2 and 4 arcs (60 and 120 degrees), so the angle is (60 + 120)/2 = 90 degrees. Then if another pair of intercepted arcs is similar, but maybe different. Wait, but vertical angles are equal. So if two chords intersect, the vertical angles are equal. So if angle x and angle y are vertical angles, they are equal, so their sum would be 2x. If x is 45, sum is 90; if x is 60, sum is 120. So depending on the intercepted arcs.Suppose two chords intersect, intercepting arcs of 60 and 120 degrees. Then the angle is (60 + 120)/2 = 90. So each angle is 90, sum is 180. Not an option. Wait, but maybe the intercepted arcs are different. For example, intercepted arcs of 30 and 90. Then angle is (30 + 90)/2 = 60. So two angles each 60, sum 120. That's possible.Alternatively, intercepted arcs of 30 and 150. Then angle is (30 + 150)/2 = 90. Sum is 180. Hmm.Wait, but the problem states "the sum of the angles x and y". If x and y are two different angles, not vertical angles. For example, two separate angles formed by different intersections. Then each angle would be calculated as half the sum of their respective intercepted arcs.Suppose angle x is formed by two chords intersecting, intercepting arcs of 30 and 90, so x = (30 + 90)/2 = 60. Angle y is formed by another intersection, intercepting arcs of 60 and 120, so y = (60 + 120)/2 = 90. Then sum x + y = 60 + 90 = 150. Option E. But is this a likely configuration?Alternatively, if angle x intercepts arcs of 60 and 60, so x = (60 + 60)/2 = 60. Angle y intercepts arcs of 90 and 90, so y = 90. Sum is 150. Hmm. Alternatively, intercepting arcs of 30 and 150 for x: (30 + 150)/2 = 90. Then y intercepts arcs of 30 and 150 as well, sum x + y = 180. Not an option.Alternatively, maybe angles x and y are external angles formed by two secants. In that case, the angle is half the difference of the intercepted arcs. For example, if two secants intersect outside the circle, angle = 1/2(difference of arcs). If angle x is formed by two secants intercepting arcs of 90 and 30, then x = 1/2(90 - 30) = 30. Angle y similarly intercepts arcs of 150 and 30, so y = 1/2(150 - 30) = 60. Then sum x + y = 90. Option C.Alternatively, if angle x intercepts arcs of 120 and 60: 1/2(120 - 60) = 30. Angle y intercepts arcs of 150 and 90: 1/2(150 - 90) = 30. Sum 60. Not an option.Alternatively, angle x intercepts arcs of 150 and 30: 1/2(150 - 30) = 60. Angle y intercepts arcs of 120 and 60: 1/2(120 - 60) = 30. Sum 90. So that's possible.But again, without the diagram, this is very speculative.Wait, perhaps considering the answer choices, 90 is a common answer in such problems. But why?Alternatively, if the problem is referring to the angles formed by the hands of a clock. Since it's 12 divisions, like a clock. If points A-L correspond to hours, then the angle between each hour is 30 degrees. If x and y are angles between certain times. For example, between 12 and 3 is 90 degrees, between 3 and 5 is 60 degrees. Sum is 150. But that's just a guess.Alternatively, maybe the problem is like a clock where x is the angle between A and G (180 degrees apart), which is 180 degrees, and y is some other angle. But 180 is not an option.Alternatively, perhaps the problem is about a star formed by connecting every fifth point, creating angles. In a 12-point circle, connecting every fifth point would create a star with certain angles. Each angle in a 12-pointed star is 150 degrees. But sum of two would be 300, not an option.Alternatively, maybe the angles are in a three-dimensional figure, but unlikely.Alternatively, maybe x and y are angles at the center, each spanning different numbers of arcs. For example, if O is the center, angle x spans from A to D (3 arcs, 90 degrees), angle y spans from D to H (4 arcs, 120 degrees), sum is 210. Not an option.Alternatively, both angles span 2 arcs each, sum 60 + 60 = 120. Option D.Alternatively, x spans 1 arc (30), y spans 3 arcs (90), sum 120.But unless the problem specifies whether the angles are central or inscribed, it's hard to know. However, given the answer choices, and the fact that each arc is 30 degrees, which is 30, the inscribed angles would be 15, 30, 45, 60, 75, 90, etc., while central angles would be 30, 60, 90, 120, 150, etc.The answer choices are 75, 80, 90, 120, 150. So 75 and 90 could be inscribed angles (75 = 150/2, 90=180/2), 120 and 150 are central angles.But the question is about the sum. If x and y are both inscribed angles, their sum could be 75 + 15=90, or 75 + 45=120, etc. If they are a mix of inscribed and central angles, but the problem doesn't specify.Wait, the problem just says "the number of degrees in the sum of the angles x and y". So maybe they are different types. But the problem is from a diagram, so likely both angles are inscribed or both are central. But since 90, 120, 150 are all possible. Hmm.Wait, since 12 arcs, 30 each. If you create a regular dodecagon, the internal angles can be calculated. The internal angle of a regular dodecagon is (n-2)*180/n = (12-2)*180/12 = 150 degrees. But that's the internal angle. If the angles x and y are internal angles of the dodecagon, then sum would be 300. Not an option. So probably not.Alternatively, the external angle of a dodecagon is 30 degrees. But sum of two external angles would be 60. Not an option.Alternatively, angles formed by connecting vertices. For example, connecting every other vertex forms a hexagon. The angles in the hexagon would be 120 degrees. Sum of two angles would be 240. Not an option.Alternatively, maybe triangle formed by three non-adjacent vertices. For example, A, E, I (each 4 arcs apart). Then the arcs between them are 120 degrees each. The angles of the triangle would be half the arcs opposite. So angle at A is half of arc EI, which is 240 degrees, so 120 degrees. Similarly, other angles are 120. Sum of two angles would be 240. Not an option.Wait, this is getting too complicated. Maybe I need to consider that the answer is likely 90 or 120. Given that the circle is divided into 12 parts, and common angles are 30, 60, 90, 120, 150.If I think about the possible inscribed angles, each can be 15 (30/2), 30 (60/2), 45 (90/2), 60 (120/2), 75 (150/2), 90 (180/2). The possible sums would be combinations of these. The answer choices are 75, 80, 90, 120, 150.75 could be 15 + 60, 30 + 45, etc. 90 could be 45 + 45, 30 + 60, 15 + 75. 120 could be 60 + 60, 45 + 75, 30 + 90. 150 would be 75 + 75, 90 + 60.But without knowing the exact arcs subtended by x and y, it's difficult. However, given that the options are there, maybe the problem is standard. Let me recall that in a regular dodecagon, the measure of an inscribed angle over three sides (90 degrees) is 45, over five sides (150 degrees) is 75. So if x and y are 45 and 75, sum is 120. Alternatively, maybe the answer is 90.Wait, perhaps looking for an SAT or AMC problem similar. Wait, I recall a problem where a circle is divided into 12 equal parts, and the sum of two angles is 90. For example, if you have a triangle inscribed in the circle with vertices at A, B, and D. The angle at B is half the arc AD, which is 90 degrees, so angle is 45. The angle at D is half the arc AB, which is 30 degrees, so 15. Then another triangle with similar angles. But adding 45 and 15 gives 60. Not 90.Alternatively, maybe angles in a star polygon. For example, a five-pointed star has angles of 36 degrees, but that's different.Alternatively, think about the coordinates. If points are every 30 degrees, coordinates can be determined. For example, point A at (1,0), point B at (cos30, sin30), etc. If angles x and y are angles between lines connecting these points. For example, angle between OA and OB is 30 degrees. But that's central angle.Alternatively, if x is the angle between chords AB and AC, which is an inscribed angle. Wait, angle at A between AB and AC. But if AB and AC are both chords from A to B and A to C, then the angle at A is between two chords. The measure of that angle is half the measure of the arc BC. Arc BC is 30 degrees (from B to C is one arc). So angle BAC is 15 degrees. Similarly, another angle at another point. If y is angle DFE, subtended by arc DE, which is 30 degrees, so 15 degrees. Sum is 30. Not an option.Alternatively, if the angle is between two chords that meet at a different point. For example, chords BD and CE intersect at point O, the center. Then the angles at O are central angles. For example, angle BOC is 60 degrees. If there are two such angles, their sum could be 60 + 30 = 90.But this is too vague. Maybe the answer is 90. Since 90 is a common right angle, and if the arcs involved are three arcs each (90 degrees), the inscribed angles would be 45 each, sum 90. Alternatively, if one angle is 30 and another is 60, sum 90. But I need to think of a configuration.Alternatively, maybe the angles x and y are angles in a rectangle inscribed in the circle. A rectangle inscribed in a circle has right angles, 90 degrees each. So if x and y are two angles of the rectangle, each 90, sum 180. Not an option.Alternatively, maybe a kite shape with two angles x and y. If the kite is symmetrical, angles x and y could be 60 and 120, sum 180. Not an option.Alternatively, if x and y are base angles of an isosceles triangle inscribed in the circle. For example, the triangle has vertices at A, B, and some other point. The base angles at A and B would be equal. The apex angle at, say, D. Arc AB is 30 degrees, so angle at D is 15 degrees. Angles at A and B would be (180 - 15)/2 = 82.5. Not an integer, so not an answer.Alternatively, maybe not a triangle. Maybe a quadrilateral with angles x and y. In cyclic quadrilaterals, opposite angles sum to 180. So if x and y are opposite angles, sum is 180. Not an option. If they are adjacent angles, sum could be anything.Alternatively, maybe x and y are exterior angles of some polygon. But I don't think so.Wait, another idea. If the circle is divided into 12 equal arcs, then each arc is 30 degrees. If you connect points such that you form an equilateral triangle, each arc between the vertices is 120 degrees (4 arcs apart). The central angle for each is 120, and the inscribed angle would be 60 degrees. So if you have two such angles, their sum is 120. But 120 is an option.Alternatively, if the triangle is formed by connecting every 4th point, making a square, each central angle 90 degrees, inscribed angle 45. Two such angles sum to 90.But without the diagram, I can't be certain. Given that 90 and 120 are both plausible, and they are answer choices, I need to figure out which is more likely.Wait, let's consider the answer choices: 75, 80, 90, 120, 150. If it's an AMC problem, often diagrams can be determined by typical configurations. For a 12-point circle, common angles are those subtended by 3, 4, or 5 arcs. For example, 3 arcs = 90 degrees (central angle), inscribed 45; 4 arcs = 120, inscribed 60; 5 arcs = 150, inscribed 75. So if angles x and y are inscribed angles subtended by 3 and 5 arcs, sum is 45 + 75 = 120. Alternatively, 4 and 4 arcs, sum 60 + 60 = 120. Or 3 and 3 arcs, 45 + 45 = 90.But since the answer choices include both 90 and 120, which is it?Alternatively, perhaps x and y are angles formed by intersecting chords, where the angle is half the sum of the intercepted arcs. For example, if two chords intersect, and the intercepted arcs are 60 and 60, the angle is 60. If another pair intercepts 60 and 60, angle is 60. Sum is 120.Alternatively, intercepted arcs 90 and 30, angle is 60. Another angle intercepts 90 and 30, sum 120.Alternatively, intercepted arcs 150 and 30, angle is 90. Another angle intercepts 150 and 30, sum 180. Not an option.Alternatively, intercepted arcs 120 and 60, angle is 90. Another angle intercepts 120 and 60, sum 180.Alternatively, intercepted arcs 90 and 90, angle is 90. Sum of two such angles is 180.Hmm. Wait, perhaps the answer is 90. If each angle is 45, sum is 90. How? If each angle is subtended by 3 arcs (90 degrees), so inscribed angle is 45. Then two such angles sum to 90. So if the diagram has two inscribed angles each subtended by three arcs, their sum is 90. Alternatively, if one angle is subtended by two arcs (60) and another by four arcs (120), sum of 30 + 60 = 90.Alternatively, given that 90 is a right angle, maybe the problem is designed such that x + y = 90. For example, if x and y are complementary angles in some right triangle.Alternatively, perhaps the problem involves a hexagon. In a regular hexagon inscribed in a circle, each central angle is 60 degrees, but here we have 12 divisions. Wait, if connecting every other point forms a hexagon, each arc is 60 degrees. The inscribed angles would be 30 degrees. But not helpful.Given that I can't see the diagram, and considering the answer choices, I think the most probable answer is 90 degrees, option C, because it's a right angle, commonly found in circle theorems when dealing with inscribed angles subtended by semicircles or combinations. However, another strong possibility is 120 degrees, as it's the sum of two 60-degree angles, which could be inscribed angles subtended by 120-degree arcs.Wait, let's think again. If the circle is divided into 12 equal arcs, each 30 degrees. If there's a triangle inscribed in the circle such that one side subtends 3 arcs (90 degrees), the angle opposite would be 45 degrees. Another triangle inscribed with a side subtending 5 arcs (150 degrees), the angle opposite would be 75 degrees. Then the sum is 45 + 75 = 120. This seems plausible. Since 3 arcs and 5 arcs are likely to be used in such problems, leading to angles of 45 and 75, summing to 120. Alternatively, if it's two angles each subtended by 2 arcs (60 degrees), each angle is 30, sum 60. Not an option. If each subtended by 4 arcs (120), each angle is 60, sum 120. Also possible.Given that 120 is an answer option and corresponds to more substantial arcs, maybe that's the intended answer. Alternatively, maybe the problem has angles x and y as vertical angles formed by intersecting chords, each being 60 degrees, so sum 120. For example, if two chords intersect, each intercepting arcs of 60 and 60 degrees, angle would be (60 + 60)/2 = 60. If there are two such angles, sum is 120. Alternatively, intercepting arcs of 90 and 30: (90 + 30)/2 = 60. Another similar angle, sum 120.Given the prevalence of 120 in the answer choices and the fact that it's a multiple of 30, which is the arc measure, I'm now leaning towards 120. However, without the diagram, it's a gamble. Wait, another way to think: in a regular 12-gon, the central angles are 30, and the internal angles are 150. But if you connect non-adjacent vertices, you can form various polygons. If x and y are internal angles of two different polygons, but this is too vague.Alternatively, if the problem is similar to the clock angle problem. For example, the angle between 12 and 3 is 90 degrees, and between 3 and 7 is 120 degrees (since from 3 to 7 is 4 hours, 4*30=120). Sum is 210. Not an option. Alternatively, 12 to 2 is 60 degrees, 2 to 5 is 90 degrees, sum 150. Option E.But this is purely speculative.Given that I have to choose between 90 and 120, and considering that inscribed angles subtended by 3 and 5 arcs give 45 and 75, summing to 120, which is an option, and that 120 is a common answer in such problems, I think the answer is 120. But I'm not entirely sure.Wait, let me check if 75 is possible. If x is 75 and y is 15, sum 90. But where would 15 come from? An inscribed angle subtended by 30 degrees. If x is subtended by 5 arcs (150), angle is 75, and y is subtended by 1 arc (30), angle is 15. Sum 90. But 15 isn't a nice number, but it's possible.Alternatively, maybe x and y are angles formed by a tangent and a chord. The angle between a tangent and a chord is half the measure of the arc. For example, if there's a tangent at point A, and a chord from A to C, the angle between the tangent and chord is half arc AC. Arc AC is 60 degrees, so angle is 30. If another tangent at point B and chord to D, angle is 30. Sum is 60. Not an option.Alternatively, tangents at A and B intersecting at some point outside, forming an angle of 1/2 the difference of arcs. The angle would be 1/2*(arc AB - arc BA). Since arc AB is 30, the difference is 360 - 30 - 30 = 300, so angle is 150. But sum of two such angles would be 300. Not an option.Alternatively, maybe x and y are angles in different intersections. One inside, one outside. For example, angle x inside is 1/2(sum arcs) = 60, angle y outside is 1/2(difference arcs) = 30. Sum 90.But this is too vague. Given the answer choices and standard problems, I think the answer is 90 or 120. I need to pick one. Given that I can't see the diagram, but considering the answer options and the problem being from a contest, I think the answer is 90 degrees, option C. Because it's a right angle, which is common in geometry, and if x and y are both 45 degrees, which is half of 90, which is three arcs, then their sum is 90. But another way, 45 + 45.Alternatively, maybe the problem is such that angles x and y are vertical angles, each 45 degrees, sum 90. But vertical angles are equal. So their sum would be 2*45=90. Alternatively, the problem states "the sum of the angles x and y". If they are vertical angles, they are equal, so the sum is twice one angle. If each is 45, sum is 90. If each is 60, sum is 120.But since 90 and 120 are both options, and given that the answer is likely to be 90 or 120, I need to make a decision. Based on common problems, I think 90 is the answer. However, I'm not sure. Wait, another approach: in a circle divided into 12 parts, if you draw lines from the center to every third division, you create a square, each angle 90 degrees. If the problem is referring to those angles, but it's sum of two, so 90 + 90=180. Not an option.Alternatively, if you connect points to form a triangle with angles x and y. For example, points A, C, E. The angle at C is half the arc AE, which is 120 degrees, so angle is 60. The angle at E is half the arc AC, which is 60 degrees, angle is 30. Sum of angles x and y (60 and 30) is 90. Then the answer is 90, option C.Yes, this seems plausible. If the problem has a triangle with vertices at every other division (like A, C, E), then the angles at C and E would be 60 and 30, sum 90. So answer C.Alternatively, another triangle. If vertices are at A, D, G (each 3 arcs apart), forming an equilateral triangle. Each central angle is 90 degrees, but the inscribed angles would be 45 each. However, in reality, in a circle, if you space three points 120 degrees apart, you get an equilateral triangle with 60-degree angles. Wait, but in this case, arcs are 90 degrees each. Wait, no. If the points are spaced every 3 arcs (90 degrees), then the triangle would have sides subtending 90 degrees each. So the central angles are 90 degrees, inscribed angles would be 45 degrees each. So each angle of the triangle is 45, sum of two angles is 90. But in reality, in a triangle inscribed in a circle, each angle is half the measure of the arc opposite. So if each side subtends a 90-degree arc, the opposite angle is 45 degrees. So a triangle with three 45-degree angles? But the sum is 135, which is impossible. Triangles must sum to 180. Therefore, this is incorrect.Wait, if each angle in the triangle is 45 degrees, the total would be 135, which is impossible. So that can't be. Therefore, my previous reasoning is flawed.Wait, the measure of an inscribed angle is half the measure of its subtended arc. So if a triangle has three arcs subtended, say, 90 each, then each angle is 45, sum 135. Impossible. Therefore, such a triangle cannot exist. Hence, the triangle must have arcs that add up to 360. So, if the three arcs subtended by the triangle's sides are, say, 90, 90, and 180, then the angles would be 45, 45, and 90, sum 180. That works. So in such a case, two angles sum to 45 + 45 = 90, and the third is 90. So the sum of two angles is 90. Hence, answer C.Therefore, the answer is 90 degrees, option C.Final Answerboxed{C}</think>The problem involves a circle divided into 12 equal arcs, each corresponding to 30 degrees. The goal is to determine the sum of the angles (x) and (y).1. Understanding the Circle Division: - The circle is divided into 12 equal arcs, each measuring (30^circ). - Points (A) to (L) are placed every 30 degrees around the circle.2. Considering Inscribed Angles: - Inscribed angles subtended by an arc are half the measure of the central angle subtended by the same arc. - For example, an arc of 60 degrees (2 arcs) subtends an inscribed angle of (30^circ), and an arc of 90 degrees (3 arcs) subtends an inscribed angle of (45^circ).3. Analyzing Possible Angles: - If angles (x) and (y) are inscribed angles subtended by arcs of 90 degrees each, they would each measure (45^circ). - Summing these angles gives (45^circ + 45^circ = 90^circ).4. Verification with Triangle Example: - A triangle inscribed in the circle with vertices at points (A), (C), and (E) (each 60 degrees apart) would have angles at (C) and (E) subtended by arcs of 60 degrees and 120 degrees, leading to angles of (30^circ) and (60^circ). - Summing these angles gives (30^circ + 60^circ = 90^circ).5. Conclusion: - The most plausible answer, considering common configurations and the need for the sum to match one of the given options, is (90^circ).Thus, the number of degrees in the sum of the angles (x) and (y) is (boxed{C}).]

🔑:For this problem, it is useful to know that the measure of an inscribed angle is half the measure of its corresponding central angle. Since each unit arc is frac{1}{12} of the circle's circumference, each unit central angle measures left( frac{360}{12} right) ^{circ}=30^{circ}. Then, we know that the central angle of x = 60, so inscribed angle = 30. Also, central angle of y = 120, so inscirbed angle = 60. Summing both inscribed angles gives 30 + 60 = boxed{(C) 90}.

🧠:[<think>Okay, so I need to prove that the points where the two parabolas Y = X² + X - 41 and X = Y² + Y - 40 lie on the same circle. Hmm, interesting. Let me start by recalling that if four points lie on a circle, then they must satisfy the equation of a circle. So, maybe I can find the points of intersection of these two parabolas and then show that there's a circle that passes through all of them.First, let's find the points of intersection. That means solving the system of equations:1) Y = X² + X - 412) X = Y² + Y - 40So, to solve this system, I can substitute Y from the first equation into the second equation. Let's try that.From equation (1), Y is expressed in terms of X. So substitute that into equation (2):X = (X² + X - 41)² + (X² + X - 41) - 40Wow, that looks complicated. Let me expand this step by step. Let's denote A = X² + X - 41. Then the equation becomes X = A² + A - 40.But A is X² + X - 41, so substituting back:X = (X² + X - 41)² + (X² + X - 41) - 40Let me compute (X² + X - 41)² first. Let's expand that:(X² + X - 41)² = (X²)^2 + (X)^2 + (-41)^2 + 2*(X²*X + X²*(-41) + X*(-41))Wait, no. Wait, the expansion of (a + b + c)^2 is a² + b² + c² + 2ab + 2ac + 2bc. So here, a = X², b = X, c = -41. Therefore:= (X²)^2 + (X)^2 + (-41)^2 + 2*(X²*X + X²*(-41) + X*(-41))= X^4 + X² + 1681 + 2*(X³ -41X² -41X)= X^4 + X² + 1681 + 2X³ -82X² -82XCombine like terms:X^4 + 2X³ + (X² -82X²) + (-82X) + 1681Which simplifies to:X^4 + 2X³ -81X² -82X + 1681So, that's the expansion of (X² + X -41)^2.Now, going back to the equation:X = [X^4 + 2X³ -81X² -82X + 1681] + [X² + X -41] -40Let's combine the terms inside the brackets:First term is X^4 + 2X³ -81X² -82X + 1681Second term is X² + X -41Third term is -40So adding them all together:X^4 + 2X³ -81X² -82X + 1681 + X² + X -41 -40Combine like terms:X^4 + 2X³ + (-81X² + X²) + (-82X + X) + (1681 -41 -40)Calculating each:X^4 + 2X³ -80X² -81X + (1681 -81) [since 41 +40 =81]1681 -81 = 1600So the equation becomes:X = X^4 + 2X³ -80X² -81X + 1600Bring all terms to one side:X^4 + 2X³ -80X² -81X + 1600 - X = 0Simplify:X^4 + 2X³ -80X² -82X + 1600 = 0So, the equation we need to solve is a quartic (fourth-degree) equation:X^4 + 2X³ -80X² -82X + 1600 = 0Hmm, quartic equations can be tough. Maybe we can factor this? Let's try to factor this polynomial.First, maybe try rational roots. The rational root theorem suggests that any rational roots, p/q, where p divides 1600 and q divides 1 (since leading coefficient is 1). So possible roots are ±1, ±2, ±4, ±5, ±8, ±10, etc. Let's test some values.Let's test x=5:5^4 + 2*5^3 -80*5^2 -82*5 +1600= 625 + 250 - 2000 -410 +1600= (625 +250) + (-2000 -410) +1600= 875 -2410 +1600= (875 +1600) -2410 = 2475 -2410 = 65 ≠0Not a root.x=4:256 + 128 - 80*16 -82*4 +1600256 +128 = 38480*16=1280; 82*4=328So,384 -1280 -328 +1600= (384 -1280) + (-328 +1600)= (-896) + (1272) = 376 ≠0Not a root.x= 10:10000 + 2000 -80*100 -82*10 +1600= 10000 +2000 -8000 -820 +1600= (10000 +2000) + (-8000 -820) +1600=12000 -8820 +1600=12000 -8820 is 3180, +1600 is 4780 ≠0x=-5:625*(-5)^4? Wait, x=-5:(-5)^4 + 2*(-5)^3 -80*(-5)^2 -82*(-5) +1600=625 + 2*(-125) -80*25 +410 +1600=625 -250 -2000 +410 +1600= (625 -250) + (-2000 +1600) +410=375 -400 +410 = 385 ≠0Hmm. How about x=8?8^4 =4096; 2*8^3=2*512=1024; -80*8^2= -80*64= -5120; -82*8= -656; +1600So total: 4096 +1024 -5120 -656 +1600Compute step by step:4096 +1024 =51205120 -5120=00 -656= -656-656 +1600=944 ≠0x= -4:256 + 2*(-64) -80*16 -82*(-4) +1600256 -128 -1280 +328 +1600= (256 -128) + (-1280 +1600) +328=128 +320 +328= 776 ≠0x= -8:4096 + 2*(-512) -80*64 -82*(-8) +16004096 -1024 -5120 +656 +1600= (4096 -1024) + (-5120 +1600) +656=3072 -3520 +656 = (3072 +656) -3520= 3728 -3520=208 ≠0x= 16:Way too big, probably. Maybe x=5 didn't work. Maybe it's factorable into quadratics.Suppose we can factor the quartic as (X² + aX + b)(X² + cX + d). Let's try to find integers a, b, c, d such that:(X² + aX + b)(X² + cX + d) = X^4 + (a + c)X³ + (ac + b + d)X² + (ad + bc)X + bdCompare to original quartic: X^4 + 2X³ -80X² -82X +1600Thus, equate coefficients:1. a + c = 22. ac + b + d = -803. ad + bc = -824. bd = 1600So, we need integers a, c such that a + c = 2. Then, find integers b and d such that bd=1600 and the other equations hold.This might take some trial and error.First, possible pairs (a,c) since a + c =2. Let's try a= 10, c= -8 (since 10 + (-8)=2). Wait, but a and c could be any integers adding to 2. Let's list some possibilities:a= 5, c= -3a= 4, c= -2a= 3, c= -1a= 2, c=0a=1, c=1But since bd=1600, which factors into pairs like (1,1600), (2,800), (4,400), (5,320), (8,200), (10,160), (16,100), (20,80), (25,64), (32,50), (40,40). Also negative pairs since 1600 is positive, so both b and d are either positive or both negative.Let's try a=10, c=-8 (since 10 + (-8)=2). Then check if this works.Then ac =10*(-8)= -80. Then equation 2: ac + b + d = -80 becomes -80 + b + d = -80, so b + d=0. So d= -b.But bd=1600. If d= -b, then b*(-b)= -b²=1600. So -b²=1600 → b²=-1600. Not possible. So discard this pair.Next try a= 1, c=1. Then equation 2: 1*1 + b + d=1 + b + d=-80 → b + d=-81bd=1600. So we need two numbers that multiply to 1600 and add to -81. Let's check factors of 1600.Looking for two numbers (positive or negative) that multiply to 1600 and add to -81. Let's see:Since their product is positive and sum is negative, both numbers are negative. So need two negative numbers whose product is 1600 and sum is -81. Let's find factors of 1600:1600= 40*40, but 40+40=80, so if -40 and -40: sum -80, not -81.32*50=1600; 32+50=82, so -32 and -50: sum -82. Close, but not -81.25*64=1600; sum 89. So -25 and -64 sum to -89.20*80=1600; sum 100. Not useful.16*100=1600; sum 116.Alternatively, maybe not. So maybe there are no such integers here. Hence a=1, c=1 is not feasible.Next try a=2, c=0. Then equation 2: 2*0 + b + d=0 + b + d=-80, so b + d=-80bd=1600So looking for two numbers that add to -80 and multiply to 1600. Again, both numbers negative. Let’s check:Possible pairs:-40 and -40: sum -80, product 1600. Perfect!So if b=-40, d=-40.Then check equation 3: ad + bc =2*(-40) + 0*(-40)= -80 +0=-80. But the original equation 3 is -82. Doesn't match. So close but not quite. So this pair is invalid.But maybe another pair?Wait, if b + d=-80 and b*d=1600, the only solution is b=d=-40, as (-40)^2=1600. So that's the only pair. So equation 3 gives -80, but we need -82. Therefore, not matching. So discard a=2, c=0.Next, a=3, c=-1. Then equation 2: 3*(-1) + b + d= -3 + b + d= -80 ⇒ b + d=-77bd=1600Looking for two numbers (positive or negative) multiplying to 1600 and adding to -77. Let's see:Factors of 1600: as before. Let's check:-25 and -64: product 1600, sum -89-32 and -50: product 1600, sum -82-40 and -40: sum -80No pair here sums to -77. So no integer solution here.Next, a=4, c=-2. Then equation 2: 4*(-2) + b + d= -8 + b + d= -80 ⇒ b + d= -72bd=1600Looking for two numbers with product 1600 and sum -72. Again, both negative. Let's check possible pairs:Looking for two numbers whose product is 1600 and sum is 72. Then their negatives would sum to -72. Let's check:Factors of 1600: 32 and 50 sum to 82, 25 and 64 sum to 89, 40 and 40 sum to 80. 16 and 100 sum to 116. No pair sums to 72. So no solution here.Next, a=5, c=-3. Then equation 2:5*(-3)+b + d= -15 +b + d= -80 ⇒ b + d= -65bd=1600Looking for two numbers that multiply to 1600 and add to -65. Let's check factors:Again, looking for positive factors summing to 65. 25 and 64 sum to 89, 32 and 50 sum to 82, 40 and 40 sum to 80. Not 65. So no.a=6, c=-4. Then equation 2:6*(-4) +b + d= -24 +b + d= -80 ⇒b +d= -56bd=1600. Looking for two numbers summing to 56 and product 1600. Factors: 40 and 40 sum 80, 32 and 50 sum 82, 25 and 64 sum 89. No. Doesn't exist.a=7, c=-5: equation 2:7*(-5) +b +d= -35 +b +d= -80 ⇒b +d= -45bd=1600. Looking for two numbers summing to 45, product 1600. 32 and 50 sum 82. 25 and 64 sum 89. No. Nope.a=8, c=-6: equation 2:8*(-6) +b +d= -48 +b +d= -80 ⇒b +d= -32bd=1600. Looking for two numbers summing to 32, product 1600. 25 and 64: sum 89. 32 and 50: sum 82. No.Hmm. This is getting frustrating. Maybe there's no factorization with integer coefficients. Maybe the quartic is irreducible. Then, perhaps I need another approach.Alternatively, perhaps the quartic can be written as a quadratic in terms of X² or something else. Let me see.Looking at the quartic equation:X^4 + 2X³ -80X² -82X +1600 =0Hmm. Not obvious. Alternatively, maybe if we make substitution Z = X + something? Let me check.Alternatively, maybe notice that the two parabolas are symmetric in some way. The first one is Y = X² + X -41, and the second is X = Y² + Y -40. There's a similarity, except the coefficients are slightly different. Maybe if we can find a substitution that relates X and Y?Alternatively, maybe add the two equations. Let's see:From equation (1): Y = X² + X -41From equation (2): X = Y² + Y -40If I add these equations: X + Y = X² + Y² + X + Y -41 -40Simplify:X + Y = X² + Y² + X + Y -81Subtract X + Y from both sides:0 = X² + Y² -81Thus, X² + Y² =81Wait, that's interesting! So, adding the two equations gives X² + Y² =81. But 81 is 9². So that's the equation of a circle centered at the origin with radius 9. Wait, but the problem says that the intersection points lie on the same circle. So according to this, X² + Y² =81 is the equation of a circle, and adding the two original equations gives X² + Y² =81. Therefore, all intersection points must satisfy X² + Y² =81, meaning they lie on this circle. Therefore, the four intersection points (since quartic can have up to four solutions) lie on the circle X² + Y² =81. So, that's the proof. Wait, is that right?Wait, let's verify. If we take the two equations:Y = X² + X -41 ...(1)X = Y² + Y -40 ...(2)If we add them, we get X + Y = X² + Y² + X + Y -81Subtracting X + Y from both sides: 0 = X² + Y² -81, so X² + Y² =81. Therefore, any solution (X,Y) to the system must satisfy X² + Y² =81. Therefore, all intersection points lie on the circle X² + Y² =81. Therefore, they lie on the same circle. Hence proved.Wait, that seems straightforward. I feel like I must have missed something earlier. Let me check with an example. Suppose we have an intersection point (x,y). Then it must satisfy both equations and hence x² + y²=81. Let's take a possible solution. For instance, let me try x=0. Then from equation (2): X=0=Y² + Y -40 ⇒Y² + Y -40=0 ⇒Y = [-1 ±√(1 +160)]/2 = [-1 ±√161]/2. But then from equation (1): Y=0 +0 -41= -41. So unless -41 is equal to [-1 ±√161]/2, which it isn't, x=0 is not a solution. So x=0 is not a solution, which is fine. Let's see if there's a solution. Let me try x=5. Then from equation (1): Y=25 +5 -41= -11. Then from equation (2): X= (-11)^2 + (-11) -40=121 -11 -40=70. So X=70 ≠5. Not a solution. So that's not a solution. Maybe another value.But regardless of whether I can find explicit solutions, the algebra shows that any solution must satisfy X² + Y²=81. Therefore, all intersection points lie on that circle. Therefore, the proof is complete.Wait, that's much simpler than solving the quartic. I was overcomplicating it by trying to find the actual points, but instead, by adding the two equations, we directly get the equation of the circle. Therefore, the intersection points must lie on that circle. Therefore, the answer is that all intersection points lie on the circle X² + Y²=81, so they lie on the same circle. Hence proved.Final Answerboxed{x^2 + y^2 = 81}</think>To prove that the points of intersection of the parabolas ( Y = X^2 + X - 41 ) and ( X = Y^2 + Y - 40 ) lie on the same circle, we start by considering the system of equations:1. ( Y = X^2 + X - 41 )2. ( X = Y^2 + Y - 40 )We add these two equations:[Y + X = X^2 + X - 41 + Y^2 + Y - 40]Simplifying the right-hand side, we get:[Y + X = X^2 + Y^2 + X + Y - 81]Subtracting ( X + Y ) from both sides, we obtain:[0 = X^2 + Y^2 - 81]This simplifies to:[X^2 + Y^2 = 81]This is the equation of a circle centered at the origin with radius 9. Therefore, any point ((X, Y)) that satisfies both equations of the parabolas must also satisfy the equation ( X^2 + Y^2 = 81 ). Hence, all points of intersection of the given parabolas lie on this circle.[boxed{x^2 + y^2 = 81}],<think>Okay, so I need to prove that the points where the two parabolas Y = X² + X - 41 and X = Y² + Y - 40 lie on the same circle. Hmm, interesting. Let me think about how to approach this. First, I know that if four points lie on a circle, they must satisfy the equation of a circle. A circle's general equation is (X - h)² + (Y - k)² = r², where (h, k) is the center and r is the radius. So, maybe if I can find four points of intersection and then show they satisfy the same circle equation, that would work. But wait, how many points of intersection do these parabolas have? Parabolas can intersect up to 4 points, right? Because each parabola is a quadratic equation, so solving them simultaneously would result in a quartic equation, which can have up to four real roots. So, if there are four intersection points, then they must lie on the same circle. But maybe there's a smarter way than finding all four points and plugging them into a circle equation. That might get messy, especially with the coordinates. Alternatively, maybe I can combine the equations of the two parabolas in a way that eliminates the quadratic terms, leading to the equation of a circle. Let me try that. So, the first parabola is Y = X² + X - 41. The second one is X = Y² + Y - 40. Let me write both equations:1) Y = X² + X - 412) X = Y² + Y - 40If I can somehow subtract these equations or manipulate them to get a linear equation, which when combined with one of the original equations would form a circle. Alternatively, maybe adding them? Let me see. Let's add equation 1 and equation 2:Y + X = X² + X - 41 + Y² + Y - 40Simplify the right-hand side: X² + Y² + X + Y - 41 - 40 = X² + Y² + X + Y - 81So, left side is X + Y, right side is X² + Y² + X + Y - 81Subtracting (X + Y) from both sides:0 = X² + Y² - 81So, X² + Y² = 81Wait, that's a circle centered at the origin with radius 9. But is that correct?Wait, let me check my steps again. Starting with:Equation 1: Y = X² + X - 41Equation 2: X = Y² + Y - 40Adding them:Y + X = X² + X - 41 + Y² + Y - 40Yes, so that gives:X + Y = X² + Y² + X + Y - 81Subtracting X + Y from both sides:0 = X² + Y² - 81So, X² + Y² = 81So, this is the equation of a circle. But does this mean that all intersection points of the two parabolas lie on this circle?But wait, when we added the two equations, we got X² + Y² = 81. However, the original equations are Y = X² + X - 41 and X = Y² + Y - 40. So, the intersection points must satisfy both equations, and therefore, also satisfy the sum of the two equations, which is X² + Y² = 81. Hence, all intersection points lie on the circle X² + Y² = 81. Therefore, they lie on the same circle.Wait, that seems straightforward. So, actually, by adding the two equations, we derived the equation of the circle. Therefore, any point that satisfies both equations (i.e., the intersection points) must lie on this circle. Therefore, all intersection points lie on the same circle. Is that the complete proof? Hmm. Let me verify with an example. Suppose we take an intersection point. Let's try to find one. Suppose we solve the two equations:From equation 1: Y = X² + X - 41Plug into equation 2: X = (X² + X - 41)^2 + (X² + X - 41) - 40That's a quartic equation. Let's compute it step by step.Let me denote Y = X² + X - 41. Then, substituting into equation 2:X = Y² + Y - 40So, X = (X² + X - 41)^2 + (X² + X - 41) - 40Let me expand (X² + X - 41)^2:= (X²)^2 + (X)^2 + (-41)^2 + 2*(X²*X) + 2*(X²*(-41)) + 2*(X*(-41))Wait, no. Wait, (a + b + c)^2 = a² + b² + c² + 2ab + 2ac + 2bc. So, in this case, a = X², b = X, c = -41.Therefore, (X² + X - 41)^2 = (X²)^2 + (X)^2 + (-41)^2 + 2*(X²*X) + 2*(X²*(-41)) + 2*(X*(-41))Compute each term:= X⁴ + X² + 1681 + 2X³ - 82X² - 82XSo, combining like terms:X⁴ + 2X³ + (X² - 82X²) + (-82X) + 1681= X⁴ + 2X³ - 81X² - 82X + 1681Then, the equation becomes:X = [X⁴ + 2X³ - 81X² - 82X + 1681] + [X² + X - 41] - 40Simplify the terms:First, expand the brackets:= X⁴ + 2X³ - 81X² - 82X + 1681 + X² + X - 41 - 40Combine like terms:X⁴ + 2X³ + (-81X² + X²) + (-82X + X) + (1681 - 41 - 40)Simplify each:= X⁴ + 2X³ - 80X² - 81X + (1681 - 81)= X⁴ + 2X³ - 80X² - 81X + 1600So, bringing the X from the left side to the right:0 = X⁴ + 2X³ - 80X² - 81X + 1600 - XSimplify:= X⁴ + 2X³ - 80X² - 82X + 1600So, the equation is X⁴ + 2X³ - 80X² - 82X + 1600 = 0Hmm, solving a quartic equation is complicated, but maybe we can factor it.Let me try possible rational roots using Rational Root Theorem. The possible roots are factors of 1600 over factors of 1, so ±1, ±2, ±4, ±5, ±8, etc.Test X=5:5⁴ + 2*5³ -80*5² -82*5 +1600= 625 + 250 - 2000 - 410 + 1600= (625 + 250) + (-2000 - 410) + 1600= 875 - 2410 + 1600= (875 + 1600) -2410= 2475 -2410 = 65 ≠0X=4:4⁴ + 2*4³ -80*4² -82*4 +1600= 256 + 128 - 1280 - 328 +1600= (256 +128) + (-1280 -328) +1600= 384 -1608 +1600= (384 +1600) -1608= 1984 -1608 = 376 ≠0X= -5:(-5)^4 +2*(-5)^3 -80*(-5)^2 -82*(-5)+1600=625 -250 -2000 +410 +1600= (625 -250) + (-2000 +410) +1600=375 -1590 +1600= (375 +1600) -1590=1975 -1590= 385 ≠0X=10:10⁴ is way too big, 10000, which is larger than 1600. So maybe not.X=8:8⁴ +2*8³ -80*8² -82*8 +1600=4096 + 1024 -5120 -656 +1600=(4096 +1024) + (-5120 -656) +1600=5120 -5776 +1600= (5120 +1600) -5776=6720 -5776=944≠0Hmm, not working.X=2:2⁴ +2*8 -80*4 -82*2 +1600=16 +16 -320 -164 +1600=32 -484 +1600=1148≠0X= -4:(-4)^4 +2*(-4)^3 -80*(-4)^2 -82*(-4) +1600=256 -128 -1280 +328 +1600=(256 -128) + (-1280 +328) +1600=128 -952 +1600= (128 +1600) -952=1728 -952=776≠0Hmm, maybe this is not factorable with integer roots. Maybe need to use another approach. Alternatively, since we know from the previous step that all intersection points lie on X² + Y² =81, so perhaps we can use that to find the coordinates.Alternatively, since we have that any intersection point (x, y) must satisfy x² + y² =81. Let's check if this holds for possible solutions.But maybe instead of solving the quartic, which is complicated, let's test a point that's on the circle x² + y²=81. For example, (0,9): 0² +9²=81. Is this an intersection point? Let's check.Plug into first equation: Y =0² +0 -41= -41≠9. So, no.Another point: (9,0). X=9, Y=0. Check first equation: 0=9² +9 -41=81+9-41=50≠0. Not.How about (sqrt(81 - y²), y). Not helpful.Alternatively, maybe take some coordinates where X and Y are integers. Let's see. Let's take X and Y such that X² + Y²=81. Possible integer solutions are (0, ±9), (±9,0), (±√(81 - Y²), Y), but integers only. So (±9,0), (0,±9), (±3√(81 - (3)^2)= sqrt(72)≈8.485, which is not integer. Maybe (±7, 4), since 7² +4²=49+16=65≠81. Hmm. (9,0) as before, which doesn't work. Maybe there are no integer solutions? But maybe non-integer.Alternatively, perhaps the intersection points are not integers, but still lie on the circle. Since we derived that any intersection point must lie on X² + Y²=81, then regardless of their coordinates, they must satisfy that. So even though solving the quartic is complicated, the key point is that the sum of the two original equations gives the equation of the circle, which must contain all intersection points.Therefore, the proof is straightforward: adding the two equations of the parabolas gives the equation of a circle, hence all intersection points lie on this circle.But let me check if there are any possible mistakes here. For instance, when we added the two equations, we assumed that the intersection points satisfy both equations, so their sum must satisfy the summed equation. Therefore, any solution (x,y) must satisfy the original two equations, and therefore their sum. The summed equation is a circle, hence all intersection points lie on that circle. Therefore, they lie on the same circle.Therefore, the proof is complete.Final AnswerThe points of intersection lie on the same circle, specifically boxed{X^2 + Y^2 = 81}.</think>To prove that the points of intersection of the parabolas ( Y = X^2 + X - 41 ) and ( X = Y^2 + Y - 40 ) lie on the same circle, we can follow these steps:1. Write down the equations of the parabolas: [ text{First parabola: } Y = X^2 + X - 41 ] [ text{Second parabola: } X = Y^2 + Y - 40 ]2. Add the two equations: [ Y + X = X^2 + X - 41 + Y^2 + Y - 40 ]3. Simplify the right-hand side: [ Y + X = X^2 + Y^2 + X + Y - 81 ]4. Subtract ( X + Y ) from both sides: [ 0 = X^2 + Y^2 - 81 ]5. Rearrange to form the equation of a circle: [ X^2 + Y^2 = 81 ]This equation represents a circle centered at the origin with radius 9. Since any point of intersection of the two parabolas must satisfy both original equations, they must also satisfy the combined equation ( X^2 + Y^2 = 81 ). Therefore, all intersection points lie on this circle. Final AnswerThe points of intersection lie on the same circle, specifically boxed{X^2 + Y^2 = 81}.]

🔑:75.16. Adding the equations, we get X^{2}+Y^{2}=81. This equation must be satisfied by the coordinates of the points of intersection of the parabolas, which means that all of them lie on the circle X^{2}+Y^{2}= =81.

🧠:[<think>Okay, so I have this problem here with vectors a and b. The given conditions are that the magnitudes of a and b are both 2, and their dot product is also 2. Then there's another condition involving a vector c: (a - c) · (b - c) = 0. The question is asking for the minimum value of |2b - c|. Hmm, let me try to visualize this.First, since |a| = |b| = 2 and a · b = 2, maybe I can find the angle between vectors a and b. The dot product formula is a · b = |a||b|cosθ. Plugging in the values, 2 = 2 * 2 * cosθ. So, cosθ = 2 / 4 = 0.5. Therefore, θ = 60 degrees. So, vectors a and b are at 60 degrees to each other, both with length 2. That helps to picture their positions.Now, the next part is (a - c) · (b - c) = 0. That equation means that the vectors (a - c) and (b - c) are perpendicular to each other. So, geometrically, if I think of points A, B, and C in space corresponding to vectors a, b, and c, then the vectors from C to A and from C to B are perpendicular. That would mean that point C lies on the circle with diameter AB, according to the circle theorem where the angle subtended by a diameter is a right angle. Wait, but in three dimensions, it's a sphere, but since all vectors are in the plane defined by a and b, maybe we can consider it in 2D.But let me check that. If vectors a and b are in a plane, then any linear combination or vector related to them would also be in that plane. So, if c is in the same plane, then the problem reduces to 2D. But the problem doesn't specify the dimensionality, so maybe we should assume it's in 3D, but since all given vectors are in a plane, perhaps c is also constrained to that plane? Hmm, not sure. Maybe the minimum occurs in the plane, so we can proceed in 2D.Alternatively, think algebraically. Let me set up coordinates to model vectors a and b. Let me place vector a along the x-axis for simplicity. So, a = (2, 0). Then vector b is at 60 degrees to a, so its components would be |b|cosθ in the x-direction and |b|sinθ in the y-direction. Since |b| = 2 and θ = 60°, then b = (2cos60°, 2sin60°) = (2*(1/2), 2*(√3/2)) = (1, √3). So, vectors a = (2, 0), b = (1, √3). Good.Now, vector c is some vector (x, y) in this plane. The condition (a - c) · (b - c) = 0. Let's write that out:(a - c) · (b - c) = 0.Expanding the dot product:(a · b) - a · c - b · c + |c|² = 0.But we know a · b = 2. So,2 - (a · c + b · c) + |c|² = 0.But a · c + b · c = (a + b) · c. Let me denote that:2 - (a + b) · c + |c|² = 0.Rearranged, this is:|c|² - (a + b) · c + 2 = 0.Hmm. That's a quadratic equation in terms of c. Maybe this represents a circle or some conic section in the plane. Let me compute a + b. Since a = (2, 0) and b = (1, √3), then a + b = (3, √3). So, (a + b) · c = 3x + √3 y.Therefore, the equation becomes:x² + y² - 3x - √3 y + 2 = 0.To find the geometric representation, let's complete the squares.For the x-terms: x² - 3x. Take half of 3, which is 1.5, square it: 2.25.For the y-terms: y² - √3 y. Take half of √3, which is √3/2, square it: (√3/2)² = 3/4.So, rewrite the equation:(x² - 3x + 2.25) - 2.25 + (y² - √3 y + 3/4) - 3/4 + 2 = 0.Simplify:(x - 1.5)² + (y - (√3)/2)² - 2.25 - 0.75 + 2 = 0.Calculating the constants: -2.25 - 0.75 = -3, so:(x - 1.5)² + (y - (√3)/2)² - 3 + 2 = 0 =>(x - 1.5)² + (y - (√3)/2)² - 1 = 0 =>(x - 1.5)² + (y - (√3)/2)² = 1.So, this is a circle with center at (1.5, √3/2) and radius 1. So, vector c must lie on this circle.Therefore, the set of possible c vectors is this circle in the plane. Now, the problem asks for the minimum value of |2b - c|. Let's compute 2b. Since b = (1, √3), then 2b = (2, 2√3). So, we need to find the minimum distance between the point (2, 2√3) and any point (x, y) on the circle centered at (1.5, √3/2) with radius 1.So, geometrically, the minimum |2b - c| is the distance from the point 2b to the center of the circle minus the radius. Because the closest point on the circle to 2b would be along the line connecting 2b to the center of the circle, minus the radius length.Let me verify that. The distance from 2b to the center of the circle is |2b - center|. Then subtract the radius 1 to get the minimum |2b - c|.First, compute the center of the circle: (1.5, √3/2). Then 2b is (2, 2√3). Let's compute the vector from the center to 2b:(2 - 1.5, 2√3 - √3/2) = (0.5, (4√3/2 - √3/2)) = (0.5, (3√3)/2).So, the distance is sqrt[(0.5)^2 + ( (3√3)/2 )^2 ].Calculating:(0.5)^2 = 0.25,( (3√3)/2 )^2 = (9*3)/4 = 27/4 = 6.75,Total: 0.25 + 6.75 = 7. So, sqrt(7).Therefore, the distance from 2b to the center is sqrt(7), and the radius of the circle is 1. Therefore, the minimum |2b - c| is sqrt(7) - 1.But wait, let me make sure. Since the point 2b is outside the circle (since sqrt(7) ≈ 2.645, which is greater than the radius 1), the minimum distance from 2b to the circle is indeed the distance to the center minus the radius. If it were inside, the minimum would be the radius minus the distance, but since it's outside, subtract the radius. Therefore, the minimum value is sqrt(7) - 1.But let me check this algebraically to be sure. Let's parameterize the circle. Let me write the circle equation:(x - 1.5)^2 + (y - √3/2)^2 = 1.Then, we can write c as:c = (1.5 + cosθ, √3/2 + sinθ), where θ varies from 0 to 2π.Then, |2b - c|^2 = (2 - (1.5 + cosθ))^2 + (2√3 - (√3/2 + sinθ))^2.Simplify each component:x-component: 2 - 1.5 - cosθ = 0.5 - cosθ.y-component: 2√3 - √3/2 - sinθ = (4√3/2 - √3/2) - sinθ = (3√3/2) - sinθ.So, |2b - c|^2 = (0.5 - cosθ)^2 + (3√3/2 - sinθ)^2.Expanding this:(0.25 - cosθ + cos²θ) + ( (3√3/2)^2 - 3√3 sinθ + sin²θ )Compute each part:First term: 0.25 - cosθ + cos²θ.Second term: (27/4) - 3√3 sinθ + sin²θ.Adding them together:0.25 - cosθ + cos²θ + 27/4 - 3√3 sinθ + sin²θ.Combine like terms:cos²θ + sin²θ = 1.0.25 + 27/4 = (0.25 + 6.75) = 7.Then, the remaining terms: -cosθ - 3√3 sinθ.So, total expression:1 + 7 - cosθ - 3√3 sinθ = 8 - (cosθ + 3√3 sinθ).Therefore, |2b - c|^2 = 8 - (cosθ + 3√3 sinθ).To find the minimum value of |2b - c|, we need to minimize this expression. Since the square root function is increasing, minimizing |2b - c| is equivalent to minimizing |2b - c|². Therefore, the minimum occurs when (cosθ + 3√3 sinθ) is maximized.So, we need to find the maximum value of cosθ + 3√3 sinθ. The maximum value of A cosθ + B sinθ is sqrt(A² + B²). Let's compute that.Here, A = 1, B = 3√3. So, sqrt(1 + (3√3)^2) = sqrt(1 + 27) = sqrt(28) = 2*sqrt(7).Therefore, the maximum of (cosθ + 3√3 sinθ) is 2*sqrt(7). Hence, the minimum of |2b - c|² is 8 - 2*sqrt(7). Then, the minimum |2b - c| is sqrt(8 - 2√7).Wait, but hold on. The calculation here gives a different result than the geometric approach. Earlier, I thought it was sqrt(7) - 1 ≈ 2.645 - 1 = 1.645, but according to this algebra, sqrt(8 - 2√7). Let's compute that numerically.First, compute 2√7 ≈ 2*2.6458 ≈ 5.2916. Then, 8 - 5.2916 ≈ 2.7084. Then sqrt(2.7084) ≈ 1.645. So, actually, both methods give the same result. Wait, sqrt(8 - 2√7) is equal to sqrt(7) - 1? Let me check.Compute (sqrt(7) - 1)^2 = 7 - 2√7 + 1 = 8 - 2√7. Yes, exactly. Therefore, sqrt(8 - 2√7) = sqrt(7) - 1. So, both approaches agree. Great, so the minimum value is sqrt(7) - 1. Therefore, the answer is sqrt{7} - 1, which can be written as boxed{sqrt{7} - 1}.But let me just recap to make sure I didn't skip any steps. So, we have two methods: one geometric, using the distance from the point to the center minus the radius, and another algebraic, parameterizing the circle and minimizing the expression. Both lead to the same answer. Therefore, confident that the minimum value is sqrt(7) - 1.Final Answerboxed{sqrt{7} - 1}</think>Given vectors (boldsymbol{a}) and (boldsymbol{b}) satisfy (|a| = |b| = 2) and (boldsymbol{a} cdot boldsymbol{b} = 2). The angle (theta) between (boldsymbol{a}) and (boldsymbol{b}) is calculated using the dot product formula:[boldsymbol{a} cdot boldsymbol{b} = |boldsymbol{a}||boldsymbol{b}| cos theta implies 2 = 2 cdot 2 cos theta implies cos theta = frac{1}{2} implies theta = 60^circ.]Vectors (boldsymbol{a}) and (boldsymbol{b}) are placed in a coordinate system with (boldsymbol{a} = (2, 0)) and (boldsymbol{b} = (1, sqrt{3})).The condition ((boldsymbol{a} - boldsymbol{c}) cdot (boldsymbol{b} - boldsymbol{c}) = 0) implies that (boldsymbol{c}) lies on a circle. Expanding this condition:[boldsymbol{a} cdot boldsymbol{b} - (boldsymbol{a} + boldsymbol{b}) cdot boldsymbol{c} + |boldsymbol{c}|^2 = 0 implies 2 - (3, sqrt{3}) cdot boldsymbol{c} + |boldsymbol{c}|^2 = 0.]Rewriting in coordinates, we find the circle equation:[x^2 + y^2 - 3x - sqrt{3}y + 2 = 0 implies (x - 1.5)^2 + left(y - frac{sqrt{3}}{2}right)^2 = 1.]This circle has center ((1.5, frac{sqrt{3}}{2})) and radius 1. We need the minimum value of (|2boldsymbol{b} - boldsymbol{c}|). The point (2boldsymbol{b}) is ((2, 2sqrt{3})). The distance from ((2, 2sqrt{3})) to the circle center is:[sqrt{(2 - 1.5)^2 + left(2sqrt{3} - frac{sqrt{3}}{2}right)^2} = sqrt{0.5^2 + left(frac{3sqrt{3}}{2}right)^2} = sqrt{0.25 + 6.75} = sqrt{7}.]Subtracting the radius, the minimum distance is (sqrt{7} - 1). Verifying algebraically, parameterizing (boldsymbol{c}) and minimizing (|2boldsymbol{b} - boldsymbol{c}|) confirms the result. Thus, the minimum value is:[boxed{sqrt{7} - 1}],<think>Okay, so I have this problem here. Let me read it again to make sure I understand. The problem says that vectors a and b satisfy |a| = |b| = a · b = 2. Then there's another condition: (a - c) · (b - c) = 0. I need to find the minimum value of |2b - c|. Alright, let's break this down step by step.First, I know that |a| and |b| are both 2. Also, the dot product of a and b is 2. That gives me some information about the angle between vectors a and b. The dot product formula is a · b = |a||b|cosθ, where θ is the angle between them. Since a · b = 2 and |a| = |b| = 2, plugging in the values, we get 2 = 2*2*cosθ. So, cosθ = 2/(2*2) = 2/4 = 1/2. Therefore, θ must be 60 degrees or π/3 radians. That's a key point; the angle between a and b is 60 degrees.Now, moving on to the next part: (a - c) · (b - c) = 0. This equation means that the vectors (a - c) and (b - c) are perpendicular to each other. So, geometrically, the point c lies somewhere such that when you connect c to a and c to b, those two vectors form a right angle. Hmm, this reminds me of the circle theorem where if you have a right angle subtended by two points, the set of all such points c lies on a circle with diameter AB. But here, the points are vectors a and b, so maybe c lies on a circle with diameter AB in the vector space? Not sure if that's the right analogy, but it's something to think about.But maybe I should approach this algebraically. Let's expand the equation (a - c) · (b - c) = 0. Expanding the dot product:a · b - a · c - b · c + c · c = 0.We know a · b = 2, so substituting that in:2 - (a · c + b · c) + |c|² = 0.Let me rewrite this:|c|² - (a · c + b · c) + 2 = 0.Hmm, maybe I can factor out the c terms. Let's see:|c|² - c · (a + b) + 2 = 0.So, this is a quadratic equation in terms of c. The equation resembles |c|² - c · (a + b) + 2 = 0. Maybe I can complete the square here to find the set of all possible c vectors that satisfy this equation.Completing the square with vectors might be a bit tricky, but let's try. Let me think of this as |c|² - c · (a + b) = -2. To complete the square, we can add and subtract the square of half the coefficient of c · (a + b). The coefficient here is 1 (since it's -c · (a + b)), so half of that is 1/2. Then, the square of that is (1/2)^2 |a + b|². Wait, maybe not exactly. Let me recall the formula for completing the square in vector terms.In general, for a vector c, |c - d|² = |c|² - 2c · d + |d|². So, if we have |c|² - c · (a + b), that's similar to |c|² - 2c · ((a + b)/2). So, completing the square would give:|c - (a + b)/2|² - |(a + b)/2|² = -2.Therefore, moving the |(a + b)/2|² term to the right side:|c - (a + b)/2|² = |(a + b)/2|² - 2.So, this is the equation of a sphere (since we're dealing with vectors in space, but since all vectors are in the plane defined by a and b, it's a circle) with center at (a + b)/2 and radius sqrt(|(a + b)/2|² - 2). Therefore, all vectors c lie on this circle.Let me compute |a + b|² first. Since |a| = |b| = 2 and a · b = 2, we can compute |a + b|² = |a|² + 2a · b + |b|² = 4 + 4 + 4 = 12. Wait, wait, hold on. |a + b|² = |a|² + 2a · b + |b|². Since |a| = |b| = 2, so |a|² = 4, |b|² = 4, and a · b = 2. Therefore, |a + b|² = 4 + 2*2 + 4 = 4 + 4 + 4 = 12. So, |a + b| = sqrt(12) = 2*sqrt(3). Therefore, |(a + b)/2|² is (|a + b|²)/4 = 12/4 = 3. Therefore, the radius squared is 3 - 2 = 1, so the radius is 1.Therefore, the set of all c satisfying (a - c) · (b - c) = 0 is a circle with center at (a + b)/2 and radius 1. So, c lies on this circle.Now, the problem asks for the minimum value of |2b - c|. So, we need to find the minimum distance between the vector c (lying on the circle centered at (a + b)/2 with radius 1) and the vector 2b. Alternatively, geometrically, this is equivalent to finding the minimum distance from the point 2b to any point on the circle.To find the minimum distance from a point to a circle, we can use the formula: if the distance from the point to the center of the circle is d, then the minimum distance is |d - r|, where r is the radius of the circle. This is because the closest point on the circle to the external point lies along the line connecting the external point to the center of the circle. So, if the distance from the external point to the center is greater than the radius, the minimum distance is d - r; if it's less, then the minimum distance would be r - d, but since the external point is outside the circle, it's d - r. Wait, but actually, if the point is inside the circle, the minimum distance would be r - d, but if the point is outside, it's d - r. But since we need the minimum distance regardless, but in our case, we need to check whether 2b is inside or outside the circle.So, let's compute the distance between 2b and the center of the circle, which is (a + b)/2. Let me compute vector from 2b to (a + b)/2: (a + b)/2 - 2b = (a + b - 4b)/2 = (a - 3b)/2. Therefore, the distance is |(a - 3b)/2|.Let me compute |a - 3b| first. |a - 3b|² = |a|² + 9|b|² - 6a · b. Since |a| = |b| = 4 (Wait, wait, no. Wait, |a| is 2, so |a|² is 4. Similarly, |b|² is 4. And a · b is 2. So, |a - 3b|² = 4 + 9*4 - 6*2 = 4 + 36 - 12 = 28. Therefore, |a - 3b| = sqrt(28) = 2*sqrt(7). Therefore, the distance from 2b to the center is |(a - 3b)/2| = sqrt(28)/2 = (2*sqrt(7))/2 = sqrt(7). So, sqrt(7) ≈ 2.645. The radius of the circle is 1. Since sqrt(7) > 1, the point 2b is outside the circle. Therefore, the minimum distance from 2b to the circle is sqrt(7) - 1. Therefore, the minimum value of |2b - c| is sqrt(7) - 1. Is that right?Wait, let me confirm. So, the center of the circle is (a + b)/2, and the radius is 1. The point in question is 2b. The distance between 2b and the center is sqrt(7). Since the radius is 1, the closest point on the circle to 2b would be along the line connecting 2b and the center. Therefore, subtract the radius from the distance: sqrt(7) - 1. Therefore, the minimal |2b - c| is sqrt(7) - 1. Therefore, the answer should be sqrt(7) - 1. So, boxed, that would be boxed{sqrt{7} - 1}.Wait, but let me double-check all steps to be sure. Let's start from the beginning.Given |a| = |b| = 2, a · b = 2. So, angle between a and b is 60 degrees. Then, (a - c) · (b - c) = 0, which expands to |c|² - c · (a + b) + 2 = 0. Then, completing the square: |c - (a + b)/2|² = |(a + b)/2|² - 2. Compute |(a + b)/2|²: |a + b|² is 12, so divided by 4 is 3. Then, 3 - 2 = 1. So, the radius is 1. So, center at (a + b)/2, radius 1. So, c is on that circle. Then, |2b - c| is the distance from c to 2b. So, center of the circle is (a + b)/2. So, compute the distance between (a + b)/2 and 2b. Let's compute vector (a + b)/2 - 2b = (a + b - 4b)/2 = (a - 3b)/2. The norm is |a - 3b|/2. Then, compute |a - 3b|²: |a|² + 9|b|² - 6a · b = 4 + 36 - 12 = 28. So, |a - 3b| is 2√7. Therefore, |(a - 3b)/2| is √7. So, distance from center to 2b is √7. Since radius is 1, then minimum |2b - c| is √7 - 1. That seems correct.But let me verify once again with coordinates. Maybe using coordinate geometry. Let's assign coordinates to vectors a and b to make it concrete. Since they have magnitude 2 and the angle between them is 60 degrees, let's place vector a along the x-axis. So, a = (2, 0). Then, vector b makes 60 degrees with a, so its components would be (2cos60°, 2sin60°) = (2*(1/2), 2*(√3/2)) = (1, √3). So, a = (2, 0), b = (1, √3). Then, center of the circle is (a + b)/2 = (2 + 1, 0 + √3)/2 = (3/2, √3/2). The radius is 1.Now, 2b would be 2*(1, √3) = (2, 2√3). The center of the circle is (3/2, √3/2). So, let's compute the distance between (2, 2√3) and (3/2, √3/2). The difference in x-coordinates: 2 - 3/2 = 1/2. The difference in y-coordinates: 2√3 - √3/2 = (4√3 - √3)/2 = (3√3)/2. So, the distance is sqrt( (1/2)^2 + ( (3√3)/2 )^2 ) = sqrt( 1/4 + (27/4) ) = sqrt(28/4) = sqrt(7). So, that confirms the distance from 2b to the center is sqrt(7). Then, subtracting the radius 1, the minimal distance is sqrt(7) - 1. Therefore, the minimal |2b - c| is sqrt(7) - 1. So, the answer is indeed boxed{sqrt{7} - 1}.Alternatively, maybe there's another approach using projections or parametric expressions. Let's see. If we parameterize c as lying on the circle, then c = (a + b)/2 + d, where |d| = 1. Then, |2b - c| = |2b - (a + b)/2 - d| = |(4b - a - b)/2 - d| = |(3b - a)/2 - d|. So, this is the norm of ( (3b - a)/2 - d ). Since d is a vector of length 1, the minimum value of this norm is | | (3b - a)/2 | - |d| |. Since |d| = 1, it's | |(3b - a)/2 | - 1 |. Since the minimum of |x - d| where |d|=1 is | |x| - 1 | if x and d are colinear. Wait, but actually, the minimal value is | |x| - 1 | if x is a vector and d is any vector of length 1. Because by the triangle inequality, |x - d| ≥ | |x| - |d| | = | |x| - 1 |. So, equality occurs when d is in the direction of x. Therefore, the minimal |x - d| is | |x| - 1 |.In this case, x is (3b - a)/2. Therefore, compute |(3b - a)/2|. We already computed |3b - a| earlier when finding |a - 3b|. Wait, |3b - a| is the same as |a - 3b|, which is sqrt(28). Therefore, |(3b - a)/2| = sqrt(28)/2 = sqrt(7). Therefore, the minimal |x - d| is |sqrt(7) - 1|. Since sqrt(7) > 1, it's sqrt(7) - 1. Therefore, the minimal |2b - c| is sqrt(7) - 1. This confirms the previous result.Alternatively, maybe using Lagrange multipliers? But that might be more complicated. Let's see. Suppose we need to minimize |2b - c| subject to the constraint (a - c) · (b - c) = 0. To use Lagrange multipliers, we can set up the function f(c) = |2b - c|² (squared to make it easier) and the constraint g(c) = (a - c) · (b - c) = 0. Then, set up the Lagrangian: L(c, λ) = |2b - c|² + λ[(a - c) · (b - c)].Taking derivatives with respect to c and λ. Let me recall that the derivative of |v - c|² with respect to c is -2(v - c). So, derivative of |2b - c|² is -2(2b - c). The derivative of the constraint term λ[(a - c)·(b - c)] with respect to c is λ[ - (a - c) - (b - c) ] = λ[ -a - b + 2c ]. Therefore, setting derivative of L with respect to c equal to zero:-2(2b - c) + λ(-a - b + 2c) = 0.So, -4b + 2c - λa - λb + 2λc = 0.Grouping terms:(2c + 2λc) + (-4b - λb - λa) = 0.Factor c and b, a:c(2 + 2λ) - b(4 + λ) - λa = 0.So,c(2 + 2λ) = b(4 + λ) + λa.Therefore,c = [ (4 + λ)b + λa ] / (2 + 2λ).Hmm, this seems a bit messy. Let's denote μ = λ/(2 + 2λ). Wait, maybe not. Let's see if we can express c in terms of a and b. Let me write c as:c = [ (4 + λ)b + λa ] / [ 2(1 + λ) ].Now, since c must satisfy the constraint (a - c)·(b - c) = 0, we can substitute this expression for c into the constraint equation.Alternatively, maybe express c as a combination of a and b. Since a and b are vectors in a plane, and given the problem's symmetry, c can be expressed as a linear combination of a and b. Let me assume c = pa + qb, where p and q are scalars. Then, we can substitute into the constraint equation.Let me try that approach. Let c = pa + qb.Then, (a - c)·(b - c) = 0.Compute a - c = a - pa - qb = (1 - p)a - qb.Compute b - c = b - pa - qb = -pa + (1 - q)b.Then, the dot product is [(1 - p)a - qb] · [-pa + (1 - q)b].Compute this:(1 - p)(-p)|a|² + (1 - p)(1 - q)a · b - q(-p)b · a - q(1 - q)|b|².Simplify term by term.First term: (1 - p)(-p)|a|² = -p(1 - p)|a|². Since |a|² = 4.Second term: (1 - p)(1 - q)(a · b) = (1 - p)(1 - q)*2.Third term: -q(-p)(a · b) = pq*2.Fourth term: -q(1 - q)|b|² = -q(1 - q)*4.Therefore, combining all terms:-4p(1 - p) + 2(1 - p)(1 - q) + 2pq - 4q(1 - q) = 0.Let me expand this:First term: -4p + 4p².Second term: 2[(1 - p)(1 - q)] = 2[1 - p - q + pq].Third term: 2pq.Fourth term: -4q + 4q².So, combining:-4p + 4p² + 2 - 2p - 2q + 2pq + 2pq -4q + 4q² = 0.Combine like terms:4p² + 4q² + (-4p -2p) + (-2q -4q) + (2pq + 2pq) + 2 = 0.Simplify:4p² + 4q² -6p -6q + 4pq + 2 = 0.Divide entire equation by 2 to simplify:2p² + 2q² -3p -3q + 2pq +1 = 0.Hmm, this seems a bit complex. Let me check my algebra again to see if I made an error.Wait, let's re-express the original expression step by step.First term: -4p(1 - p) = -4p + 4p².Second term: 2(1 - p)(1 - q) = 2(1 - p - q + pq) = 2 - 2p - 2q + 2pq.Third term: 2pq.Fourth term: -4q(1 - q) = -4q + 4q².Now, summing all these:(-4p + 4p²) + (2 - 2p - 2q + 2pq) + (2pq) + (-4q + 4q²) =4p² + 4q² + (-4p -2p) + (-2q -4q) + (2pq + 2pq) + 2 =4p² + 4q² -6p -6q + 4pq + 2 = 0. Yes, that's correct.So, 4p² + 4q² -6p -6q + 4pq + 2 = 0.Divide by 2: 2p² + 2q² -3p -3q + 2pq +1 =0.Hmm, tricky. Let me see if we can find p and q such that this equation holds. Maybe by assuming some symmetry? For instance, perhaps p = q? Let's test p = q.If p = q, then:2p² + 2p² -3p -3p + 2p² +1 =0 ⇒ 6p² -6p +1 =0.Solving quadratic equation: 6p² -6p +1=0. Discriminant: 36 -24=12. So, p=(6±sqrt(12))/12=(6±2*sqrt(3))/12=(3±sqrt(3))/6. Hmm, possible solutions, but not sure if they lead to minimal |2b - c|.Alternatively, maybe set up another equation. Since we need to minimize |2b - c| where c = pa + qb. So, |2b - c|² = |2b - pa - qb|² = | -pa + (2 - q)b |².Compute this squared magnitude:p²|a|² + (2 - q)²|b|² - 2p(2 - q)a · b.Given |a|² = |b|² =4, a · b=2.Therefore:4p² + 4(2 - q)² - 4p(2 - q).So, 4p² + 4(4 -4q + q²) -8p +4pq.Simplify:4p² +16 -16q +4q² -8p +4pq.Combine like terms:4p² +4q² +4pq -8p -16q +16.So, that's the expression we need to minimize subject to 2p² + 2q² -3p -3q + 2pq +1 =0.This seems complicated, but maybe we can use Lagrange multipliers here. Let me set up the Lagrangian:L(p, q, λ) = 4p² +4q² +4pq -8p -16q +16 + λ(2p² + 2q² -3p -3q + 2pq +1).Take partial derivatives with respect to p, q, and λ, set them to zero.First, derivative with respect to p:dL/dp = 8p +4q -8 + λ(4p -3 +2q) =0.Derivative with respect to q:dL/dq = 8q +4p -16 + λ(4q -3 +2p) =0.Derivative with respect to λ:2p² + 2q² -3p -3q + 2pq +1 =0.So, we have three equations:1. 8p +4q -8 + λ(4p -3 +2q) =0.2. 8q +4p -16 + λ(4q -3 +2p) =0.3. 2p² + 2q² -3p -3q + 2pq +1 =0.This system of equations looks quite involved, but perhaps we can subtract the first equation from the second to eliminate some terms.Let me subtract equation 1 from equation 2:[8q +4p -16 + λ(4q -3 +2p)] - [8p +4q -8 + λ(4p -3 +2q)] =0.Simplify:8q +4p -16 -8p -4q +8 + λ(4q -3 +2p -4p +3 -2q)=0.Simplify term by term:(8q -4q) + (4p -8p) + (-16 +8) + λ( (4q -2q) + (-3 +3) + (2p -4p) ) =0.Which is:4q -4p -8 + λ(2q -2p) =0.Factor:-4(p - q) -8 + 2λ(q - p)=0.Let me factor out (p - q):-4(p - q) -8 -2λ(p - q)=0.Factor (p - q):[-4 -2λ](p - q) -8=0.Let me write this as:(-4 -2λ)(p - q) =8.So,(p - q) = 8 / (-4 -2λ) = -4/(2 + λ).Hmm, this gives a relationship between p and q in terms of λ. Let's denote this as equation 4.Now, let's look back at equations 1 and 2. Maybe express them in terms of (p - q). Let's see.From equation 1:8p +4q -8 + λ(4p -3 +2q)=0.Let me express this as:8p +4q -8 +4λ p +2λ q -3λ=0.Group terms:(8 +4λ)p + (4 +2λ)q -8 -3λ=0.Similarly, equation 2:8q +4p -16 +4λ q +2λ p -3λ=0.Group terms:(4 +2λ)p + (8 +4λ)q -16 -3λ=0.So, equations 1 and 2 can be written as:(8 +4λ)p + (4 +2λ)q =8 +3λ. (Equation 1a)(4 +2λ)p + (8 +4λ)q =16 +3λ. (Equation 2a)Now, let's write these two equations as a system:Let me denote coefficients:A =8 +4λ, B=4 +2λ, C=4 +2λ, D=8 +4λ.Wait, the system is:A p + B q =8 +3λ,C p + D q =16 +3λ.But A =8 +4λ, B=4 +2λ,C=4 +2λ, D=8 +4λ.So, the system is:[8 +4λ, 4 +2λ] [p] = [8 +3λ][4 +2λ, 8 +4λ] [q] [16 +3λ]To solve this system, we can use the method of determinants (Cramer's Rule) or subtract equations. Let's subtract equation 1a multiplied by some factor from equation 2a to eliminate variables.Alternatively, notice that the matrix is symmetric. Let me denote:Let’s write equation 1a: (8 +4λ)p + (4 +2λ)q =8 +3λ.Equation 2a: (4 +2λ)p + (8 +4λ)q =16 +3λ.Let’s multiply equation 1a by 2:2*(8 +4λ)p + 2*(4 +2λ)q = 2*(8 +3λ).Which is:(16 +8λ)p + (8 +4λ)q =16 +6λ.Now, subtract equation 2a from this:[(16 +8λ)p + (8 +4λ)q] - [(4 +2λ)p + (8 +4λ)q] = (16 +6λ) - (16 +3λ).Simplifying left side:(16 +8λ -4 -2λ)p + (8 +4λ -8 -4λ)q = (12 +6λ)p.Right side:16 +6λ -16 -3λ =3λ.Therefore, (12 +6λ)p =3λ.Divide both sides by 3:(4 +2λ)p = λ.Thus,p = λ/(4 +2λ).Similarly, from equation 4:p - q = -4/(2 + λ).Thus, q = p +4/(2 + λ).Substitute p = λ/(4 +2λ):q = λ/(4 +2λ) +4/(2 + λ).Let me compute this:q = [λ +4*(4 +2λ)/(2 + λ)]/(4 +2λ).Wait, no, wait. Let me do step by step:p = λ/(4 + 2λ) = λ/(2*(2 + λ)) = (λ/2)/(2 + λ).Then, q = p +4/(2 + λ) = (λ/2)/(2 + λ) +4/(2 + λ) = [λ/2 +4]/(2 + λ) = (λ +8)/[2(2 + λ)].So, q = (λ +8)/(2*(2 + λ)).Now, substitute p and q into equation 1a or 2a. Let's pick equation 1a:(8 +4λ)p + (4 +2λ)q =8 +3λ.Substitute p = λ/(4 +2λ) and q = (λ +8)/(2*(2 + λ)).First, compute each term:(8 +4λ)p = (8 +4λ)*(λ)/(4 +2λ) = [ (8 +4λ)/ (4 +2λ) ) ] * λ = [ (4*(2 + λ))/(2*(2 + λ)) ) ] * λ = [4/2] * λ = 2λ.Similarly, (4 +2λ)q = (4 +2λ)*(λ +8)/(2*(2 + λ)) = [ (2*(2 + λ))*(λ +8) ) / (2*(2 + λ)) ) ] = (λ +8).Therefore, left side: 2λ + (λ +8) =3λ +8.Equation 1a says this equals8 +3λ.So,3λ +8 =8 +3λ.Which is an identity. So, no new information. Therefore, our expressions for p and q in terms of λ satisfy equation 1a. So, we need to use the constraint equation 3: 2p² + 2q² -3p -3q + 2pq +1 =0.Substitute p = λ/(4 +2λ) and q = (λ +8)/(2*(2 + λ)) into this equation.This seems complicated, but perhaps we can let t =2 + λ for substitution.Let t =2 + λ. Then, λ = t -2.Then, p = (t -2)/(4 +2(t -2)) = (t -2)/(4 +2t -4) = (t -2)/(2t).Similarly, q = ( (t -2) +8 )/(2*t ) = (t +6)/(2t).So, p = (t -2)/(2t), q = (t +6)/(2t).Now, substitute into equation 3:2p² + 2q² -3p -3q + 2pq +1 =0.Compute each term:2p² =2*( (t -2)^2 / (4t²) )= ( (t -2)^2 )/(2t² )2q²=2*( (t +6)^2 / (4t²) )= ( (t +6)^2 )/(2t² )-3p= -3*(t -2)/(2t )-3q= -3*(t +6)/(2t )2pq=2*( (t -2)/2t )*( (t +6)/2t )=2*( (t -2)(t +6) )/(4t² )= ( (t -2)(t +6) )/(2t² )Adding all terms:[ (t -2)^2 + (t +6)^2 ]/(2t² ) -3*(t -2 + t +6)/(2t ) + ( (t -2)(t +6) )/(2t² ) +1 =0.Simplify each part:First part: numerator (t -2)^2 + (t +6)^2 = t² -4t +4 + t² +12t +36 =2t² +8t +40.Second part: -3*(2t +4)/(2t )= -3*(t +2)/t.Third part: (t -2)(t +6)=t² +4t -12.Fourth part: +1.So, putting together:(2t² +8t +40)/(2t² ) -3(t +2)/t + (t² +4t -12)/(2t² ) +1 =0.Simplify each term:First term: (2t² +8t +40)/(2t² ) =1 +4/t +20/t².Wait, no, let's divide each term by 2t²:= (2t²)/(2t²) +8t/(2t²) +40/(2t²)=1 +4/t +20/t².Second term: -3(t +2)/t = -3 -6/t.Third term: (t² +4t -12)/(2t² )=1/2 +2/t -6/t².Fourth term: +1.Adding all terms together:1 +4/t +20/t² -3 -6/t +1/2 +2/t -6/t² +1.Combine like terms:Constants:1 -3 +1/2 +1 = (1 -3 +1) +1/2 = (-1) +1/2 = -1/2.Terms with 1/t:4/t -6/t +2/t =0.Terms with 1/t²:20/t² -6/t² =14/t².So, the entire expression becomes:-1/2 +14/t² =0.Therefore,14/t² =1/2 ⇒ t²=28 ⇒ t=±√28=±2√7.Since t=2 + λ, and we are dealing with real numbers, t can be positive or negative. However, let's check the original problem. When we set c=pa + qb, there's no restriction on p and q being positive or negative. However, we need to see if t=2 + λ can be negative. If t is negative, then λ =t -2 would be even more negative. Let's check if these solutions make sense.So, t²=28 ⇒ t=±2√7. Let's take t=2√7 first. Then, λ =t -2=2√7 -2.Then, p=(t -2)/(2t )=(2√7 -2)/(2*2√7 )=(2(√7 -1 ))/(4√7 )=(√7 -1 )/(2√7 ).Similarly, q=(t +6 )/(2t )=(2√7 +6 )/(2*2√7 )=(2(√7 +3 ))/(4√7 )=(√7 +3 )/(2√7 ).Alternatively, rationalizing the denominators:p=(√7 -1 )/(2√7 )=( (√7 -1 )√7 )/(2*7 )=(7 -√7 )/14.Similarly, q=(√7 +3 )/(2√7 )=( (√7 +3 )√7 )/(2*7 )=(7 +3√7 )/14.So, p=(7 -√7 )/14, q=(7 +3√7 )/14.Similarly, if t=-2√7, then λ=t -2= -2√7 -2.Then, p=(t -2 )/(2t )=( -2√7 -2 -2 )/(2*(-2√7 ))=( -2√7 -4 )/( -4√7 )=(2√7 +4 )/(4√7 )=(√7 +2 )/(2√7 ).Similarly, q=(t +6 )/(2t )=( -2√7 +6 )/(2*(-2√7 ))=(6 -2√7 )/( -4√7 )=(2√7 -6 )/(4√7 )=(√7 -3 )/(2√7 ).But with t negative, does this lead to valid solutions? Let's check.First, let's take t=2√7. Then, c=pa +qb.Compute c:c = [(7 -√7)/14]a + [(7 +3√7)/14]b.Since a and b are vectors with |a|=|b|=2 and angle 60°, we could compute |2b -c|, but this might be tedious. Alternatively, since we already know from the previous methods that the minimal distance is sqrt(7)-1, let's check if this solution gives the same.But perhaps not necessary. Since t=2√7 gives a valid solution, and t=-2√7 gives another solution. However, since the original problem is about minimizing |2b -c|, which is a distance, it must have a unique minimum. However, depending on the position, there might be two points on the circle at minimal and maximal distance, but in our case, since 2b is outside the circle, the minimal distance is unique.But in our Lagrangian approach, we obtained two possible solutions. However, when t is negative, the point c might be in the opposite direction, leading to a larger distance, so perhaps only the positive t gives the minimal distance.But given that our previous geometric approach gave sqrt(7) -1, which is approximately 2.645 -1=1.645, and t=2√7≈5.291, which is a scalar in substitution, but the actual value of |2b -c| would be sqrt(7)-1≈1.645.Alternatively, since we have t=2√7, and with t=2 +λ, then lambda=2√7 -2. Then, using p and q expressions, compute c= pa + qb.But maybe it's easier to compute |2b -c| with c expressed in terms of a and b.But given that we already have two methods giving the same answer, and the coordinate geometry approach also confirms it, I think the answer is indeed sqrt(7)-1.Therefore, I think I've verified it sufficiently. The minimum value is sqrt(7) -1.Final AnswerThe minimum value of ( |2boldsymbol{b} - boldsymbol{c}| ) is boxed{sqrt{7} - 1}.</think>Given vectors (boldsymbol{a}) and (boldsymbol{b}) satisfy (|boldsymbol{a}| = |boldsymbol{b}| = boldsymbol{a} cdot boldsymbol{b} = 2). We need to find the minimum value of (|2boldsymbol{b} - boldsymbol{c}|) given that ((boldsymbol{a} - boldsymbol{c}) cdot (boldsymbol{b} - boldsymbol{c}) = 0).1. Determine the angle between (boldsymbol{a}) and (boldsymbol{b}): - Using the dot product formula, (boldsymbol{a} cdot boldsymbol{b} = |boldsymbol{a}||boldsymbol{b}|costheta), we find (costheta = frac{1}{2}), so (theta = frac{pi}{3}).2. Expand the given condition: - Expanding ((boldsymbol{a} - boldsymbol{c}) cdot (boldsymbol{b} - boldsymbol{c}) = 0) gives (|boldsymbol{c}|^2 - boldsymbol{c} cdot (boldsymbol{a} + boldsymbol{b}) + 2 = 0).3. Complete the square: - Rewrite the equation as (|boldsymbol{c} - frac{boldsymbol{a} + boldsymbol{b}}{2}|^2 = 1), indicating (boldsymbol{c}) lies on a circle with center (frac{boldsymbol{a} + boldsymbol{b}}{2}) and radius 1.4. Calculate the distance from (2boldsymbol{b}) to the center of the circle: - The center is (frac{boldsymbol{a} + boldsymbol{b}}{2}). The distance from (2boldsymbol{b}) to this center is (|frac{boldsymbol{a} - 3boldsymbol{b}}{2}|). - Compute (|boldsymbol{a} - 3boldsymbol{b}|^2 = 28), so the distance is (sqrt{7}).5. Determine the minimum distance: - The minimum distance from (2boldsymbol{b}) to the circle is (sqrt{7} - 1).Thus, the minimum value of (|2boldsymbol{b} - boldsymbol{c}|) is (boxed{sqrt{7} - 1}).]

🔑:2. sqrt{7}-1.Notice,cos langleboldsymbol{a}, boldsymbol{b}rangle=frac{boldsymbol{a} cdot boldsymbol{b}}{|boldsymbol{a}||boldsymbol{b}|}=frac{1}{2} Rightarrowlangleboldsymbol{a}, boldsymbol{b}rangle=frac{pi}{3} text {. }From this, we can set boldsymbol{b}=(2,0), boldsymbol{a}=(1, sqrt{3}).Let boldsymbol{c}=(m, n).begin{array}{l}text { By }(boldsymbol{a}-boldsymbol{c}) cdot(boldsymbol{b}-boldsymbol{c})=0 Rightarrow(1-m)(2-m)+(0-n)(sqrt{3}-n)=0 Rightarrowleft(m-frac{3}{2}right)^{2}+left(n-frac{sqrt{3}}{2}right)^{2}=1 .end{array}Let m=frac{3}{2}+cos alpha, n=frac{sqrt{3}}{2}+sin alpha.Also, 2 boldsymbol{b}-boldsymbol{c}=(4-m,-n), thenbegin{array}{l}|2 b-c|=sqrt{(4-m)^{2}+n^{2}} =sqrt{8-(5 cos alpha-sqrt{3} sin alpha)} =sqrt{8-2 sqrt{7} sin left(-alpha+arctan frac{5}{sqrt{3}}right)} geqslant sqrt{8-2 sqrt{7}}=sqrt{7}-1 .end{array}Therefore, |2 b-c|_{min }=sqrt{7}-1.

🧠:[<think>Okay, let's tackle this problem. So, we have a sequence defined by a₀ = 2019, and each subsequent term is the previous term raised to the 2019th power. We need to find the remainder when the sum of the first 52 terms (from a₀ to a₅₁) is divided by 856. Hmm, that seems pretty big, especially considering how quickly exponentials grow. But since we're working modulo 856, maybe there's a pattern or some property we can use to simplify this.First, let me note that 856 is the modulus here. Let's factorize 856 to see if we can apply the Chinese Remainder Theorem (CRT) later. Breaking it down: 856 ÷ 2 = 428, ÷2 again = 214, ÷2 again = 107. 107 is a prime number, right? So, 856 = 8 × 107. Therefore, if we can compute the remainder modulo 8 and modulo 107 separately, we can combine them using CRT to get the final result modulo 856. That might make things easier.So, the plan is to compute the sum S = a₀ + a₁ + ... + a₅₁ modulo 8 and modulo 107, then combine those results. Let's start with modulo 8.Modulo 8 Calculation:First, compute each aₖ mod 8. Let's start with a₀ = 2019. 2019 ÷ 8 is 252×8 = 2016, so 2019 ≡ 3 mod 8. So, a₀ ≡ 3 mod 8.Now, a₁ = a₀^2019. So, modulo 8, that's 3^2019 mod 8. Let's find the pattern for powers of 3 mod 8:3^1 ≡ 3 mod 83^2 ≡ 9 ≡ 1 mod 83^3 ≡ 3×1 ≡ 3 mod 83^4 ≡ 1 mod 8So, the cycle length is 2. Therefore, 3^even ≡ 1 mod 8, 3^odd ≡ 3 mod 8.2019 is odd, so 3^2019 ≡ 3 mod 8. Therefore, a₁ ≡ 3 mod 8.Wait, but a₁ = a₀^2019 ≡ 3^2019 ≡ 3 mod 8. Similarly, a₂ = a₁^2019 ≡ 3^2019 ≡ 3 mod 8. Hmm, so it looks like every term in the sequence is ≡ 3 mod 8. So, all aₖ ≡ 3 mod 8. Therefore, the sum S = a₀ + a₁ + ... + a₅₁ ≡ 52×3 mod 8.52×3 = 156. 156 ÷ 8 = 19×8 = 152, so 156 ≡ 4 mod 8. Therefore, S ≡ 4 mod 8.Alright, that's modulo 8 done. Now, modulo 107. 107 is a prime number, which might be helpful because we can use Fermat's Little Theorem.Modulo 107 Calculation:First, compute each aₖ mod 107. Let's start with a₀ = 2019. 2019 ÷ 107: 107×18 = 1926, 2019 - 1926 = 93. So, a₀ ≡ 93 mod 107.Now, a₁ = a₀^2019 mod 107. Since 107 is prime, by Fermat's Little Theorem, for any integer x not divisible by 107, x^(106) ≡ 1 mod 107. So, exponents can be reduced modulo 106.So, to compute 93^2019 mod 107, first find 2019 mod 106. Let's calculate 106×19 = 2014, so 2019 - 2014 = 5. Therefore, 2019 ≡ 5 mod 106. Thus, 93^2019 ≡ 93^5 mod 107.Now compute 93^5 mod 107. Let's compute step by step:First, 93 mod 107 is 93.Compute 93^2: 93×93. Let's compute 90×90 = 8100, 90×3 = 270, 3×90 = 270, 3×3=9. So, 93^2 = 8649. Now, 8649 mod 107. Let's divide 8649 by 107.107×80 = 8560. 8649 - 8560 = 89. So, 93^2 ≡ 89 mod 107.Now, 93^3 = 93^2 × 93 ≡ 89 × 93 mod 107. Let's compute 89×93. 90×90 = 8100, subtract 1×90 and 89×3: 8100 - 90 - 267 = 8100 - 357 = 7743. Wait, maybe better to compute directly:89×93: 80×93 + 9×93 = 7440 + 837 = 8277. Now, 8277 ÷ 107. Let's see 107×77 = 107×70 + 107×7 = 7490 + 749 = 8239. 8277 - 8239 = 38. So, 89×93 ≡ 38 mod 107. Thus, 93^3 ≡ 38 mod 107.Next, 93^4 = 93^3 × 93 ≡ 38 × 93 mod 107. 38×93: 30×93 = 2790, 8×93 = 744. 2790 + 744 = 3534. 3534 ÷ 107: 107×33 = 3531. 3534 - 3531 = 3. So, 38×93 ≡ 3 mod 107. Thus, 93^4 ≡ 3 mod 107.Then, 93^5 = 93^4 × 93 ≡ 3 × 93 ≡ 279 mod 107. 279 ÷ 107 = 2×107 = 214. 279 - 214 = 65. So, 93^5 ≡ 65 mod 107. Therefore, a₁ ≡ 65 mod 107.Alright, so a₁ ≡ 65 mod 107. Now, a₂ = a₁^2019 mod 107. Let's compute a₂.Again, using Fermat's Little Theorem, exponent 2019 can be reduced modulo 106. 2019 mod 106: Wait, we already computed earlier that 2019 ≡ 5 mod 106. Wait, no, actually, when we computed for a₁, we had the exponent 2019 for a₀^2019. But here, a₁ is 65, so a₂ = 65^2019 mod 107. So, again, exponent 2019 mod 106. But 2019 mod 106 is still 5, as before. Wait, actually, hold on. Let's verify.106 × 19 = 2014. 2019 - 2014 = 5. So yes, 2019 ≡ 5 mod 106. Therefore, 65^2019 ≡ 65^5 mod 107.Compute 65^5 mod 107. Let's compute step by step:First, 65 mod 107 is 65.65^2 = 4225. 4225 ÷ 107: 107×39 = 4173. 4225 - 4173 = 52. So, 65^2 ≡ 52 mod 107.65^3 = 65^2 × 65 ≡ 52 × 65 mod 107. 52×60 = 3120, 52×5 = 260, total 3120 + 260 = 3380. 3380 ÷ 107: 107×31 = 3317. 3380 - 3317 = 63. So, 65^3 ≡ 63 mod 107.65^4 = 65^3 × 65 ≡ 63 × 65 mod 107. 60×65 = 3900, 3×65 = 195, total 3900 + 195 = 4095. 4095 ÷ 107: 107×38 = 4066. 4095 - 4066 = 29. So, 65^4 ≡ 29 mod 107.65^5 = 65^4 × 65 ≡ 29 × 65 mod 107. 20×65=1300, 9×65=585. 1300 + 585 = 1885. 1885 ÷ 107: 107×17 = 1819. 1885 - 1819 = 66. So, 65^5 ≡ 66 mod 107. Therefore, a₂ ≡ 66 mod 107.Wait, interesting. So, a₀ ≡ 93, a₁ ≡ 65, a₂ ≡ 66. Let's see a₃. Let's compute a₃ = a₂^2019 mod 107. Again, exponent 2019 ≡ 5 mod 106. So, 66^5 mod 107.Compute 66^5 mod 107.66^2: 66×66 = 4356. 4356 ÷ 107: 107×40 = 4280. 4356 - 4280 = 76. So, 66^2 ≡ 76 mod 107.66^3 = 66^2 × 66 ≡ 76 × 66 mod 107. 70×66=4620, 6×66=396. Total 4620 + 396 = 5016. 5016 ÷ 107: 107×46 = 4922. 5016 - 4922 = 94. So, 66^3 ≡ 94 mod 107.66^4 = 66^3 × 66 ≡ 94 × 66 mod 107. 90×66=5940, 4×66=264. 5940 + 264 = 6204. 6204 ÷ 107: 107×57 = 107×50 + 107×7 = 5350 + 749 = 6099. 6204 - 6099 = 105. So, 66^4 ≡ 105 mod 107.66^5 = 66^4 × 66 ≡ 105 × 66 mod 107. 100×66=6600, 5×66=330. 6600 + 330 = 6930. 6930 ÷ 107: 107×64 = 6848. 6930 - 6848 = 82. So, 66^5 ≡ 82 mod 107. Therefore, a₃ ≡ 82 mod 107.Continuing, a₄ = a₃^2019 mod 107 ≡ 82^5 mod 107.Compute 82^5 mod 107.82^2: 82×82 = 6724. 6724 ÷ 107: 107×62 = 6634. 6724 - 6634 = 90. So, 82^2 ≡ 90 mod 107.82^3 = 82^2 × 82 ≡ 90 × 82 mod 107. 90×80=7200, 90×2=180. Total 7200 + 180 = 7380. 7380 ÷ 107: 107×68 = 7276. 7380 - 7276 = 104. So, 82^3 ≡ 104 mod 107.82^4 = 82^3 × 82 ≡ 104 × 82 mod 107. 100×82=8200, 4×82=328. Total 8200 + 328 = 8528. 8528 ÷ 107: 107×79 = 107×(70 + 9) = 7490 + 963 = 8453. 8528 - 8453 = 75. So, 82^4 ≡ 75 mod 107.82^5 = 82^4 × 82 ≡ 75 × 82 mod 107. 70×82=5740, 5×82=410. Total 5740 + 410 = 6150. 6150 ÷ 107: 107×57 = 6099. 6150 - 6099 = 51. So, 82^5 ≡ 51 mod 107. Therefore, a₄ ≡ 51 mod 107.a₅ = a₄^2019 mod 107 ≡ 51^5 mod 107.Compute 51^5 mod 107.51^2 = 2601. 2601 ÷ 107: 107×24 = 2568. 2601 - 2568 = 33. So, 51^2 ≡ 33 mod 107.51^3 = 51^2 × 51 ≡ 33 × 51 mod 107. 30×51=1530, 3×51=153. Total 1530 + 153 = 1683. 1683 ÷ 107: 107×15 = 1605. 1683 - 1605 = 78. So, 51^3 ≡ 78 mod 107.51^4 = 51^3 × 51 ≡ 78 × 51 mod 107. 70×51=3570, 8×51=408. Total 3570 + 408 = 3978. 3978 ÷ 107: 107×37 = 3959. 3978 - 3959 = 19. So, 51^4 ≡ 19 mod 107.51^5 = 51^4 × 51 ≡ 19 × 51 mod 107. 10×51=510, 9×51=459. Total 510 + 459 = 969. 969 ÷ 107: 107×9 = 963. 969 - 963 = 6. So, 51^5 ≡ 6 mod 107. Therefore, a₅ ≡ 6 mod 107.a₆ = a₅^2019 mod 107 ≡ 6^5 mod 107.Compute 6^5:6^2 = 36.6^3 = 36×6 = 216 ≡ 216 - 2×107 = 216 - 214 = 2 mod 107.6^4 = 2×6 = 12 mod 107.6^5 = 12×6 = 72 mod 107. So, a₆ ≡ 72 mod 107.a₇ = a₆^2019 mod 107 ≡ 72^5 mod 107.Compute 72^5 mod 107.First, 72^2: 72×72 = 5184. 5184 ÷ 107: 107×48 = 5136. 5184 - 5136 = 48. So, 72^2 ≡ 48 mod 107.72^3 = 72^2 × 72 ≡ 48 × 72 mod 107. 40×72=2880, 8×72=576. Total 2880 + 576 = 3456. 3456 ÷ 107: 107×32 = 3424. 3456 - 3424 = 32. So, 72^3 ≡ 32 mod 107.72^4 = 72^3 × 72 ≡ 32 × 72 mod 107. 30×72=2160, 2×72=144. Total 2160 + 144 = 2304. 2304 ÷ 107: 107×21 = 2247. 2304 - 2247 = 57. So, 72^4 ≡ 57 mod 107.72^5 = 72^4 × 72 ≡ 57 × 72 mod 107. 50×72=3600, 7×72=504. Total 3600 + 504 = 4104. 4104 ÷ 107: 107×38 = 4066. 4104 - 4066 = 38. So, 72^5 ≡ 38 mod 107. Therefore, a₇ ≡ 38 mod 107.a₈ = a₇^2019 mod 107 ≡ 38^5 mod 107.Compute 38^5 mod 107.38^2: 38×38 = 1444. 1444 ÷ 107: 107×13 = 1391. 1444 - 1391 = 53. So, 38^2 ≡ 53 mod 107.38^3 = 38^2 × 38 ≡ 53 × 38 mod 107. 50×38=1900, 3×38=114. Total 1900 + 114 = 2014. 2014 ÷ 107 = 18×107 = 1926. 2014 - 1926 = 88. So, 38^3 ≡ 88 mod 107.38^4 = 38^3 × 38 ≡ 88 × 38 mod 107. 80×38=3040, 8×38=304. Total 3040 + 304 = 3344. 3344 ÷ 107: 107×31 = 3317. 3344 - 3317 = 27. So, 38^4 ≡ 27 mod 107.38^5 = 38^4 × 38 ≡ 27 × 38 mod 107. 20×38=760, 7×38=266. Total 760 + 266 = 1026. 1026 ÷ 107: 107×9 = 963. 1026 - 963 = 63. So, 38^5 ≡ 63 mod 107. Therefore, a₈ ≡ 63 mod 107.a₉ = a₈^2019 mod 107 ≡ 63^5 mod 107.Compute 63^5 mod 107.63^2: 63×63 = 3969. 3969 ÷ 107: 107×37 = 3959. 3969 - 3959 = 10. So, 63^2 ≡ 10 mod 107.63^3 = 63^2 × 63 ≡ 10 × 63 = 630 ≡ 630 - 5×107 = 630 - 535 = 95 mod 107.63^4 = 63^3 × 63 ≡ 95 × 63 mod 107. 90×63=5670, 5×63=315. Total 5670 + 315 = 5985. 5985 ÷ 107: 107×55 = 5885. 5985 - 5885 = 100. So, 63^4 ≡ 100 mod 107.63^5 = 63^4 × 63 ≡ 100 × 63 = 6300 mod 107. 6300 ÷ 107: 107×58 = 6206. 6300 - 6206 = 94. So, 63^5 ≡ 94 mod 107. Therefore, a₉ ≡ 94 mod 107.a₁₀ = a₉^2019 mod 107 ≡ 94^5 mod 107.Compute 94^5 mod 107. Note that 94 ≡ -13 mod 107. Maybe using negative numbers could make computations easier.Alternatively, proceed step by step:94^2: 94×94. Let's compute 90×90=8100, 90×4=360, 4×90=360, 4×4=16. So, 94²=8100+360+360+16=8836. 8836 mod 107. Let's divide 8836 by 107:107×82 = 8774. 8836 - 8774 = 62. So, 94^2 ≡ 62 mod 107.94^3 = 94^2 × 94 ≡ 62 × 94 mod 107. 60×94=5640, 2×94=188. Total 5640 + 188 = 5828. 5828 ÷ 107: 107×54 = 5778. 5828 - 5778 = 50. So, 94^3 ≡ 50 mod 107.94^4 = 94^3 × 94 ≡ 50 × 94 mod 107. 50×90=4500, 50×4=200. Total 4500 + 200 = 4700. 4700 ÷ 107: 107×43 = 4601. 4700 - 4601 = 99. So, 94^4 ≡ 99 mod 107.94^5 = 94^4 × 94 ≡ 99 × 94 mod 107. 90×94=8460, 9×94=846. Total 8460 + 846 = 9306. 9306 ÷ 107: 107×86 = 9202. 9306 - 9202 = 104. So, 94^5 ≡ 104 mod 107. Therefore, a₁₀ ≡ 104 mod 107.Hmm, this is getting tedious. There must be a pattern or cycle here. Let me see the terms we have so far modulo 107:a₀: 93a₁: 65a₂: 66a₃: 82a₄: 51a₅: 6a₆: 72a₇: 38a₈: 63a₉: 94a₁₀: 104I wonder if this sequence starts repeating after some point. Let's compute a few more terms to check.a₁₀ ≡ 104 mod 107. Then a₁₁ = a₁₀^2019 mod 107 ≡ 104^5 mod 107.Compute 104^5 mod 107. 104 ≡ -3 mod 107. So, (-3)^5 mod 107 = -243 mod 107. 243 ÷ 107 = 2×107 = 214. 243 - 214 = 29. So, -243 ≡ -29 mod 107 ≡ 78 mod 107. Therefore, a₁₁ ≡ 78 mod 107.a₁₁ ≡ 78 mod 107. Then a₁₂ = 78^5 mod 107.78 mod 107 is 78.78^2: 78×78 = 6084. 6084 ÷ 107: 107×56 = 5992. 6084 - 5992 = 92. So, 78^2 ≡ 92 mod 107.78^3 = 78^2 × 78 ≡ 92 × 78 mod 107. 90×78=7020, 2×78=156. Total 7020 + 156 = 7176. 7176 ÷ 107: 107×67 = 7169. 7176 - 7169 = 7. So, 78^3 ≡ 7 mod 107.78^4 = 78^3 × 78 ≡ 7 × 78 = 546 ≡ 546 - 5×107 = 546 - 535 = 11 mod 107.78^5 = 78^4 × 78 ≡ 11 × 78 = 858 ≡ 858 - 8×107 = 858 - 856 = 2 mod 107. So, a₁₂ ≡ 2 mod 107.a₁₂ ≡ 2 mod 107. Then a₁₃ = 2^5 mod 107 = 32 mod 107.a₁₃ ≡ 32 mod 107. Then a₁₄ = 32^5 mod 107.Compute 32^5 mod 107.32^2 = 1024. 1024 mod 107: 107×9=963. 1024 - 963 = 61. So, 32^2 ≡ 61 mod 107.32^3 = 32^2 × 32 ≡ 61 × 32 = 1952 mod 107. 1952 ÷ 107: 107×18 = 1926. 1952 - 1926 = 26. So, 32^3 ≡ 26 mod 107.32^4 = 32^3 × 32 ≡ 26 × 32 = 832 mod 107. 832 ÷ 107: 107×7 = 749. 832 - 749 = 83. So, 32^4 ≡ 83 mod 107.32^5 = 32^4 × 32 ≡ 83 × 32 = 2656 mod 107. 2656 ÷ 107: 107×24 = 2568. 2656 - 2568 = 88. So, 32^5 ≡ 88 mod 107. Therefore, a₁₄ ≡ 88 mod 107.a₁₄ ≡ 88 mod 107. Then a₁₅ = 88^5 mod 107.88 ≡ -19 mod 107. Let's compute (-19)^5 mod 107.(-19)^2 = 361 ≡ 361 - 3×107 = 361 - 321 = 40 mod 107.(-19)^3 = (-19)^2 × (-19) ≡ 40 × (-19) = -760 mod 107. 760 ÷ 107 = 7×107 = 749. 760 - 749 = 11. So, -760 ≡ -11 mod 107 ≡ 96 mod 107.(-19)^4 = 96 × (-19) = -1824 mod 107. 1824 ÷ 107: 107×17 = 1819. 1824 - 1819 = 5. So, -1824 ≡ -5 mod 107 ≡ 102 mod 107.(-19)^5 = 102 × (-19) = -1938 mod 107. 1938 ÷ 107: 107×18 = 1926. 1938 - 1926 = 12. So, -1938 ≡ -12 mod 107 ≡ 95 mod 107. Therefore, a₁₅ ≡ 95 mod 107.Hmm, this seems like the terms are not repeating yet. Let's check if there's a cycle. Let's list the terms again:a₀:93, a₁:65, a₂:66, a₃:82, a₄:51, a₅:6, a₆:72, a₇:38, a₈:63, a₉:94, a₁₀:104, a₁₁:78, a₁₂:2, a₁₃:32, a₁₄:88, a₁₅:95, ...I don't see an obvious repetition here. Maybe we need to compute all terms up to a₅₁? That would take forever manually. There must be a smarter approach.Wait, each term aₖ is defined as aₖ₋₁^2019 mod 856. But since we're working modulo 107, and given that 856 = 8×107, we've split the modulus using CRT. However, since we already handled modulo 8, we can focus on modulo 107.But even better, perhaps we can find the period of the sequence modulo 107. Since each term is the previous term raised to the 2019th power modulo 107. Because 107 is prime, we can use Euler's theorem: for any a not divisible by 107, a^φ(107) ≡ 1 mod 107, where φ(107) = 106. Therefore, exponents can be reduced modulo 106. But in this case, the exponent is 2019, which we've previously established is ≡5 mod 106. So, each term is the previous term raised to the 5th power mod 107.Therefore, the sequence is a₀, a₀^5, (a₀^5)^5 = a₀^(5^2), ..., aₖ = a₀^(5^k) mod 107.Therefore, the general term is aₖ ≡ 93^(5^k) mod 107. So, we need to compute exponents of 93 raised to 5^k mod φ(107) = 106. Wait, but φ(107) = 106, so exponents mod 106. But 93 and 107 are coprime, so by Euler's theorem, 93^106 ≡ 1 mod 107. Therefore, the exponent 5^k can be reduced modulo 106. So, perhaps we can find the exponent 5^k mod 106, and then compute 93^(5^k mod 106) mod 107.Wait, but 93 and 107 are coprime, so exponents can be reduced modulo 106. Therefore, to compute 93^(5^k) mod 107, we can first compute 5^k mod 106, then compute 93^(that exponent) mod 107. Therefore, if we can find the exponent 5^k mod 106, then we can compute the term aₖ.Therefore, the problem reduces to computing exponents 5^k mod 106, for k from 0 to 51, and then computing 93^(those exponents) mod 107, summing them up, and then combining with the modulo 8 result.So, first, let's compute 5^k mod 106 for k from 0 to 51. Then, compute 93^(those exponents) mod 107. Then sum them.But 5^k mod 106. Let's see if there's a cycle in the exponents. Let's compute the multiplicative order of 5 modulo 106.First, factorize 106: 106 = 2×53. φ(106) = φ(2×53) = φ(2)×φ(53) = 1×52 = 52. Therefore, the multiplicative order of 5 modulo 106 divides 52. Let's compute the order.We need the smallest positive integer d such that 5^d ≡ 1 mod 106.Check divisors of 52: 1, 2, 4, 13, 26, 52.5^1 mod 106 = 5 ≠ 15^2 = 25 ≠ 15^4 = 625 mod 106. 106×5 = 530. 625 - 530 = 95 ≠ 15^13: Hmm, this might be tedious. Alternatively, maybe compute 5^d mod 2 and mod 53 separately.Since 106 = 2×53, 5^d ≡ 1 mod 2 and 5^d ≡ 1 mod 53.5 mod 2 ≡ 1, so 5^d ≡ 1 mod 2 for any d. So, the condition is automatically satisfied for modulus 2.Now, mod 53: We need 5^d ≡ 1 mod 53. The multiplicative order of 5 mod 53 divides φ(53) = 52. Let's find the order.Check divisors of 52: 1, 2, 4, 13, 26, 52.5^1 ≡ 5 ≠1 mod 535^2 ≡25 ≠15^4 = 625 ≡ 625 - 11×53 = 625 - 583 = 42 ≠15^13: Let's compute 5^13 mod 53.First, 5^1 = 55^2 =255^4 = (5^2)^2 =25^2=625≡625-11×53=625-583=42 mod535^8 = (5^4)^2=42^2=1764≡1764-33×53=1764-1749=15 mod535^13=5^8 × 5^4 × 5^1 ≡15×42×5 mod53. 15×42=630≡630-11×53=630-583=47. 47×5=235≡235-4×53=235-212=23 mod53≠15^26=(5^13)^2≡23^2=529≡529-9×53=529-477=52≡-1 mod53. So, 5^26 ≡ -1 mod53. Therefore, 5^52 ≡1 mod53. So, the order is 52.Therefore, the multiplicative order of 5 modulo 53 is 52. Hence, the multiplicative order modulo 106 is lcm(ord_2(5), ord_53(5)) = lcm(1,52) =52.Therefore, 5^52 ≡1 mod106. Therefore, the multiplicative order of 5 mod 106 is 52. Therefore, the exponents 5^k mod 106 will cycle every 52 terms. Wait, but 5^k mod 106 cycles with period 52. Therefore, since we need exponents for k from 0 to 51, 5^0, 5^1, ..., 5^51 mod106. Since the cycle is 52, these exponents will all be distinct modulo 106. So, the exponents 5^k mod106 for k=0 to 51 are all distinct.Therefore, we can't really find a cycle shorter than 52. That complicates things. Because for each term aₖ, we need to compute 93^(5^k mod106) mod107. Since each exponent is unique modulo 106, each term is a different power of 93 modulo 107. However, 93 is a primitive root modulo 107? Or does it have a smaller order?Wait, 107 is prime, so the multiplicative group mod 107 has order 106. The order of 93 modulo 107 divides 106. Let's compute the order. If 93 is a primitive root, then its order is 106. Otherwise, it's a factor of 106.Let's check. 93 mod107 =93. Let's compute 93^k mod107 and see if we can find the minimal d such that 93^d ≡1 mod107.But 93≡-14 mod107. Maybe that helps. Let's compute the order of -14 mod107.We need the smallest d where (-14)^d ≡1 mod107. The order must divide 106=2×53. So possible orders: 1,2,53,106.Check (-14)^1≡-14≡93≠1.(-14)^2=196≡196-1×107=89≡89≠1.(-14)^53. Since 53 is prime, and 106=2×53. If (-14)^53 ≡1 mod107, then order is 53. Otherwise, order is 106.Compute (-14)^53 mod107. Let's use Fermat's little theorem: since 107 is prime, (-14)^106≡1 mod107. Therefore, [(-14)^53]^2≡1 mod107. Therefore, (-14)^53≡±1 mod107.Compute (-14)^53 mod107. Let's compute exponent 53. Let's note that 14 is a quadratic residue mod107? Let's check if 14^53 ≡1 or -1.But since (-14)^53 ≡ (-1)^53 ×14^53 ≡ -14^53 mod107.So, if 14^53 ≡1 mod107, then (-14)^53≡-1 mod107.If 14^53≡-1 mod107, then (-14)^53≡1 mod107.So, let's compute 14^53 mod107. Note that 14^106≡1 mod107. Therefore, 14^53 ≡ ±1 mod107.Compute 14^53 mod107. Let's compute 14^53.Using exponentiation by squaring:First, express 53 in binary: 32 + 16 + 4 + 1 = 53. So, compute 14^1, 14^2, 14^4, 14^8, 14^16, 14^32.Compute:14^1 ≡14 mod107.14^2=196≡196-1×107=89 mod107.14^4=(14^2)^2=89^2=7921 mod107. 107×74=7918. 7921-7918=3. So, 14^4≡3 mod107.14^8=(14^4)^2=3^2=9 mod107.14^16=(14^8)^2=9^2=81 mod107.14^32=(14^16)^2=81^2=6561 mod107. 107×61=6527. 6561-6527=34. So, 14^32≡34 mod107.Now, 14^53=14^32 ×14^16 ×14^4 ×14^1 ≡34×81×3×14 mod107.Compute step by step:34×81=2754. 2754 ÷107: 107×25=2675. 2754 -2675=79. So, 34×81≡79 mod107.79×3=237. 237 -2×107=237-214=23. So, 79×3≡23 mod107.23×14=322. 322 -3×107=322-321=1. So, 23×14≡1 mod107.Therefore, 14^53≡1 mod107. Therefore, (-14)^53≡-1 mod107. So, (-14)^53≡-1 mod107. Therefore, the order of -14 modulo107 is 106, since (-14)^106≡1 mod107 and no smaller exponent. Therefore, 93 is a primitive root modulo107. So, the multiplicative order of 93 modulo107 is 106. Therefore, 93 generates all residues modulo107 except 0. Therefore, the powers of 93 modulo107 cycle with period 106.But in our case, each term aₖ ≡93^(5^k) mod107. Since 93 is a primitive root, each exponent 5^k mod106 gives a unique exponent, and since 5 and 106 are coprime? Wait, 5 and 106: gcd(5,106)=1? 106=2×53. 5 is coprime with both 2 and 53, so yes, gcd(5,106)=1. Therefore, the exponents 5^k mod106 cycle with period equal to the multiplicative order of 5 modulo106, which we found earlier is 52. Therefore, the exponents 5^k mod106 for k=0,1,...,51 are all distinct. Therefore, the terms aₖ =93^(5^k mod106) mod107 for k=0,...,51 are 52 distinct powers of 93 modulo107. Since 93 is a primitive root, each of these exponents corresponds to a unique residue modulo107. Therefore, the sum S = sum_{k=0}^{51} aₖ mod107 is the sum of 52 distinct powers of 93 modulo107.However, summing 52 distinct terms from a cyclic group of order 106... Hmm, this seems complex. Maybe there's a geometric series aspect here. Wait, since aₖ =93^(5^k) mod107, and aₖ₊₁ =aₖ^5 mod107. So, each term is the previous term raised to the 5th power. So, the sequence is a geometric progression in exponents: each term is the 5th power of the previous. Therefore, the exponents form a geometric progression: starting with 93^1, then 93^5, then 93^(5^2), ..., up to 93^(5^51). Therefore, the exponents are 5^0,5^1,...,5^51 mod106 (since 93 has order 106). So, the exponents are 5^k mod106 for k from0 to51.But since 5 and 106 are coprime, and the multiplicative order of 5 modulo106 is52, the exponents5^k mod106 cycle every52 terms. However, we are summing the first52 terms (k=0 to51). Since the exponents cycle every52, the exponents5^0 to5^51 mod106 are all distinct, as the cycle length is52. So, we have52 distinct exponents. Therefore, the terms aₖ are93 raised to52 distinct exponents modulo106, which correspond to52 distinct residues modulo107.Summing these terms seems challenging because they are all different. Maybe there's a pattern or a property we can use here. Alternatively, perhaps the sum telescopes or there's some symmetry.Alternatively, note that since each aₖ =a_{k-1}^5 mod107, and the sequence is a geometric progression of exponents, maybe we can relate the sum S =a₀ +a₁ +... +a_{51} mod107.But I don't see an immediate telescoping here. Alternatively, if we let S be the sum, then perhaps consider multiplying S by 93^5 and subtracting? But since each term is multiplied by 93^5 to get the next term. Wait, let's see:S = a₀ + a₁ + a₂ + ... + a_{51}Multiply both sides by 93^5:93^5 * S = a₀^5 + a₁^5 + a₂^5 + ... + a_{51}^5 = a₁ + a₂ + ... + a_{52}But S = a₀ + a₁ + ... +a_{51}, so 93^5*S - S = a_{52} - a₀Therefore, S*(93^5 -1) ≡ a_{52} - a₀ mod107But S = [a_{52} - a₀]/(93^5 -1) mod107Wait, but this requires that 93^5 -1 is invertible modulo107. Let's check if 93^5 ≡1 mod107.Earlier, we computed 93^5 ≡65 mod107 (since a₁ ≡65 mod107). So, 93^5 -1 ≡65 -1=64 mod107. 64 and107 are coprime (gcd(64,107)=1), so 64 has an inverse modulo107. Therefore, we can write S ≡ (a_{52} -a₀) * (64^{-1}) mod107.But we need to compute a_{52} mod107. However, in the original problem, we need the sum up to a_{51}. Wait, here, in the expression above, 93^5*S - S = a_{52} -a₀. So, S*(93^5 -1) ≡a_{52} - a₀ mod107. Therefore, S ≡(a_{52} -a₀)*(93^5 -1)^{-1} mod107.But we need a_{52}, which is term 52. However, our sum only goes up to a_{51}. So, the indices might be shifted. Let me check:Wait, S = a₀ +a₁ +...+a_{51}Multiply by 93^5: 93^5*S = a₀^5 +a₁^5 +...+a_{51}^5 =a₁ +a₂ +...+a_{52}Therefore, 93^5*S - S = (a₁ +a₂ +...+a_{52}) - (a₀ +a₁ +...+a_{51}) )= a_{52} -a₀Therefore, S*(93^5 -1) ≡a_{52} -a₀ mod107.Hence, S ≡(a_{52} -a₀)*(93^5 -1)^{-1} mod107.We can compute this. Let's compute each component:First, compute a_{52} mod107. Since aₖ =93^(5^k) mod107, so a_{52} =93^(5^{52}) mod107. But 5^{52} mod106: Since 5 has order52 modulo106, 5^{52} ≡1 mod106. Therefore, a_{52} ≡93^1 ≡93 mod107.Therefore, a_{52} ≡93 mod107.a₀ ≡93 mod107. Therefore, a_{52} -a₀ ≡93 -93 ≡0 mod107.Therefore, S*(93^5 -1) ≡0 mod107. But 93^5 -1 ≡64 mod107, which is invertible. Therefore, S ≡0 mod107.Wait, that's a key insight! Therefore, the sum S ≡0 mod107.But let's verify this carefully. If a_{52} ≡a₀, then S*(93^5 -1)≡0 mod107, and since 93^5 -1 is invertible, then S≡0 mod107.Yes, because 93^5 -1 ≡64 mod107, which has inverse. Therefore, S must be ≡0 mod107.Therefore, the sum modulo107 is 0. Therefore, combining with the earlier result that S ≡4 mod8, we need to find a number that is ≡0 mod107 and ≡4 mod8. Then, use Chinese Remainder Theorem to solve for x ≡0 mod107 and x≡4 mod8.Find x such that x ≡0 mod107 and x ≡4 mod8.Let x =107k. Then 107k ≡4 mod8. 107 mod8: 107 ÷13×8=104, 107-104=3. So, 107≡3 mod8. Therefore, 3k ≡4 mod8. Solve for k: 3k ≡4 mod8. Multiply both sides by inverse of 3 mod8. Since 3×3=9≡1 mod8, inverse is3. Therefore, k≡4×3=12≡4 mod8. So, k=8m +4. Therefore, x=107*(8m +4)=856m +428. Therefore, the minimal positive solution is x≡428 mod856.Therefore, the remainder when S is divided by856 is428.But let's confirm this. Since S≡0 mod107 and S≡4 mod8. So, x ≡0 mod107 and x≡4 mod8. Let’s check x=428.428 ÷107=4, 4×107=428. So, 428≡0 mod107.428 mod8: 428 ÷8=53×8=424, 428-424=4. So, 428≡4 mod8. Correct.Therefore, the remainder is428.Final Answerboxed{428}</think>To solve the problem, we need to compute the remainder when the sum (a_0 + a_1 + a_2 + dots + a_{51}) is divided by 856. The sequence is defined by (a_0 = 2019) and (a_n = a_{n-1}^{2019}) for all positive integers (n).First, we factorize 856 as (856 = 8 times 107). We use the Chinese Remainder Theorem (CRT) to compute the remainder modulo 8 and modulo 107 separately, then combine the results.Modulo 8 Calculation:1. (a_0 equiv 2019 equiv 3 mod 8)2. Each subsequent term (a_n) is (a_{n-1}^{2019} mod 8). Since (3^2 equiv 1 mod 8), and 2019 is odd, all terms (a_n equiv 3 mod 8).3. The sum of 52 terms each congruent to 3 modulo 8 is (52 times 3 = 156 equiv 4 mod 8).Modulo 107 Calculation:1. (a_0 equiv 2019 equiv 93 mod 107).2. Using Fermat's Little Theorem, exponents modulo 106 (since 107 is prime, (phi(107) = 106)). Each term (a_n equiv 93^{5^n} mod 107).3. The sequence of exponents (5^n mod 106) has a period of 52. Since we need terms up to (a_{51}), the exponents cover all residues modulo 106 once.4. The sum (S = a_0 + a_1 + dots + a_{51}) modulo 107 is found using properties of geometric series and CRT. It turns out (S equiv 0 mod 107).Combining Results with CRT:1. We need (x equiv 4 mod 8) and (x equiv 0 mod 107).2. Solving (107k equiv 4 mod 8), we find (k equiv 4 mod 8). Thus, (x = 107 times 4 = 428).The final remainder when the sum is divided by 856 is (boxed{428}).,<think>Okay, so I need to compute the remainder when the sum a₀ + a₁ + a₂ + ... + a₅₁ is divided by 856. The sequence is defined by a₀ = 2019 and each subsequent term is the previous term raised to the 2019th power. That means a₁ = a₀^2019 = 2019^2019, a₂ = a₁^2019 = (2019^2019)^2019 = 2019^(2019²), and so on up to a₅₁. That's a massive number, but since we're dealing with modular arithmetic, maybe there's a pattern or a way to simplify these exponents modulo 856.First, let me note that 856 factors into prime factors. Let me check: 856 divided by 2 is 428, divided by 2 again is 214, and again by 2 is 107. 107 is a prime number. So 856 = 8 × 107. Since 8 and 107 are coprime (their GCD is 1), I can use the Chinese Remainder Theorem. That is, compute the remainder modulo 8 and modulo 107 separately, then combine the results to find the remainder modulo 856.So the plan is:1. Compute the sum S = a₀ + a₁ + ... + a₅₁ modulo 8.2. Compute the same sum S modulo 107.3. Use the Chinese Remainder Theorem to find a number congruent to the computed remainders modulo 8 and 107, which will give the remainder modulo 856.Let's start with modulo 8.Step 1: Compute S mod 8First, compute each aₖ mod 8, then sum them up.Given a₀ = 2019. 2019 divided by 8: 8×252 = 2016, so 2019 - 2016 = 3. So a₀ ≡ 3 mod 8.Now, a₁ = a₀^2019. So 3^2019 mod 8. Let's find the pattern of powers of 3 modulo 8.3¹ ≡ 3 mod 83² ≡ 9 ≡ 1 mod 83³ ≡ 3×1 ≡ 3 mod 83⁴ ≡ 1 mod 8So the pattern cycles every 2 exponents: 3, 1, 3, 1,...2019 divided by 2: 2019 is odd, so exponent 2019 ≡ 1 mod 2. Therefore, 3^2019 ≡ 3^1 ≡ 3 mod 8. So a₁ ≡ 3 mod 8.Similarly, a₂ = a₁^2019 ≡ 3^2019 ≡ 3 mod 8. So in fact, all aₖ ≡ 3 mod 8. Therefore, each term in the sum is 3 mod 8.There are 52 terms (from a₀ to a₅₁). So the sum S ≡ 52 × 3 mod 8.52 × 3 = 156. 156 divided by 8: 8×19=152, so 156 - 152 = 4. Therefore, S ≡ 4 mod 8.Step 2: Compute S mod 107This seems more complicated. Let's compute each aₖ mod 107 and sum them up.First, a₀ = 2019. 2019 mod 107: Let's divide 2019 by 107.107 × 18 = 1926. 2019 - 1926 = 93. So a₀ ≡ 93 mod 107.Next, a₁ = a₀^2019 mod 107. That is, 93^2019 mod 107. Since 107 is prime, by Fermat's Little Theorem, 93^(106) ≡ 1 mod 107. So we can reduce the exponent 2019 modulo 106.First, compute 2019 mod 106.106 × 19 = 2014. 2019 - 2014 = 5. So 2019 ≡ 5 mod 106. Therefore, 93^2019 ≡ 93^5 mod 107.Compute 93^5 mod 107. Let's compute step by step:First, note that 93 mod 107 is 93, which is equivalent to -14 mod 107 (since 107 - 93 = 14). So maybe easier to compute (-14)^5 mod 107.Compute (-14)^2 = 196. 196 mod 107: 107×1=107, 196-107=89. So (-14)^2 ≡ 89 mod 107.(-14)^3 = (-14)^2 × (-14) ≡ 89 × (-14) mod 107. 89×14=1246. 1246 mod 107: Let's divide 1246 by 107. 107×11=1177, 1246-1177=69. So 1246 ≡ 69 mod 107. Therefore, 89 × (-14) ≡ -69 mod 107. But -69 mod 107 is 107 - 69 = 38. So (-14)^3 ≡ 38 mod 107.(-14)^4 = (-14)^3 × (-14) ≡ 38 × (-14) mod 107. 38×14=532. 532 mod 107: 107×4=428, 532-428=104. So 532 ≡ 104 mod 107. Therefore, 38 × (-14) ≡ -104 mod 107, which is 107 - 104 = 3. So (-14)^4 ≡ 3 mod 107.(-14)^5 = (-14)^4 × (-14) ≡ 3 × (-14) ≡ -42 mod 107. Which is 107 - 42 = 65 mod 107. Therefore, 93^5 ≡ 65 mod 107. So a₁ ≡ 65 mod 107.Now, moving on to a₂ = a₁^2019 mod 107. That is, 65^2019 mod 107. Again, using Fermat's Little Theorem, exponent modulo 106. 2019 mod 106 is 5 as before. So 65^5 mod 107.Compute 65^5 mod 107. Let's compute:65 mod 107 is 65.65^2 = 4225. 4225 ÷ 107: 107×39=4173. 4225 - 4173 = 52. So 65^2 ≡ 52 mod 107.65^3 = 65^2 × 65 ≡ 52 × 65 mod 107. 52×65=3380. 3380 ÷ 107: 107×31=3317, 3380 - 3317 = 63. So 65^3 ≡ 63 mod 107.65^4 = 65^3 × 65 ≡ 63 × 65 mod 107. 63×65=4095. 4095 ÷ 107: 107×38=4066, 4095 - 4066=29. So 65^4 ≡ 29 mod 107.65^5 = 65^4 × 65 ≡ 29 × 65 mod 107. 29×65=1885. 1885 ÷ 107: 107×17=1819, 1885 - 1819=66. So 65^5 ≡ 66 mod 107. Therefore, a₂ ≡ 66 mod 107.Next, a₃ = a₂^2019 mod 107. So 66^2019 mod 107. Again, exponent 2019 mod 106 = 5. Compute 66^5 mod 107.66^2 = 4356. 4356 ÷ 107: 107×40=4280, 4356 - 4280=76. So 66^2 ≡ 76 mod 107.66^3 = 66^2 × 66 ≡ 76 × 66 mod 107. 76×66=5016. 5016 ÷ 107: 107×46=4922, 5016 - 4922=94. So 66^3 ≡ 94 mod 107.66^4 = 66^3 × 66 ≡ 94 × 66 mod 107. 94×66=6204. 6204 ÷ 107: 107×57=6099, 6204 - 6099=105. So 66^4 ≡ 105 mod 107.66^5 = 66^4 × 66 ≡ 105 × 66 mod 107. 105×66=6930. 6930 ÷ 107: 107×64=6848, 6930 - 6848=82. So 66^5 ≡ 82 mod 107. Therefore, a₃ ≡ 82 mod 107.Continuing, a₄ = a₃^2019 mod 107 = 82^5 mod 107.Compute 82^5 mod 107:82 mod 107 = 82.82^2 = 6724. 6724 ÷ 107: 107×62=6634, 6724 - 6634=90. So 82^2 ≡ 90 mod 107.82^3 = 82^2 × 82 ≡ 90 × 82 mod 107. 90×82=7380. 7380 ÷ 107: 107×68=7276, 7380 - 7276=104. So 82^3 ≡ 104 mod 107.82^4 = 82^3 × 82 ≡ 104 × 82 mod 107. 104×82=8528. 8528 ÷ 107: 107×79=8453, 8528 - 8453=75. So 82^4 ≡ 75 mod 107.82^5 = 82^4 × 82 ≡ 75 × 82 mod 107. 75×82=6150. 6150 ÷ 107: 107×57=6099, 6150 - 6099=51. So 82^5 ≡ 51 mod 107. Thus, a₄ ≡ 51 mod 107.Next, a₅ = a₄^2019 mod 107 = 51^5 mod 107.Compute 51^5 mod 107:51 mod 107 = 51.51^2 = 2601. 2601 ÷ 107: 107×24=2568, 2601 - 2568=33. So 51^2 ≡ 33 mod 107.51^3 = 51^2 × 51 ≡ 33 × 51 mod 107. 33×51=1683. 1683 ÷ 107: 107×15=1605, 1683 - 1605=78. So 51^3 ≡ 78 mod 107.51^4 = 51^3 × 51 ≡ 78 × 51 mod 107. 78×51=3978. 3978 ÷ 107: 107×37=3959, 3978 - 3959=19. So 51^4 ≡ 19 mod 107.51^5 = 51^4 × 51 ≡ 19 × 51 mod 107. 19×51=969. 969 ÷ 107: 107×9=963, 969 - 963=6. So 51^5 ≡ 6 mod 107. Therefore, a₅ ≡ 6 mod 107.a₆ = a₅^2019 mod 107 = 6^5 mod 107.Compute 6^5 mod 107:6^2 = 366^3 = 36 × 6 = 216 ≡ 216 - 2×107 = 216 - 214 = 2 mod 1076^4 = 2 × 6 = 12 mod 1076^5 = 12 × 6 = 72 mod 107. So a₆ ≡ 72 mod 107.a₇ = a₆^2019 mod 107 = 72^5 mod 107.Compute 72^5 mod 107:72 mod 107 = 72.72^2 = 5184. 5184 ÷ 107: 107×48=5136, 5184 - 5136=48. So 72^2 ≡ 48 mod 107.72^3 = 72^2 × 72 ≡ 48 × 72 mod 107. 48×72=3456. 3456 ÷ 107: 107×32=3424, 3456 - 3424=32. So 72^3 ≡ 32 mod 107.72^4 = 72^3 × 72 ≡ 32 × 72 mod 107. 32×72=2304. 2304 ÷ 107: 107×21=2247, 2304 - 2247=57. So 72^4 ≡ 57 mod 107.72^5 = 72^4 × 72 ≡ 57 × 72 mod 107. 57×72=4104. 4104 ÷ 107: 107×38=4066, 4104 - 4066=38. So 72^5 ≡ 38 mod 107. Therefore, a₇ ≡ 38 mod 107.a₈ = a₇^2019 mod 107 = 38^5 mod 107.Compute 38^5 mod 107:38 mod 107 = 38.38^2 = 1444. 1444 ÷ 107: 107×13=1391, 1444 - 1391=53. So 38^2 ≡ 53 mod 107.38^3 = 38^2 × 38 ≡ 53 × 38 mod 107. 53×38=2014. 2014 ÷ 107: 107×18=1926, 2014 - 1926=88. So 38^3 ≡ 88 mod 107.38^4 = 38^3 × 38 ≡ 88 × 38 mod 107. 88×38=3344. 3344 ÷ 107: 107×31=3317, 3344 - 3317=27. So 38^4 ≡ 27 mod 107.38^5 = 38^4 × 38 ≡ 27 × 38 mod 107. 27×38=1026. 1026 ÷ 107: 107×9=963, 1026 - 963=63. So 38^5 ≡ 63 mod 107. Therefore, a₈ ≡ 63 mod 107.a₉ = a₈^2019 mod 107 = 63^5 mod 107.Compute 63^5 mod 107:63 mod 107 = 63.63^2 = 3969. 3969 ÷ 107: 107×37=3959, 3969 - 3959=10. So 63^2 ≡ 10 mod 107.63^3 = 63^2 × 63 ≡ 10 × 63 = 630 mod 107. 630 ÷ 107: 107×5=535, 630 - 535=95. So 63^3 ≡ 95 mod 107.63^4 = 63^3 × 63 ≡ 95 × 63 mod 107. 95×63=5985. 5985 ÷ 107: 107×55=5885, 5985 - 5885=100. So 63^4 ≡ 100 mod 107.63^5 = 63^4 × 63 ≡ 100 × 63 mod 107. 100×63=6300. 6300 ÷ 107: 107×58=6206, 6300 - 6206=94. So 63^5 ≡ 94 mod 107. Therefore, a₉ ≡ 94 mod 107.a₁₀ = a₉^2019 mod 107 = 94^5 mod 107.Compute 94^5 mod 107. Note that 94 ≡ -13 mod 107.(-13)^2 = 169 ≡ 169 - 107 = 62 mod 107.(-13)^3 = (-13)^2 × (-13) ≡ 62 × (-13) = -806 mod 107. Let's compute -806 mod 107:107×7=749, 806 - 749=57, so -806 ≡ -57 mod 107 ≡ 50 mod 107.(-13)^4 = (-13)^3 × (-13) ≡ 50 × (-13) = -650 mod 107. 650 ÷ 107: 107×6=642, 650 - 642=8, so -650 ≡ -8 mod 107 ≡ 99 mod 107.(-13)^5 = (-13)^4 × (-13) ≡ 99 × (-13) = -1287 mod 107. 1287 ÷ 107: 107×12=1284, 1287 - 1284=3, so -1287 ≡ -3 mod 107 ≡ 104 mod 107. Therefore, 94^5 ≡ 104 mod 107. So a₁₀ ≡ 104 mod 107.a₁₁ = a₁₀^2019 mod 107 = 104^5 mod 107.104 ≡ -3 mod 107. So compute (-3)^5 mod 107.(-3)^5 = -243. -243 mod 107: 107×2=214, 243 - 214=29, so -243 ≡ -29 mod 107 ≡ 78 mod 107. Therefore, a₁₁ ≡ 78 mod 107.a₁₂ = a₁₁^2019 mod 107 = 78^5 mod 107.Compute 78^5 mod 107:78 ≡ -29 mod 107.(-29)^2 = 841. 841 ÷ 107: 107×7=749, 841 - 749=92. So (-29)^2 ≡ 92 mod 107.(-29)^3 = (-29)^2 × (-29) ≡ 92 × (-29) mod 107. 92×29=2668. 2668 ÷ 107: 107×24=2568, 2668 - 2568=100. So 92×29=2668 ≡ 100 mod 107. Therefore, (-29)^3 ≡ -100 mod 107 ≡ 7 mod 107.(-29)^4 ≡ 7 × (-29) ≡ -203 mod 107. 203 ÷ 107=1×107=107, 203 - 107=96, so -203 ≡ -96 mod 107 ≡ 11 mod 107.(-29)^5 ≡ 11 × (-29) ≡ -319 mod 107. 319 ÷ 107: 107×2=214, 319 - 214=105. So -319 ≡ -105 mod 107 ≡ 2 mod 107. Therefore, 78^5 ≡ 2 mod 107. So a₁₂ ≡ 2 mod 107.a₁₃ = a₁₂^2019 mod 107 = 2^5 mod 107. 2^5=32 mod 107. So a₁₃ ≡ 32 mod 107.a₁₄ = a₁₃^2019 mod 107 = 32^5 mod 107.Compute 32^5 mod 107:32^2=1024 mod 107. 1024 ÷ 107: 107×9=963, 1024 - 963=61. So 32^2 ≡ 61 mod 107.32^3=32×61=1952 mod 107. 1952 ÷ 107: 107×18=1926, 1952 - 1926=26. So 32^3 ≡ 26 mod 107.32^4=32×26=832 mod 107. 832 ÷ 107: 107×7=749, 832 - 749=83. So 32^4 ≡ 83 mod 107.32^5=32×83=2656 mod 107. 2656 ÷ 107: 107×24=2568, 2656 - 2568=88. So 32^5 ≡ 88 mod 107. Therefore, a₁₄ ≡ 88 mod 107.a₁₅ = a₁₄^2019 mod 107 = 88^5 mod 107.88 ≡ -19 mod 107. Compute (-19)^5 mod 107.(-19)^2=361 ≡ 361 - 3×107=361 - 321=40 mod 107.(-19)^3=(-19)^2 × (-19)=40×(-19)= -760 mod 107. 760 ÷ 107: 107×7=749, 760 - 749=11. So -760 ≡ -11 mod 107 ≡ 96 mod 107.(-19)^4=96 × (-19)= -1824 mod 107. 1824 ÷ 107: 107×17=1819, 1824 - 1819=5. So -1824 ≡ -5 mod 107 ≡ 102 mod 107.(-19)^5=102 × (-19)= -1938 mod 107. 1938 ÷ 107: 107×18=1926, 1938 - 1926=12. So -1938 ≡ -12 mod 107 ≡ 95 mod 107. Therefore, a₁₅ ≡ 95 mod 107.a₁₆ = a₁₅^2019 mod 107 = 95^5 mod 107.95 ≡ -12 mod 107.(-12)^2=144 ≡ 144 - 107=37 mod 107.(-12)^3=37×(-12)= -444 mod 107. 444 ÷ 107=4×107=428, 444 - 428=16. So -444 ≡ -16 mod 107 ≡ 91 mod 107.(-12)^4=91×(-12)= -1092 mod 107. 1092 ÷ 107: 107×10=1070, 1092 - 1070=22. So -1092 ≡ -22 mod 107 ≡ 85 mod 107.(-12)^5=85×(-12)= -1020 mod 107. 1020 ÷ 107: 107×9=963, 1020 - 963=57. So -1020 ≡ -57 mod 107 ≡ 50 mod 107. Therefore, a₁₆ ≡ 50 mod 107.a₁₇ = a₁₆^2019 mod 107 = 50^5 mod 107.Compute 50^5 mod 107:50^2=2500 mod 107. 2500 ÷ 107: 107×23=2461, 2500 - 2461=39. So 50^2 ≡ 39 mod 107.50^3=50×39=1950 mod 107. 1950 ÷ 107=18×107=1926, 1950 - 1926=24. So 50^3 ≡ 24 mod 107.50^4=50×24=1200 mod 107. 1200 ÷ 107=11×107=1177, 1200 - 1177=23. So 50^4 ≡ 23 mod 107.50^5=50×23=1150 mod 107. 1150 ÷ 107: 107×10=1070, 1150 - 1070=80. So 50^5 ≡ 80 mod 107. Therefore, a₁₇ ≡ 80 mod 107.a₁₈ = a₁₇^2019 mod 107 = 80^5 mod 107.80 ≡ -27 mod 107.(-27)^2=729 mod 107. 729 ÷ 107: 107×6=642, 729 - 642=87. So (-27)^2 ≡ 87 mod 107.(-27)^3=87×(-27)= -2349 mod 107. 2349 ÷ 107: 107×21=2247, 2349 - 2247=102. So -2349 ≡ -102 mod 107 ≡ 5 mod 107.(-27)^4=5×(-27)= -135 mod 107 ≡ -135 + 2×107= -135 + 214=79 mod 107.(-27)^5=79×(-27)= -2133 mod 107. 2133 ÷ 107: 107×19=2033, 2133 - 2033=100. So -2133 ≡ -100 mod 107 ≡ 7 mod 107. Therefore, a₁₈ ≡ 7 mod 107.a₁₉ = a₁₈^2019 mod 107 = 7^5 mod 107.Compute 7^5 mod 107:7^2=497^3=49×7=343 mod 107. 343 ÷ 107=3×107=321, 343 - 321=22. So 7^3 ≡ 22 mod 107.7^4=22×7=154 mod 107. 154 - 107=47. So 7^4 ≡ 47 mod 107.7^5=47×7=329 mod 107. 329 ÷ 107=3×107=321, 329 - 321=8. So 7^5 ≡ 8 mod 107. Therefore, a₁₉ ≡ 8 mod 107.a₂₀ = a₁₉^2019 mod 107 = 8^5 mod 107.8^2=648^3=64×8=512 mod 107. 512 ÷ 107=4×107=428, 512 - 428=84. So 8^3 ≡ 84 mod 107.8^4=84×8=672 mod 107. 672 ÷ 107=6×107=642, 672 - 642=30. So 8^4 ≡ 30 mod 107.8^5=30×8=240 mod 107. 240 - 2×107=240 - 214=26. So 8^5 ≡ 26 mod 107. Therefore, a₂₀ ≡ 26 mod 107.a₂₁ = a₂₀^2019 mod 107 = 26^5 mod 107.26^2=676 mod 107. 676 ÷ 107=6×107=642, 676 - 642=34. So 26^2 ≡ 34 mod 107.26^3=34×26=884 mod 107. 884 ÷ 107=8×107=856, 884 - 856=28. So 26^3 ≡ 28 mod 107.26^4=28×26=728 mod 107. 728 ÷ 107=6×107=642, 728 - 642=86. So 26^4 ≡ 86 mod 107.26^5=86×26=2236 mod 107. 2236 ÷ 107=20×107=2140, 2236 - 2140=96. So 26^5 ≡ 96 mod 107. Therefore, a₂₁ ≡ 96 mod 107.a₂₂ = a₂₁^2019 mod 107 = 96^5 mod 107.96 ≡ -11 mod 107.(-11)^2=121 ≡ 121 - 107=14 mod 107.(-11)^3=14×(-11)= -154 mod 107 ≡ -154 + 2×107= -154 + 214=60 mod 107.(-11)^4=60×(-11)= -660 mod 107. 660 ÷ 107=6×107=642, 660 - 642=18. So -660 ≡ -18 mod 107 ≡ 89 mod 107.(-11)^5=89×(-11)= -979 mod 107. 979 ÷ 107=9×107=963, 979 - 963=16. So -979 ≡ -16 mod 107 ≡ 91 mod 107. Therefore, a₂₂ ≡ 91 mod 107.a₂₃ = a₂₂^2019 mod 107 = 91^5 mod 107.91 ≡ -16 mod 107.(-16)^2=256 mod 107. 256 - 2×107=256 - 214=42 mod 107.(-16)^3=42×(-16)= -672 mod 107. 672 ÷ 107=6×107=642, 672 - 642=30. So -672 ≡ -30 mod 107 ≡ 77 mod 107.(-16)^4=77×(-16)= -1232 mod 107. 1232 ÷ 107=11×107=1177, 1232 - 1177=55. So -1232 ≡ -55 mod 107 ≡ 52 mod 107.(-16)^5=52×(-16)= -832 mod 107. 832 ÷ 107=7×107=749, 832 - 749=83. So -832 ≡ -83 mod 107 ≡ 24 mod 107. Therefore, a₂₃ ≡ 24 mod 107.a₂₄ = a₂₃^2019 mod 107 = 24^5 mod 107.24^2=576 mod 107. 576 ÷ 107=5×107=535, 576 - 535=41. So 24^2 ≡ 41 mod 107.24^3=41×24=984 mod 107. 984 ÷ 107=9×107=963, 984 - 963=21. So 24^3 ≡ 21 mod 107.24^4=21×24=504 mod 107. 504 ÷ 107=4×107=428, 504 - 428=76. So 24^4 ≡ 76 mod 107.24^5=76×24=1824 mod 107. 1824 ÷ 107=17×107=1819, 1824 - 1819=5. So 24^5 ≡ 5 mod 107. Therefore, a₂₄ ≡ 5 mod 107.a₂₅ = a₂₄^2019 mod 107 = 5^5 mod 107.5^2=255^3=25×5=125 ≡ 125 - 107=18 mod 107.5^4=18×5=90 mod 107.5^5=90×5=450 mod 107. 450 ÷ 107=4×107=428, 450 - 428=22. So 5^5 ≡ 22 mod 107. Therefore, a₂₅ ≡ 22 mod 107.a₂₆ = a₂₅^2019 mod 107 = 22^5 mod 107.22^2=484 mod 107. 484 ÷ 107=4×107=428, 484 - 428=56. So 22^2 ≡ 56 mod 107.22^3=56×22=1232 mod 107. 1232 ÷ 107=11×107=1177, 1232 - 1177=55. So 22^3 ≡ 55 mod 107.22^4=55×22=1210 mod 107. 1210 ÷ 107=11×107=1177, 1210 - 1177=33. So 22^4 ≡ 33 mod 107.22^5=33×22=726 mod 107. 726 ÷ 107=6×107=642, 726 - 642=84. So 22^5 ≡ 84 mod 107. Therefore, a₂₆ ≡ 84 mod 107.a₂₇ = a₂₆^2019 mod 107 = 84^5 mod 107.84 ≡ -23 mod 107.(-23)^2=529 mod 107. 529 ÷ 107=4×107=428, 529 - 428=101. So (-23)^2 ≡ 101 mod 107.(-23)^3=101×(-23)= -2323 mod 107. 2323 ÷ 107=21×107=2247, 2323 - 2247=76. So -2323 ≡ -76 mod 107 ≡ 31 mod 107.(-23)^4=31×(-23)= -713 mod 107. 713 ÷ 107=6×107=642, 713 - 642=71. So -713 ≡ -71 mod 107 ≡ 36 mod 107.(-23)^5=36×(-23)= -828 mod 107. 828 ÷ 107=7×107=749, 828 - 749=79. So -828 ≡ -79 mod 107 ≡ 28 mod 107. Therefore, a₂₇ ≡ 28 mod 107.a₂₈ = a₂₇^2019 mod 107 = 28^5 mod 107.Compute 28^5 mod 107:28^2=784 mod 107. 784 ÷ 107=7×107=749, 784 - 749=35. So 28^2 ≡ 35 mod 107.28^3=35×28=980 mod 107. 980 ÷ 107=9×107=963, 980 - 963=17. So 28^3 ≡ 17 mod 107.28^4=17×28=476 mod 107. 476 ÷ 107=4×107=428, 476 - 428=48. So 28^4 ≡ 48 mod 107.28^5=48×28=1344 mod 107. 1344 ÷ 107=12×107=1284, 1344 - 1284=60. So 28^5 ≡ 60 mod 107. Therefore, a₂₈ ≡ 60 mod 107.a₂₉ = a₂₈^2019 mod 107 = 60^5 mod 107.60^2=3600 mod 107. 3600 ÷ 107: 107×33=3531, 3600 - 3531=69. So 60^2 ≡ 69 mod 107.60^3=69×60=4140 mod 107. 4140 ÷ 107=38×107=4066, 4140 - 4066=74. So 60^3 ≡ 74 mod 107.60^4=74×60=4440 mod 107. 4440 ÷ 107=41×107=4387, 4440 - 4387=53. So 60^4 ≡ 53 mod 107.60^5=53×60=3180 mod 107. 3180 ÷ 107=29×107=3103, 3180 - 3103=77. So 60^5 ≡ 77 mod 107. Therefore, a₂₉ ≡ 77 mod 107.a₃₀ = a₂₉^2019 mod 107 = 77^5 mod 107.77 ≡ -30 mod 107.(-30)^2=900 mod 107. 900 ÷ 107=8×107=856, 900 - 856=44. So (-30)^2 ≡ 44 mod 107.(-30)^3=44×(-30)= -1320 mod 107. 1320 ÷ 107=12×107=1284, 1320 - 1284=36. So -1320 ≡ -36 mod 107 ≡ 71 mod 107.(-30)^4=71×(-30)= -2130 mod 107. 2130 ÷ 107=19×107=2033, 2130 - 2033=97. So -2130 ≡ -97 mod 107 ≡ 10 mod 107.(-30)^5=10×(-30)= -300 mod 107. 300 ÷ 107=2×107=214, 300 - 214=86. So -300 ≡ -86 mod 107 ≡ 21 mod 107. Therefore, a₃₀ ≡ 21 mod 107.a₃₁ = a₃₀^2019 mod 107 = 21^5 mod 107.21^2=441 mod 107. 441 ÷ 107=4×107=428, 441 - 428=13. So 21^2 ≡ 13 mod 107.21^3=13×21=273 mod 107. 273 ÷ 107=2×107=214, 273 - 214=59. So 21^3 ≡ 59 mod 107.21^4=59×21=1239 mod 107. 1239 ÷ 107=11×107=1177, 1239 - 1177=62. So 21^4 ≡ 62 mod 107.21^5=62×21=1302 mod 107. 1302 ÷ 107=12×107=1284, 1302 - 1284=18. So 21^5 ≡ 18 mod 107. Therefore, a₃₁ ≡ 18 mod 107.a₃₂ = a₃₁^2019 mod 107 = 18^5 mod 107.18^2=324 mod 107. 324 ÷ 107=3×107=321, 324 - 321=3. So 18^2 ≡ 3 mod 107.18^3=3×18=54 mod 107.18^4=54×18=972 mod 107. 972 ÷ 107=9×107=963, 972 - 963=9. So 18^4 ≡ 9 mod 107.18^5=9×18=162 mod 107. 162 - 107=55. So 18^5 ≡ 55 mod 107. Therefore, a₃₂ ≡ 55 mod 107.a₃₃ = a₃₂^2019 mod 107 = 55^5 mod 107.55^2=3025 mod 107. 3025 ÷ 107=28×107=2996, 3025 - 2996=29. So 55^2 ≡ 29 mod 107.55^3=29×55=1595 mod 107. 1595 ÷ 107=14×107=1498, 1595 - 1498=97. So 55^3 ≡ 97 mod 107.55^4=97×55=5335 mod 107. 5335 ÷ 107=49×107=5243, 5335 - 5243=92. So 55^4 ≡ 92 mod 107.55^5=92×55=5060 mod 107. 5060 ÷ 107=47×107=5029, 5060 - 5029=31. So 55^5 ≡ 31 mod 107. Therefore, a₃₃ ≡ 31 mod 107.a₃₄ = a₃₃^2019 mod 107 = 31^5 mod 107.31^2=961 mod 107. 961 ÷ 107=8×107=856, 961 - 856=105. So 31^2 ≡ 105 mod 107.31^3=105×31=3255 mod 107. 3255 ÷ 107=30×107=3210, 3255 - 3210=45. So 31^3 ≡ 45 mod 107.31^4=45×31=1395 mod 107. 1395 ÷ 107=13×107=1391, 1395 - 1391=4. So 31^4 ≡ 4 mod 107.31^5=4×31=124 mod 107. 124 - 107=17. So 31^5 ≡ 17 mod 107. Therefore, a₃₄ ≡ 17 mod 107.a₃₅ = a₃₄^2019 mod 107 = 17^5 mod 107.17^2=289 mod 107. 289 ÷ 107=2×107=214, 289 - 214=75. So 17^2 ≡ 75 mod 107.17^3=75×17=1275 mod 107. 1275 ÷ 107=11×107=1177, 1275 - 1177=98. So 17^3 ≡ 98 mod 107.17^4=98×17=1666 mod 107. 1666 ÷ 107=15×107=1605, 1666 - 1605=61. So 17^4 ≡ 61 mod 107.17^5=61×17=1037 mod 107. 1037 ÷ 107=9×107=963, 1037 - 963=74. So 17^5 ≡ 74 mod 107. Therefore, a₃₅ ≡ 74 mod 107.a₃₆ = a₃₅^2019 mod 107 = 74^5 mod 107.74 ≡ -33 mod 107.(-33)^2=1089 mod 107. 1089 ÷ 107=10×107=1070, 1089 - 1070=19. So (-33)^2 ≡ 19 mod 107.(-33)^3=19×(-33)= -627 mod 107. 627 ÷ 107=5×107=535, 627 - 535=92. So -627 ≡ -92 mod 107 ≡ 15 mod 107.(-33)^4=15×(-33)= -495 mod 107. 495 ÷ 107=4×107=428, 495 - 428=67. So -495 ≡ -67 mod 107 ≡ 40 mod 107.(-33)^5=40×(-33)= -1320 mod 107. 1320 ÷ 107=12×107=1284, 1320 - 1284=36. So -1320 ≡ -36 mod 107 ≡ 71 mod 107. Therefore, a₃₆ ≡ 71 mod 107.a₃₇ = a₃₆^2019 mod 107 = 71^5 mod 107.71 ≡ -36 mod 107.(-36)^2=1296 mod 107. 1296 ÷ 107=12×107=1284, 1296 - 1284=12. So (-36)^2 ≡ 12 mod 107.(-36)^3=12×(-36)= -432 mod 107. 432 ÷ 107=4×107=428, 432 - 428=4. So -432 ≡ -4 mod 107 ≡ 103 mod 107.(-36)^4=103×(-36)= -3708 mod 107. 3708 ÷ 107=34×107=3638, 3708 - 3638=70. So -3708 ≡ -70 mod 107 ≡ 37 mod 107.(-36)^5=37×(-36)= -1332 mod 107. 1332 ÷ 107=12×107=1284, 1332 - 1284=48. So -1332 ≡ -48 mod 107 ≡ 59 mod 107. Therefore, a₃₇ ≡ 59 mod 107.a₃₈ = a₃₇^2019 mod 107 = 59^5 mod 107.59^2=3481 mod 107. 3481 ÷ 107=32×107=3424, 3481 - 3424=57. So 59^2 ≡ 57 mod 107.59^3=57×59=3363 mod 107. 3363 ÷ 107=31×107=3317, 3363 - 3317=46. So 59^3 ≡ 46 mod 107.59^4=46×59=2714 mod 107. 2714 ÷ 107=25×107=2675, 2714 - 2675=39. So 59^4 ≡ 39 mod 107.59^5=39×59=2301 mod 107. 2301 ÷ 107=21×107=2247, 2301 - 2247=54. So 59^5 ≡ 54 mod 107. Therefore, a₃₈ ≡ 54 mod 107.a₃₉ = a₃₈^2019 mod 107 = 54^5 mod 107.54 ≡ -53 mod 107.(-53)^2=2809 mod 107. 2809 ÷ 107=26×107=2782, 2809 - 2782=27. So (-53)^2 ≡ 27 mod 107.(-53)^3=27×(-53)= -1431 mod 107. 1431 ÷ 107=13×107=1391, 1431 - 1391=40. So -1431 ≡ -40 mod 107 ≡ 67 mod 107.(-53)^4=67×(-53)= -3551 mod 107. 3551 ÷ 107=33×107=3531, 3551 - 3531=20. So -3551 ≡ -20 mod 107 ≡ 87 mod 107.(-53)^5=87×(-53)= -4611 mod 107. 4611 ÷ 107=43×107=4601, 4611 - 4601=10. So -4611 ≡ -10 mod 107 ≡ 97 mod 107. Therefore, a₃₉ ≡ 97 mod 107.a₄₀ = a₃₉^2019 mod 107 = 97^5 mod 107.97 ≡ -10 mod 107.(-10)^2=100 mod 107.(-10)^3=100×(-10)= -1000 mod 107. 1000 ÷ 107=9×107=963, 1000 - 963=37. So -1000 ≡ -37 mod 107 ≡ 70 mod 107.(-10)^4=70×(-10)= -700 mod 107. 700 ÷ 107=6×107=642, 700 - 642=58. So -700 ≡ -58 mod 107 ≡ 49 mod 107.(-10)^5=49×(-10)= -490 mod 107. 490 ÷ 107=4×107=428, 490 - 428=62. So -490 ≡ -62 mod 107 ≡ 45 mod 107. Therefore, a₄₀ ≡ 45 mod 107.a₄₁ = a₄₀^2019 mod 107 = 45^5 mod 107.45^2=2025 mod 107. 2025 ÷ 107=18×107=1926, 2025 - 1926=99. So 45^2 ≡ 99 mod 107.45^3=99×45=4455 mod 107. 4455 ÷ 107=41×107=4387, 4455 - 4387=68. So 45^3 ≡ 68 mod 107.45^4=68×45=3060 mod 107. 3060 ÷ 107=28×107=2996, 3060 - 2996=64. So 45^4 ≡ 64 mod 107.45^5=64×45=2880 mod 107. 2880 ÷ 107=26×107=2782, 2880 - 2782=98. So 45^5 ≡ 98 mod 107. Therefore, a₄₁ ≡ 98 mod 107.a₄₂ = a₄₁^2019 mod 107 = 98^5 mod 107.98 ≡ -9 mod 107.(-9)^2=81 mod 107.(-9)^3=81×(-9)= -729 mod 107. 729 ÷ 107=6×107=642, 729 - 642=87. So -729 ≡ -87 mod 107 ≡ 20 mod 107.(-9)^4=20×(-9)= -180 mod 107 ≡ -180 + 2×107= -180 + 214=34 mod 107.(-9)^5=34×(-9)= -306 mod 107 ≡ -306 + 3×107= -306 + 321=15 mod 107. Therefore, a₄₂ ≡ 15 mod 107.a₄₃ = a₄₂^2019 mod 107 = 15^5 mod 107.15^2=225 mod 107. 225 - 2×107=225 - 214=11. So 15^2 ≡ 11 mod 107.15^3=11×15=165 mod 107. 165 - 107=58. So 15^3 ≡ 58 mod 107.15^4=58×15=870 mod 107. 870 ÷ 107=8×107=856, 870 - 856=14. So 15^4 ≡ 14 mod 107.15^5=14×15=210 mod 107. 210 - 2×107=210 - 214= -4 mod 107 ≡ 103 mod 107. Therefore, a₄₃ ≡ 103 mod 107.a₄₄ = a₄₃^2019 mod 107 = 103^5 mod 107.103 ≡ -4 mod 107.(-4)^2=16 mod 107.(-4)^3=16×(-4)= -64 mod 107 ≡ 43 mod 107.(-4)^4=43×(-4)= -172 mod 107 ≡ -172 + 2×107= -172 + 214=42 mod 107.(-4)^5=42×(-4)= -168 mod 107 ≡ -168 + 2×107= -168 + 214=46 mod 107. Therefore, a₄₄ ≡ 46 mod 107.a₄₅ = a₄₄^2019 mod 107 = 46^5 mod 107.46^2=2116 mod 107. 2116 ÷ 107=19×107=2033, 2116 - 2033=83. So 46^2 ≡ 83 mod 107.46^3=83×46=3818 mod 107. 3818 ÷ 107=35×107=3745, 3818 - 3745=73. So 46^3 ≡ 73 mod 107.46^4=73×46=3358 mod 107. 3358 ÷ 107=31×107=3317, 3358 - 3317=41. So 46^4 ≡ 41 mod 107.46^5=41×46=1886 mod 107. 1886 ÷ 107=17×107=1819, 1886 - 1819=67. So 46^5 ≡ 67 mod 107. Therefore, a₄₅ ≡ 67 mod 107.a₄₆ = a₄₅^2019 mod 107 = 67^5 mod 107.67 ≡ -40 mod 107.(-40)^2=1600 mod 107. 1600 ÷ 107=14×107=1498, 1600 - 1498=102. So (-40)^2 ≡ 102 mod 107.(-40)^3=102×(-40)= -4080 mod 107. 4080 ÷ 107=38×107=4066, 4080 - 4066=14. So -4080 ≡ -14 mod 107 ≡ 93 mod 107.(-40)^4=93×(-40)= -3720 mod 107. 3720 ÷ 107=34×107=3638, 3720 - 3638=82. So -3720 ≡ -82 mod 107 ≡ 25 mod 107.(-40)^5=25×(-40)= -1000 mod 107. 1000 ÷ 107=9×107=963, 1000 - 963=37. So -1000 ≡ -37 mod 107 ≡ 70 mod 107. Therefore, a₄₆ ≡ 70 mod 107.a₄₇ = a₄₆^2019 mod 107 = 70^5 mod 107.70^2=4900 mod 107. 4900 ÷ 107=45×107=4815, 4900 - 4815=85. So 70^2 ≡ 85 mod 107.70^3=85×70=5950 mod 107. 5950 ÷ 107=55×107=5885, 5950 - 5885=65. So 70^3 ≡ 65 mod 107.70^4=65×70=4550 mod 107. 4550 ÷ 107=42×107=4494, 4550 - 4494=56. So 70^4 ≡ 56 mod 107.70^5=56×70=3920 mod 107. 3920 ÷ 107=36×107=3852, 3920 - 3852=68. So 70^5 ≡ 68 mod 107. Therefore, a₄₇ ≡ 68 mod 107.a₄₈ = a₄₇^2019 mod 107 = 68^5 mod 107.68 ≡ -39 mod 107.(-39)^2=1521 mod 107. 1521 ÷ 107=14×107=1498, 1521 - 1498=23. So (-39)^2 ≡ 23 mod 107.(-39)^3=23×(-39)= -897 mod 107. 897 ÷ 107=8×107=856, 897 - 856=41. So -897 ≡ -41 mod 107 ≡ 66 mod 107.(-39)^4=66×(-39)= -2574 mod 107. 2574 ÷ 107=24×107=2568, 2574 - 2568=6. So -2574 ≡ -6 mod 107 ≡ 101 mod 107.(-39)^5=101×(-39)= -3939 mod 107. 3939 ÷ 107=36×107=3852, 3939 - 3852=87. So -3939 ≡ -87 mod 107 ≡ 20 mod 107. Therefore, a₄₈ ≡ 20 mod 107.a₄₉ = a₄₈^2019 mod 107 = 20^5 mod 107.20^2=400 mod 107. 400 - 3×107=400 - 321=79. So 20^2 ≡ 79 mod 107.20^3=79×20=1580 mod 107. 1580 ÷ 107=14×107=1498, 1580 - 1498=82. So 20^3 ≡ 82 mod 107.20^4=82×20=1640 mod 107. 1640 ÷ 107=15×107=1605, 1640 - 1605=35. So 20^4 ≡ 35 mod 107.20^5=35×20=700 mod 107. 700 ÷ 107=6×107=642, 700 - 642=58. So 20^5 ≡ 58 mod 107. Therefore, a₄₉ ≡ 58 mod 107.a₅₀ = a₄₉^2019 mod 107 = 58^5 mod 107.58^2=3364 mod 107. 3364 ÷ 107=31×107=3317, 3364 - 3317=47. So 58^2 ≡ 47 mod 107.58^3=47×58=2726 mod 107. 2726 ÷ 107=25×107=2675, 2726 - 2675=51. So 58^3 ≡ 51 mod 107.58^4=51×58=2958 mod 107. 2958 ÷ 107=27×107=2889, 2958 - 2889=69. So 58^4 ≡ 69 mod 107.58^5=69×58=4002 mod 107. 4002 ÷ 107=37×107=3959, 4002 - 3959=43. So 58^5 ≡ 43 mod 107. Therefore, a₅₀ ≡ 43 mod 107.a₅₁ = a₅₀^2019 mod 107 = 43^5 mod 107.43^2=1849 mod 107. 1849 ÷ 107=17×107=1819, 1849 - 1819=30. So 43^2 ≡ 30 mod 107.43^3=30×43=1290 mod 107. 1290 ÷ 107=12×107=1284, 1290 - 1284=6. So 43^3 ≡ 6 mod 107.43^4=6×43=258 mod 107. 258 ÷ 107=2×107=214, 258 - 214=44. So 43^4 ≡ 44 mod 107.43^5=44×43=1892 mod 107. 1892 ÷ 107=17×107=1819, 1892 - 1819=73. So 43^5 ≡ 73 mod 107. Therefore, a₅₁ ≡ 73 mod 107.Now, we have all aₖ mod 107 for k from 0 to 51. Let's list them:a₀: 93a₁: 65a₂: 66a₃: 82a₄: 51a₅: 6a₆: 72a₇: 38a₈: 63a₉: 94a₁₀: 104a₁₁: 78a₁₂: 2a₁₃: 32a₁₄: 88a₁₅: 95a₁₆: 50a₁₇: 80a₁₈: 7a₁₉: 8a₂₀: 26a₂₁: 96a₂₂: 91a₂₃: 24a₂₄: 5a₂₅: 22a₂₆: 84a₂₇: 28a₂₈: 60a₂₉: 77a₃₀: 21a₃₁: 18a₃₂: 55a₃₃: 31a₃₄: 17a₃₅: 74a₃₆: 71a₃₇: 59a₃₈: 54a₃₉: 97a₄₀: 45a₄₁: 98a₄₂: 15a₄₃: 103a₄₄: 46a₄₅: 67a₄₆: 70a₄₇: 68a₄₈: 20a₄₉: 58a₅₀: 43a₅₁: 73Now, sum all these remainders mod 107. Let's compute this step by step:Starting with 93 (a₀):Total = 93Add a₁: 93 + 65 = 158 → 158 - 107 = 51Total = 51Add a₂: 51 + 66 = 117 → 117 - 107 = 10Total = 10Add a₃: 10 + 82 = 92Total = 92Add a₄: 92 + 51 = 143 → 143 - 107 = 36Total = 36Add a₅: 36 + 6 = 42Total = 42Add a₆: 42 + 72 = 114 → 114 - 107 = 7Total = 7Add a₇: 7 + 38 = 45Total = 45Add a₈: 45 + 63 = 108 → 108 - 107 = 1Total = 1Add a₉: 1 + 94 = 95Total = 95Add a₁₀: 95 + 104 = 199 → 199 - 107×1=92Total = 92Add a₁₁: 92 + 78 = 170 → 170 - 107=63Total = 63Add a₁₂: 63 + 2 = 65Total = 65Add a₁₃: 65 + 32 = 97Total = 97Add a₁₄: 97 + 88 = 185 → 185 - 107=78Total = 78Add a₁₅: 78 + 95 = 173 → 173 - 107=66Total = 66Add a₁₆: 66 + 50 = 116 → 116 - 107=9Total = 9Add a₁₇: 9 + 80 = 89Total = 89Add a₁₈: 89 + 7 = 96Total = 96Add a₁₉: 96 + 8 = 104Total = 104Add a₂₀: 104 + 26 = 130 → 130 - 107=23Total = 23Add a₂₁: 23 + 96 = 119 → 119 - 107=12Total = 12Add a₂₂: 12 + 91 = 103Total = 103Add a₂₃: 103 + 24 = 127 → 127 - 107=20Total = 20Add a₂₄: 20 + 5 = 25Total = 25Add a₂₅: 25 + 22 = 47Total = 47Add a₂₆: 47 + 84 = 131 → 131 - 107=24Total = 24Add a₂₇: 24 + 28 = 52Total = 52Add a₂₈: 52 + 60 = 112 → 112 - 107=5Total = 5Add a₂₉: 5 + 77 = 82Total = 82Add a₃₀: 82 + 21 = 103Total = 103Add a₃₁: 103 + 18 = 121 → 121 - 107=14Total = 14Add a₃₂: 14 + 55 = 69Total = 69Add a₃₃: 69 + 31 = 100Total = 100Add a₃₄: 100 + 17 = 117 → 117 - 107=10Total = 10Add a₃₅: 10 + 74 = 84Total = 84Add a₃₆: 84 + 71 = 155 → 155 - 107=48Total = 48Add a₃₇: 48 + 59 = 107 → 107 - 107=0Total = 0Add a₃₈: 0 + 54 = 54Total = 54Add a₃₉: 54 + 97 = 151 → 151 - 107=44Total = 44Add a₄₀: 44 + 45 = 89Total = 89Add a₄₁: 89 + 98 = 187 → 187 - 107=80Total = 80Add a₄₂: 80 + 15 = 95Total = 95Add a₄₃: 95 + 103 = 198 → 198 - 107×1=91Total = 91Add a₄₄: 91 + 46 = 137 → 137 - 107=30Total = 30Add a₄₅: 30 + 67 = 97Total = 97Add a₄₆: 97 + 70 = 167 → 167 - 107=60Total = 60Add a₄₇: 60 + 68 = 128 → 128 - 107=21Total = 21Add a₄₈: 21 + 20 = 41Total = 41Add a₄₉: 41 + 58 = 99Total = 99Add a₅₀: 99 + 43 = 142 → 142 - 107=35Total = 35Add a₅₁: 35 + 73 = 108 → 108 - 107=1Total = 1 mod 107.Therefore, the sum S ≡ 1 mod 107.Step 3: Combine using Chinese Remainder TheoremWe need to find a number x such that:x ≡ 4 mod 8x ≡ 1 mod 107We can write x = 107k + 1 for some integer k. Substitute into the first congruence:107k + 1 ≡ 4 mod 8107 mod 8: 107 ÷ 8=13×8=104, 107 - 104=3. So 107 ≡ 3 mod 8.Therefore:3k + 1 ≡ 4 mod 83k ≡ 3 mod 8Divide both sides by 3 (since 3 and 8 are coprime):k ≡ 1 mod 8So k = 8m + 1 for some integer m.Therefore, x = 107(8m + 1) + 1 = 856m + 107 + 1 = 856m + 108.Therefore, the smallest non-negative solution is x = 108 mod 856.So the remainder when the sum is divided by 856 is 108.Final Answerboxed{108}</think>Given the sequence defined by (a_0 = 2019) and (a_n = a_{n-1}^{2019}) for all positive integers (n), we need to compute the remainder when (a_0 + a_1 + a_2 + dots + a_{51}) is divided by 856.First, we factorize 856 as (856 = 8 times 107). Using the Chinese Remainder Theorem, we compute the sum modulo 8 and modulo 107 separately.Step 1: Compute the sum modulo 8- (a_0 equiv 2019 equiv 3 mod 8)- All subsequent terms (a_n) are powers of 3 modulo 8. Since (3^2 equiv 1 mod 8), any odd power of 3 is congruent to 3 modulo 8. Hence, all (a_n equiv 3 mod 8).- The sum of 52 terms each congruent to 3 modulo 8 is (52 times 3 = 156 equiv 4 mod 8).Step 2: Compute the sum modulo 107- (a_0 equiv 2019 equiv 93 mod 107)- Using Fermat's Little Theorem, we reduce exponents modulo 106. Each term (a_n) is computed by raising the previous term to the 2019th power modulo 107. This results in a cycle of residues modulo 107.After computing each term modulo 107 and summing them up, we find the total sum modulo 107 is 1.Step 3: Combine results using Chinese Remainder TheoremWe solve the system:[begin{cases}x equiv 4 mod 8 x equiv 1 mod 107end{cases}]Express (x) as (x = 107k + 1). Substituting into the first congruence:[107k + 1 equiv 4 mod 8 implies 3k + 1 equiv 4 mod 8 implies 3k equiv 3 mod 8 implies k equiv 1 mod 8]Thus, (k = 8m + 1) and (x = 107(8m + 1) + 1 = 856m + 108). The smallest non-negative solution is (x = 108).The remainder when the sum is divided by 856 is (boxed{108}).]

🔑:To solve the problem, we need to compute the remainder when the sum (a_0 + a_1 + a_2 + dots + a_{51}) is divided by 856, where the sequence is defined by (a_0 = 2019) and (a_n = a_{n-1}^{2019}) for all positive integers (n).1. Understanding the Sequence: - The sequence starts with (a_0 = 2019). - Each subsequent term is the previous term raised to the power of 2019: (a_n = a_{n-1}^{2019}).2. Modulo 107: - We first consider the sum modulo 107. Using Euler's theorem, since (phi(107) = 106), we have (a_{n-1}^{106} equiv 1 pmod{107}) for any (a_{n-1}) coprime with 107. - Since (2019 equiv 93 pmod{107}), we can rewrite the sequence in terms of 93 modulo 107. - The sequence modulo 107 becomes (93^{5^0}, 93^{5^1}, 93^{5^2}, ldots, 93^{5^{51}}).3. Sum Modulo 107: - We need to compute the sum (93^{5^0} + 93^{5^1} + 93^{5^2} + cdots + 93^{5^{51}} pmod{107}). - Since 93 is a primitive root modulo 107, the powers of 93 will cycle through all residues modulo 107. - The sum of all residues modulo 107 is zero because they form a complete residue system. - Therefore, the sum modulo 107 is (0 pmod{107}).4. Modulo 8: - Next, we consider the sum modulo 8. Since (2019 equiv 3 pmod{8}), the sequence modulo 8 becomes (3^{1^0}, 3^{1^1}, 3^{1^2}, ldots, 3^{1^{51}}). - Since (3^2 equiv 1 pmod{8}), the sequence modulo 8 cycles every 2 terms: (3, 3, 3, 3, ldots). - There are 52 terms in the sum, and each term is 3 modulo 8. - Therefore, the sum modulo 8 is (3 times 52 = 156 equiv 4 pmod{8}).5. Chinese Remainder Theorem: - We now have the sum modulo 107 and modulo 8: [ S equiv 0 pmod{107} ] [ S equiv 4 pmod{8} ] - Using the Chinese Remainder Theorem, we solve the system of congruences: [ S = 107k quad text{for some integer } k ] [ 107k equiv 4 pmod{8} ] - Since (107 equiv 3 pmod{8}), we have: [ 3k equiv 4 pmod{8} ] - The multiplicative inverse of 3 modulo 8 is 3 (since (3 times 3 = 9 equiv 1 pmod{8})), so: [ k equiv 4 times 3 equiv 12 equiv 4 pmod{8} ] - Therefore, (k = 8m + 4) for some integer (m), and: [ S = 107(8m + 4) = 856m + 428 ] - Thus, the remainder when (S) is divided by 856 is: [ S equiv 428 pmod{856} ]The final answer is (boxed{428})