🧠:[<think>Okay, so I need to find the probability that when I throw three dice, the sum of the points on two of them is 7. Hmm, let's break this down step by step.First, when they say "the sum of the points on 2 of the dice is 7," does that mean exactly two of them add up to 7, and the third is something else? Or could it be any pair among the three? The problem says "the sum of the points on 2 of the dice is 7," so I think it means that there exists at least one pair among the three dice that adds up to 7. The third die can be any value. So, the event we're interested in is that among the three dice, at least one pair sums to 7.So, to find the probability, I need to calculate the number of favorable outcomes (where at least one pair sums to 7) divided by the total number of possible outcomes when throwing three dice.Total number of possible outcomes when throwing three dice is 6^3 = 216. That's straightforward.Now, how to calculate the number of favorable outcomes? That's the tricky part. Let's think about how many ways we can have at least one pair summing to 7 in three dice.First, let's consider all possible pairs in three dice. Since there are three dice, there are C(3,2) = 3 pairs. Each pair could potentially sum to 7. So, the favorable outcomes are those where at least one of these three pairs sums to 7.But wait, calculating "at least one" can sometimes be tricky because of overlapping cases where two pairs might both sum to 7. For example, if two different pairs both sum to 7, how do we count that? So, maybe inclusion-exclusion principle is needed here.Let me recall that inclusion-exclusion for three events. But here we have three pairs, each of which could be a possible event where the pair sums to 7. Let's denote the three pairs as AB, AC, and BC (assuming the dice are labeled A, B, C). Then, the number of favorable outcomes is the number of outcomes where AB sums to 7, or AC sums to 7, or BC sums to 7.So, using inclusion-exclusion:Number of favorable outcomes = N(AB) + N(AC) + N(BC) - N(AB and AC) - N(AB and BC) - N(AC and BC) + N(AB and AC and BC)Where N(AB) is the number of outcomes where pair AB sums to 7, and similarly for others. Then, N(AB and AC) is the number of outcomes where both AB and AC sum to 7, and so on.First, compute N(AB). For a specific pair, say AB, the number of outcomes where A + B = 7. Since A and B are two dice, the number of possible (A,B) pairs that sum to 7 is 6: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So, for each pair, there are 6 possibilities. Then, the third die, C, can be any of 6 numbers. Therefore, N(AB) = 6 * 6 = 36. Similarly, N(AC) = 36 and N(BC) = 36.So, the first three terms sum to 3 * 36 = 108.Next, subtract the overlaps. Let's compute N(AB and AC). If both AB and AC sum to 7, then A + B = 7 and A + C = 7. Let's see what this implies. Let me solve these equations.From A + B = 7 and A + C = 7, subtracting the two equations gives B - C = 0, so B = C. Also, A = 7 - B. So, A is determined once B is chosen. However, since A, B, and C are dice, they must each be between 1 and 6.So, possible values for B (and hence C) can be from 1 to 6, but A must be 7 - B. However, since A must also be between 1 and 6, B can be from 1 to 6, but 7 - B must also be between 1 and 6, which is true for B from 1 to 6. Therefore, B can be any value from 1 to 6, then A = 7 - B, and C = B. Therefore, for each B from 1 to 6, we have a unique solution. So, there are 6 possible outcomes where both AB and AC sum to 7. However, wait: in this case, die A is determined by die B, and die C is equal to die B. So, for example, if B is 1, then A is 6 and C is 1. So, the triplet is (6,1,1). Similarly, B=2 gives (5,2,2), etc.So, each of these triplets is determined by B, and there are 6 such triplets. Therefore, N(AB and AC) = 6. Similarly, N(AB and BC) and N(AC and BC) would also be 6 each. So, the next three terms are 3 * 6 = 18.Now, subtracting this from the previous total: 108 - 18 = 90.Now, add back N(AB and AC and BC). That is, the case where all three pairs sum to 7. Let's see what that would imply. If AB, AC, and BC all sum to 7, then:A + B = 7,A + C = 7,B + C = 7.From the first two equations: A + B = A + C ⇒ B = C.From the first and third equations: A + B = 7 and B + C = 7. Since B = C, substitute into the third equation: B + B = 7 ⇒ 2B = 7 ⇒ B = 3.5. But B must be an integer between 1 and 6. So, there is no solution where all three pairs sum to 7. Therefore, N(AB and AC and BC) = 0.Therefore, applying inclusion-exclusion, the total number of favorable outcomes is 108 - 18 + 0 = 90.Wait, but hold on. Let me verify that. If we have 3 pairs each overlapping, but when considering all three overlapping, there's no solution. So, the inclusion-exclusion formula gives 3*36 - 3*6 + 0 = 108 - 18 = 90. Therefore, the number of favorable outcomes is 90.Therefore, the probability is 90 / 216. Simplify that: divide numerator and denominator by 18: 5 / 12. So, 5/12 ≈ 0.4167.But wait, hold on. Let me check again if this is correct. Let me think through an example. Suppose we have three dice. Let's say die A, B, C. The pairs are AB, AC, BC. Each pair has 6 possibilities to sum to 7, but when we consider overlaps, we subtract the overlaps where two pairs sum to 7. But in those overlaps, like AB and AC both summing to 7, which forces B=C and A=7-B. So, there are 6 such cases where two pairs sum to 7. Similarly for the other pairs. So, each overlap is 6, three overlaps, so total overlapping cases are 18. Therefore, total favorable is 108 - 18 = 90. That seems correct.But let's take a concrete example. For example, in the case where all three dice are 3.5, which is impossible. But as we saw, there are no cases where all three pairs sum to 7. So, that part is fine.But another way to think about this problem is to calculate the probability that at least one pair sums to 7, which is equivalent to 1 - probability(no pair sums to 7). Maybe that approach is simpler? Let me try that.Total outcomes: 216. Probability that no pair sums to 7. So, how many outcomes where none of the three pairs sum to 7.So, let's compute the number of such outcomes.First, for three dice, A, B, C, we need that A+B ≠ 7, A+C ≠ 7, and B+C ≠7.To compute this, we can use inclusion-exclusion again, but in reverse. Wait, perhaps not. Alternatively, maybe we can model it step by step.First, choose a value for die A. There are 6 possibilities. Then, for die B, it cannot be 7 - A. So, die B has 5 possibilities (since 7 - A is excluded). Then, for die C, it cannot be 7 - A and cannot be 7 - B. But need to be careful here because 7 - A and 7 - B might be the same or different.Wait, this might get complicated. Let's see.Alternatively, using inclusion-exclusion:Number of outcomes with no pair summing to 7 = Total outcomes - number of outcomes with at least one pair summing to 7. But that's circular because we're trying to compute the complement. So, if we can compute the number of outcomes with at least one pair summing to 7 (which we did as 90), then the number with no pairs summing to 7 is 216 - 90 = 126. Therefore, probability of no pairs is 126 / 216 = 7 / 12, so the probability of at least one pair is 1 - 7/12 = 5/12. So, same result. So, that's good, cross-checked.Alternatively, trying to compute the number of outcomes with no pairs summing to 7 directly:First, choose A: 6 possibilities.For each A, choose B such that B ≠ 7 - A. So, for each A, there are 5 choices for B.Then, choose C such that C ≠ 7 - A and C ≠ 7 - B.But here, 7 - A and 7 - B may or may not be the same.So, depending on whether 7 - A equals 7 - B, which occurs when A = B. But since B ≠ 7 - A, we need to check if A can equal B. Wait, B is chosen after A, and B ≠ 7 - A. So, if A is 1, then B can be 2-6, but not 6. So, if A is 1, B can be 1,2,3,4,5. Wait, no, if A is 1, 7 - A is 6, so B can be any value except 6. So, B can be 1,2,3,4,5. So, B can be equal to A (1 in this case) or not. Therefore, when choosing C, C cannot be 7 - A or 7 - B.But 7 - A and 7 - B might overlap if A and B are such that 7 - A = 7 - B, which would mean A = B.But if A and B are different, then 7 - A and 7 - B are different. If A and B are the same, then 7 - A = 7 - B.Wait, but in the case where A = B, but B was chosen such that B ≠ 7 - A. So, A = B, but since B ≠ 7 - A, so A ≠ 7 - A, so 2A ≠7 ⇒ A ≠ 3.5, which is always true since A is integer. So, A can equal B only if B ≠ 7 - A. For example, if A is 1, B can be 1,2,3,4,5. So, in this case, B can be 1. So, when A = 1 and B =1, then 7 - A =6, 7 - B =6, so C cannot be 6. Therefore, in this case, C can be 1-5, so 5 choices.If A and B are different, say A=1, B=2, then 7 - A=6 and 7 - B=5, so C cannot be 5 or 6, so C has 4 choices.Therefore, depending on whether A and B are the same or different, the number of choices for C changes.Therefore, the total number of outcomes with no pairs summing to 7 can be calculated as:For each A from 1 to 6:- Choose B ≠ 7 - A. If B = A, then when choosing C, C ≠ 7 - A (since 7 - A =7 - B in this case). If B ≠ A, then C ≠7 - A and C ≠7 - B.So, let's split into two cases:Case 1: B = A. How many such cases?For each A, B can be A only if A ≠7 - A. Wait, but B can be A as long as B ≠7 - A. Since B is chosen as B ≠7 - A, and B can be any value except 7 - A. So, B can be A only if A ≠7 - A. But 7 - A = A ⇒ 7=2A ⇒ A=3.5, which is not possible. So, B can always be A, because 7 - A ≠ A for integer A from 1-6. Therefore, for each A, B can be A or any other except 7 - A. Therefore, the number of choices for B when B = A is 1 (only A itself), and when B ≠ A, it's 5 -1=4? Wait, wait. Let's see.Wait, when you choose B ≠7 - A, so for each A, there are 5 choices for B (since one value, 7 - A, is excluded). Among these 5 choices, how many are equal to A?If A is not 7 - A, which is always true since 7 - A ≠ A for any integer A, then B can be A or other numbers. For example, if A=1, 7 - A=6, so B can be 1,2,3,4,5. So, B=1 is allowed. Therefore, for each A, exactly 1 of the 5 choices is equal to A, and the remaining 4 are different from A.Therefore, for each A, when choosing B:- 1 way to choose B = A.- 4 ways to choose B ≠ A.Then, for each of these, we have different numbers of choices for C.Case 1: B = A. Then, C cannot be 7 - A (since C ≠7 - A and C ≠7 - B, but since B=A, 7 - B =7 - A). So, C has 5 choices: all except 7 - A.Case 2: B ≠ A. Then, C cannot be 7 - A or 7 - B. Since A ≠ B, 7 - A and 7 - B are two distinct numbers (since A ≠ B ⇒ 7 - A ≠7 - B). Therefore, C has 6 - 2 =4 choices.Therefore, total number of outcomes with no pairs summing to 7 is:For each A (6 choices):- Case 1: B = A. There is 1 choice for B, and 5 choices for C. So, 1*5=5.- Case 2: B ≠ A. There are 4 choices for B, and 4 choices for C. So, 4*4=16.Therefore, total per A: 5 +16=21.Since there are 6 choices for A, total outcomes: 6*21=126. Which matches the previous result of 216 -90=126.Therefore, the number of favorable outcomes (at least one pair sums to 7) is 90, as before.Therefore, probability is 90/216 = 5/12 ≈0.4167.So, the answer should be 5/12.But just to make sure, let's try another approach. Let's enumerate all possible triplets and count how many have at least one pair summing to 7.But enumerating all 216 is impractical, but perhaps we can think of the possible pairs.For three dice, each die can be from 1 to 6. For each possible pair, the number of triplets where that pair sums to 7 is 6*6=36 as before. But since there are three pairs, 3*36=108, but this counts overlaps multiple times.For example, a triplet where two different pairs sum to 7 is counted twice. For instance, if we have (6,1,1), then AB=6+1=7 and AC=6+1=7, so this triplet is counted in both N(AB) and N(AC). Similarly, triplet (3,4,3) would have BC=4+3=7 and AC=3+3=6, so only one pair. Wait, no. Wait, (3,4,3): A=3, B=4, C=3. Then AB=7, AC=6, BC=7. So, both AB and BC sum to 7. So, this triplet is counted in N(AB) and N(BC). Therefore, such triplets where two pairs sum to 7 are counted twice in the initial 108. So, to correct for overcounting, we need to subtract the overlaps.Each overlap where two pairs sum to 7 has been counted twice, so we subtract them once. There are three such overlaps (AB & AC, AB & BC, AC & BC), each with 6 triplets, so 3*6=18. So, 108 -18=90. Then, triplets where all three pairs sum to 7 would be added back in, but there are none. So, total 90.Therefore, this confirms the previous result. Therefore, the probability is 90/216=5/12.Therefore, the answer is 5/12.Final AnswerThe probability is boxed{dfrac{5}{12}}.</think>To find the probability that the sum of the points on two of the dice is 7 when three dice are thrown, we use the inclusion-exclusion principle.1. Total number of possible outcomes: (6^3 = 216).2. Number of favorable outcomes: - Each pair of dice (AB, AC, BC) can sum to 7. There are 6 possible pairs for each sum (e.g., (1,6), (2,5), etc.), and the third die can be any of 6 values. Thus, each pair contributes (6 times 6 = 36) outcomes. - There are 3 pairs, so initially, we have (3 times 36 = 108) outcomes.3. Subtract overlapping cases: - If two pairs sum to 7 (e.g., AB and AC), the third die is determined, leading to 6 outcomes per pair. There are 3 such pairs, contributing (3 times 6 = 18) overlapping outcomes. - No triplet exists where all three pairs sum to 7, so this term is 0.4. Applying inclusion-exclusion: - Total favorable outcomes: (108 - 18 + 0 = 90).5. Probability calculation: - The probability is (frac{90}{216} = frac{5}{12}).Thus, the probability that the sum of the points on two of the dice is 7 is (boxed{dfrac{5}{12}}).,<think>Okay, so the problem is: If 3 dice are thrown randomly, then what is the probability that the sum of the points on 2 of the dice is 7? Hmm, let's try to figure this out step by step. First, I need to understand what exactly is being asked. We have three dice, each with faces numbered from 1 to 6. When we throw them, we get three numbers, say a, b, and c. The question is asking for the probability that among these three numbers, any two of them add up to 7. So, either a + b = 7, or a + c = 7, or b + c = 7. We need to find the probability that at least one of these pairs sums to 7. Alright, so the total number of possible outcomes when throwing three dice is 6 * 6 * 6 = 216. That's straightforward because each die has 6 possible outcomes, and they're independent.Now, the tricky part is figuring out how many of these 216 outcomes have at least one pair summing to 7. To calculate this, maybe I can use the principle of inclusion-exclusion. Let's denote the events:- A: The sum of the first and second die is 7.- B: The sum of the first and third die is 7.- C: The sum of the second and third die is 7.We need to find the probability of A ∪ B ∪ C, which is the probability that at least one of these events occurs. According to inclusion-exclusion principle:P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)So, first, let's calculate P(A). Since we're considering two dice, the number of favorable outcomes where their sum is 7. For two dice, the pairs that add up to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So there are 6 possible outcomes. However, since we have three dice, the third die can be any number from 1 to 6. So for event A, the number of favorable outcomes is 6 * 6 = 36. Similarly for events B and C. Therefore, each of P(A), P(B), P(C) is 36/216 = 1/6.So, the first part of the inclusion-exclusion formula is 3*(1/6) = 1/2.Now, we need to subtract the probabilities of the intersections. Let's calculate P(A ∩ B). This is the probability that both the first and second die sum to 7 with the first die, and the first and third die also sum to 7. Wait, so if both A and B occur, then the first die is part of two different pairs that each sum to 7. Let me think.Suppose the first die is x, the second is y, and the third is z. If both A and B occur, then x + y = 7 and x + z = 7. From these two equations, we can subtract them: (x + y) - (x + z) = 0 => y - z = 0 => y = z. So, if both A and B occur, then the second and third dice must be equal. Also, x must be such that x + y = 7 and x + z =7. Since y = z, x is fixed once y is chosen. So, how many outcomes are there in A ∩ B? Let's break it down. Let x be any value such that there exists y and z where x + y =7 and x + z =7. But since y = z, this is possible for any x. Let's choose x first. For each x (from 1 to 6), y and z must both be 7 - x. However, y and z have to be valid die faces, so 7 - x must be between 1 and 6. Which it is, since x is between 1 and 6. So for each x, there is exactly one value of y and z (which are equal). Therefore, the number of outcomes where both A and B occur is 6 (for each x, y = z =7 -x). Therefore, P(A ∩ B) = 6/216 = 1/36.Similarly, the same logic applies to P(A ∩ C) and P(B ∩ C). For example, in P(A ∩ C), if both A (x + y =7) and C (y + z =7) occur, then similar equations: x + y =7 and y + z=7. Subtracting, x - z =0 => x = z. So, x and z are equal, and y is 7 -x. So again, for each x (which equals z), y is determined. So again, 6 outcomes. Similarly for P(B ∩ C). Therefore, each intersection of two events has probability 1/36, and there are three such intersections. So total subtraction is 3*(1/36) = 1/12.Now, the next term in inclusion-exclusion is adding back P(A ∩ B ∩ C). That is, the probability that all three events occur simultaneously. So, all three pairs (x+y, x+z, y+z) sum to 7. Let's see if this is possible. If x + y =7, x + z=7, and y + z=7, then we have three equations:1. x + y =72. x + z =73. y + z =7From equations 1 and 2, subtracting gives y - z =0 => y = z.Substituting into equation 3: y + y =7 => 2y =7 => y=3.5. But y must be an integer between 1 and 6. Therefore, there are no solutions where all three pairs sum to 7. Therefore, P(A ∩ B ∩ C) = 0.Therefore, putting it all together:P(A ∪ B ∪ C) = 1/2 - 1/12 + 0 = (6/12 - 1/12) = 5/12.Wait, 1/2 is 6/12, minus 1/12 is 5/12. So the probability is 5/12. Is that right?Let me verify this another way, to make sure I didn't make a mistake.Alternative approach: Instead of inclusion-exclusion, maybe calculate the number of favorable outcomes directly.We need to count the number of triples (a, b, c) where at least one pair sums to 7.Alternatively, compute the total number of such triples. Let's consider each pair:- Pair (a, b): sum to 7. The number of such triples: For each of the 6 possible (a, b) pairs that sum to 7, c can be anything from 1 to 6. So 6*6=36.- Pair (a, c): Similarly, 6*6=36.- Pair (b, c): Similarly, 6*6=36.Total so far: 36*3=108. But wait, this counts triples where two different pairs sum to 7 multiple times. For example, if both (a, b) and (a, c) sum to 7, then such a triple is counted in both the first and second group. Similarly, if all three pairs sum to 7 (but we saw earlier that this is impossible), but since it's impossible, we only have overlaps where two pairs sum to 7. So, we need to subtract the overlaps. How many triples are counted in two of the groups? For example, triples where both (a, b) and (a, c) sum to 7. As we calculated earlier, in such cases, (a, b) =7 and (a, c)=7, so b = c. Therefore, the triple is (a, b, b) where a + b =7. For each a, b is determined as 7 -a, and c =b. Therefore, there are 6 such triples (a from 1 to 6). Similarly, triples where both (a, b) and (b, c) sum to 7: same logic, a and c must both equal 7 - b, so a = c. Therefore, triples (a, b, a) where a + b =7. Again 6 triples. Similarly, triples where both (a, c) and (b, c) sum to 7: same thing, so (a, b, c) where a =7 - c and b=7 -c, so a =b. So triples (a, a, c) where a + c=7. Again 6 triples. Each of these three overlaps has 6 triples, so total overlaps are 3*6=18. But since we subtracted these overlaps, we need to check if there are any triples that are counted in all three groups, but we already determined that there are none. So the inclusion-exclusion formula gives total favorable outcomes as 108 - 18 = 90. Therefore, the probability is 90 / 216 = 15 / 36 = 5 / 12. So that matches our previous result. Therefore, 5/12 is the correct probability. Alternatively, another way to check: Let's compute the probability that none of the pairs sum to 7, and then subtract that from 1. Maybe that's another approach.So, probability that no two dice sum to 7. How many such triples are there? Let's compute that. For three dice, each die can be from 1 to 6. But we need to ensure that for all pairs, their sum is not 7. So, given a triple (a, b, c), we need a + b ≠7, a + c ≠7, and b + c ≠7.How can we count these triples? This might be a bit more complicated, but let's try. First, let's fix the value of the first die, a. Then, for each a, we need to count the number of (b, c) such that b ≠7 -a, c ≠7 -a, and b + c ≠7.Wait, but this might get too involved. Let me see. Alternatively, since all three pairs must not sum to 7, perhaps we can model this as a graph where each die is a node, and edges represent pairs that cannot sum to 7. But maybe that's overcomplicating. Let's try another approach.Alternatively, for the first die, a, there are 6 choices. For the second die, b, it cannot be 7 -a. So if a is fixed, b can be any of 5 values. Then, for the third die, c, it cannot be 7 -a, cannot be 7 -b, and also c cannot be such that b + c =7. Wait, but this is getting complex. Let's think step by step.Let’s fix a value for a. For each a from 1 to 6:- b can be any value except 7 - a. So 5 choices.- For each such b, c cannot be 7 - a, 7 - b, or 7 - b (since b + c ≠7). Wait, actually, the constraints on c are:1. c ≠7 - a (to avoid a + c =7)2. c ≠7 - b (to avoid b + c =7)Additionally, we already fixed that a + b ≠7 since we chose b ≠7 - a. So, c just needs to avoid 7 - a and 7 - b.So, for each a, and each b ≠7 - a, the number of c's is 6 (total) minus 2 (if 7 -a and 7 -b are distinct) or minus 1 if 7 -a =7 -b.Wait, when would 7 -a =7 -b? That would mean a =b. But b ≠7 -a. If a =b, then since b ≠7 -a, substituting a =b gives a ≠7 -a => 2a ≠7 => a ≠3.5. But a is an integer from 1 to6, so this is always true. Therefore, 7 -a and 7 -b are distinct unless a =b. But in our case, since b ≠7 -a, even if a =b, as long as a ≠7 -a. Wait, this is getting a bit confusing.Alternatively, let's proceed with concrete examples.Take a specific value of a, say a=1. Then, b cannot be 6 (since 1 +6=7). So b can be 1,2,3,4,5. For each of these b's, c cannot be 7 -1=6 or 7 -b. Let's check each possible b:Case 1: a=1, b=1. Then c cannot be 6 or 6 (since 7 -1=6 and 7 -1=6). So c cannot be 6. So c can be 1,2,3,4,5. So 5 choices.Case 2: a=1, b=2. Then c cannot be 6 or 5 (since 7 -2=5). So c can be 1,2,3,4, which is 4 choices.Wait, wait, c cannot be 6 or 5. So numbers from 1-6 excluding 5 and6: 1,2,3,4. So 4 choices.Case3: a=1, b=3. Then c cannot be6 or4. So c can be1,2,3,5. 4 choices.Case4: a=1, b=4. c cannot be6 or3. So c can be1,2,4,5. 4 choices.Case5: a=1, b=5. c cannot be6 or2. So c can be1,3,4,5. 4 choices.So total for a=1: 5 (when b=1) + 4+4+4+4=5 +16=21. Wait, but 5 choices for b (since b can be1,2,3,4,5). For each b, number of c's: when b=1, 5; when b=2,3,4,5, each 4. So total c's: 5 + 4*4=5+16=21. So for a=1, 21 possible (b,c) pairs.Similarly, let's check a=2.a=2. Then b cannot be5. So b can be1,2,3,4,6.For each b:b=1: c cannot be5 (7 -2=5) and 6 (7 -1=6). So c can be1,2,3,4. 4 choices.b=2: c cannot be5 and5. So c cannot be5. So c can be1,2,3,4,6. 5 choices.b=3: c cannot be5 and4. So c can be1,2,3,6. 4 choices.b=4: c cannot be5 and3. So c can be1,2,4,6. 4 choices.b=6: c cannot be5 and1 (7 -6=1). So c can be2,3,4,5,6. Wait, 7 -6=1, so c ≠1 and 5. So c can be2,3,4,6. Wait, 7 -6=1, so c ≠1. Also, c ≠5. So c can be2,3,4,6. 4 choices.Wait, so for a=2:b=1:4, b=2:5, b=3:4, b=4:4, b=6:4. Total:4+5+4+4+4=21.Same as a=1. Hmm.Similarly, let me check a=3.a=3. Then b cannot be4. So b can be1,2,3,5,6.For each b:b=1: c cannot be4 (7 -3=4) and6 (7 -1=6). So c can be1,2,3,5. 4 choices.b=2: c cannot be4 and5 (7 -2=5). So c can be1,2,3,6. 4 choices.b=3: c cannot be4 and4. So c cannot be4. Thus, c can be1,2,3,5,6. 5 choices.b=5: c cannot be4 and2 (7 -5=2). So c can be1,3,4,5,6. Wait, c cannot be4 and2. So exclude 4 and2: c can be1,3,5,6. 4 choices.b=6: c cannot be4 and1 (7 -6=1). So c cannot be4 and1. Thus, c can be2,3,5,6. 4 choices.So total: b=1:4, b=2:4, b=3:5, b=5:4, b=6:4. Total:4+4+5+4+4=21. Same as before.Interesting. So perhaps for each a, there are 21 possible (b,c) pairs. Let me check a=4.a=4. Then b cannot be3 (since 4 +3=7). So b can be1,2,4,5,6.For each b:b=1: c cannot be3 (7 -4=3) and6 (7 -1=6). So c can be1,2,4,5. 4 choices.b=2: c cannot be3 and5 (7 -2=5). So c can be1,2,4,6. 4 choices.b=4: c cannot be3 and3 (7 -4=3). So c cannot be3. Thus, c can be1,2,4,5,6. 5 choices.b=5: c cannot be3 and2 (7 -5=2). So c can be1,3,4,5,6. Excluding 3 and2: 1,4,5,6. Wait, no. Wait, c cannot be3 (from 7 -4=3) and2 (from7 -5=2). So c ≠3 and c ≠2. So c can be1,4,5,6. 4 choices.b=6: c cannot be3 and1 (7 -6=1). So c cannot be3 and1. Thus, c can be2,4,5,6. 4 choices.Total:4+4+5+4+4=21. Again same.Similarly, for a=5 and a=6, I can check, but maybe it's safe to assume that for each a, the number of valid (b,c) pairs is 21. Thus, total number of triples where no pair sums to7 is 6 *21=126. Therefore, the number of triples where at least one pair sums to7 is 216 -126=90. Therefore, probability is90/216=15/36=5/12≈0.4167.So this method also gives 5/12. Therefore, confirms the previous result.Therefore, the answer should be 5/12.But let me just check once more for a=6.a=6. Then b cannot be1 (since6+1=7). So b can be2,3,4,5,6.For each b:b=2: c cannot be1 (7 -6=1) and5 (7 -2=5). So c can be2,3,4,6. 4 choices.b=3: c cannot be1 and4 (7 -3=4). So c can be2,3,5,6. 4 choices.b=4: c cannot be1 and3 (7 -4=3). So c can be2,4,5,6. 4 choices.b=5: c cannot be1 and2 (7 -5=2). So c can be3,4,5,6. 4 choices.b=6: c cannot be1 and1 (7 -6=1). So c cannot be1. Thus, c can be2,3,4,5,6. 5 choices.So total:4+4+4+4+5=21. Yep, same as before. Therefore, 6*21=126. 216-126=90. 90/216=5/12. Correct.So both methods lead to 5/12. Therefore, I can be confident that the probability is 5/12.Final AnswerThe probability is boxed{dfrac{5}{12}}.</think>To determine the probability that the sum of the points on 2 of the 3 dice is 7, we use the principle of inclusion-exclusion.1. Total number of outcomes: (6 times 6 times 6 = 216).2. Events: - (A): The sum of the first and second die is 7. - (B): The sum of the first and third die is 7. - (C): The sum of the second and third die is 7.3. Calculating individual probabilities: - Each pair (e.g., (1,6), (2,5), etc.) that sums to 7 has 6 outcomes. For three dice, each pair can be combined with any value of the third die, giving (6 times 6 = 36) favorable outcomes for each event. - Probability for each event: ( frac{36}{216} = frac{1}{6} ). - Total for individual events: (3 times frac{1}{6} = frac{1}{2}).4. Subtracting intersections: - For intersections of two events (e.g., (A cap B)), the third die must be such that the other two pairs sum to 7. This results in 6 favorable outcomes for each intersection. - Probability for each intersection: ( frac{6}{216} = frac{1}{36} ). - Total for intersections: (3 times frac{1}{36} = frac{1}{12}).5. Adding back the intersection of all three events: - There are no outcomes where all three pairs sum to 7, so this probability is 0.6. Applying inclusion-exclusion principle: [ P(A cup B cup C) = frac{1}{2} - frac{1}{12} + 0 = frac{6}{12} - frac{1}{12} = frac{5}{12} ]7. Verification by complementary counting: - Counting triples where no pair sums to 7 results in 126 outcomes. - Probability of complementary event: ( frac{126}{216} = frac{7}{12} ). - Probability of desired event: (1 - frac{7}{12} = frac{5}{12}).Thus, the probability that the sum of the points on 2 of the dice is 7 is (boxed{dfrac{5}{12}}).]

🔑:6. frac{5}{12}.Rolling 3 dice has a total of 6^{3}=216 possibilities. Consider 7=1+6=2+5=3+4.When rolling three dice, the possibility of two dice showing 1 and 6 is 6 times 6-6=30 (divided into (1,6, *), (1, *, 6),(6,1, *),(6, *, 1),(*, 1,6),(*, 6,1), these 6 possibilities, each with 6 cases. Among them, (1,6,1), (1,6,6),(1,1,6),(6,1,1),(6,1,6),(6,6,1) are repeated).Similarly, the possibility of two dice showing 2 and 5 when rolling three dice, and the possibility of two dice showing 3 and 4, are both 30.Therefore, the number of possibilities where 2 out of 3 dice show a sum of 7 is 3 times 30=90. The required probability is frac{90}{216}=frac{5}{12}.

🧠:[<think>Okay, let's tackle this problem step by step. The question is about finding the minimum value of |MA| + |MF| where M is a point on the parabola y² = 4x, F is the focus of the parabola, and A is a point on the circle C: (x - 4)² + (y - 1)² = 1. The options given are A: 2, B: 3, C: 4, D: 5.First, I need to recall some properties of parabolas and circles. The standard form of a parabola that opens to the right is y² = 4ax, where 'a' is the distance from the vertex to the focus. Comparing this to the given equation y² = 4x, it looks like 4a = 4, so a = 1. Therefore, the focus F of this parabola is at (a, 0) which is (1, 0).Next, the circle C is given by (x - 4)² + (y - 1)² = 1. This is a circle with center at (4, 1) and radius 1. So, point A is any point on the circumference of this circle.The problem is to find the minimum value of |MA| + |MF| where M is on the parabola and A is on the circle. Hmm, this seems like a problem where we might need to use some geometric insights or optimization techniques.Let me visualize this. The parabola y² = 4x opens to the right with vertex at the origin, focus at (1,0). The circle is centered at (4,1) with radius 1. So, the circle is located to the right of the parabola's focus. We need to find points M on the parabola and A on the circle such that the sum of the distance from M to A and from M to F is minimized.This reminds me of the reflection property of parabolas. For a parabola, any ray coming from the focus reflects off the parabola and travels parallel to the axis of the parabola. But I'm not sure if that's directly applicable here. Alternatively, maybe we can use the idea that the shortest path from a point to another via a curve is achieved by reflecting one of the points.Wait, maybe there's a way to convert this problem into a more manageable form. Let me think. If we fix point M, then |MA| + |MF| is the sum of distances from M to A and from M to F. But since A is on the circle, maybe we can think of minimizing |MA| + |MF| over both M and A.Alternatively, since A is variable on the circle, for a fixed M, the minimum |MA| would be |MC| - radius, where C is the center of the circle. Wait, that might be useful. Let's see.The circle has center at (4,1) and radius 1. So, the distance from any point M to a point A on the circle is at least |MC| - 1, by the triangle inequality. Therefore, |MA| ≥ |MC| - 1. Therefore, |MA| + |MF| ≥ |MC| - 1 + |MF|. So, maybe instead of considering both M and A, we can transform the problem into minimizing |MC| + |MF| - 1. Then, the minimum of |MA| + |MF| would be the minimum of |MC| + |MF| - 1.Therefore, if we can find the minimum of |MC| + |MF|, then subtract 1 to get the answer. Let me check if this logic holds. For any M, the minimal |MA| is |MC| - 1, so the minimal |MA| + |MF| is at least |MC| + |MF| - 1. Therefore, if we can find the minimum of |MC| + |MF| over M on the parabola, then subtract 1, that would give the minimal |MA| + |MF|.So, now the problem reduces to minimizing |MC| + |MF| where M is on the parabola y² = 4x, C is (4,1), F is (1,0). Then subtract 1.So, how can we minimize |MC| + |MF|? This is like finding a point M on the parabola such that the sum of distances from M to C and from M to F is minimized. This is a classic problem in optimization, perhaps related to Fermat-Torricelli problem, but with constraints.Alternatively, maybe reflection can help here. Let me recall that in some problems involving parabolas, reflecting a point across the focus or the directrix can help find minimal paths.Wait, the parabola's definition is the set of points equidistant from the focus and the directrix. The directrix of the parabola y² = 4x is x = -1. So, for any point M on the parabola, the distance from M to F is equal to the distance from M to the directrix x = -1.But how does that help here? Maybe if we can express |MC| + |MF| in terms that can utilize this property.Alternatively, if we consider that |MF| is equal to the distance from M to the directrix, which is x + 1 (since the directrix is x = -1, the distance from a point (x, y) to the directrix is x - (-1) = x + 1). Therefore, |MF| = x + 1 for any point M(x, y) on the parabola.Therefore, the expression |MC| + |MF| becomes |MC| + x + 1. Since M is on the parabola, x is related to y via x = y² / 4.So, maybe we can express |MC| in terms of y and then add x + 1. Let me try that.Let M be (y²/4, y). Then, the distance from M to C(4,1) is sqrt[(y²/4 - 4)² + (y - 1)²]. The distance from M to F(1,0) is sqrt[(y²/4 - 1)² + y²], but actually, we already know that |MF| is equal to x + 1, which is y²/4 + 1. Therefore, |MF| = y²/4 + 1.Wait, that's a key point. Since M is on the parabola, |MF| is equal to x + 1. So, instead of computing sqrt[(x - 1)^2 + y^2], which is the distance from M(x,y) to F(1,0), for points on the parabola y² = 4x, we can use the definition that |MF| = x + 1. Let me verify that.For a parabola y² = 4ax, the distance from any point (x, y) on the parabola to the focus (a, 0) is equal to x + a. In our case, a = 1, so |MF| = x + 1. So, yes, that's correct. So instead of calculating the actual distance, we can substitute x + 1. That's simpler.Therefore, |MF| = x + 1, where x is the x-coordinate of point M on the parabola.Therefore, the expression |MC| + |MF| becomes |MC| + x + 1. But since x = y² / 4, this becomes |MC| + y² / 4 + 1.But we need to compute |MC|. Let's compute |MC| where C is (4,1) and M is (y²/4, y). The distance is sqrt[(y²/4 - 4)^2 + (y - 1)^2]. Therefore, |MC| = sqrt[(y²/4 - 4)^2 + (y - 1)^2].So, our expression to minimize is sqrt[(y²/4 - 4)^2 + (y - 1)^2] + y²/4 + 1. Hmm, that seems a bit complicated. Maybe there's a smarter way.Alternatively, since we need to minimize |MC| + |MF|, and |MF| = x + 1, maybe we can parameterize M in terms of x or y and then take derivatives to find the minimum. But that might get messy.Alternatively, maybe using reflection. Let me think. If we need to minimize the sum of distances from M to C and from M to F, this is equivalent to finding a point M such that the path from C to M to F is minimized. In some cases, reflecting one of the points over a line or curve can help find such minimal paths.In optics, the law of reflection states that the angle of incidence equals the angle of reflection. Similarly, in optimization problems, reflecting a point across a curve can sometimes help in finding minimal paths that interact with the curve.Given that the parabola has a reflection property related to its focus, perhaps reflecting point C across the directrix or the axis of the parabola might help?Wait, the reflection property of a parabola is that any ray coming from the focus reflects off the parabola parallel to the axis, and conversely, any incoming ray parallel to the axis reflects through the focus. But how does that help here?Alternatively, maybe we can reflect point F over the parabola's directrix, but since the directrix is x = -1, reflecting F(1,0) over x = -1 would give (-3, 0). Not sure if that's helpful.Alternatively, perhaps reflecting point C over the focus F or something else.Wait, another idea. Let me recall that in some problems involving minimizing the sum of distances, you can use the method of images. For example, if you need to minimize |MC| + |MF|, you might reflect one point over the parabola and then find the straight line distance.But how does reflecting over a parabola work? It's more complicated than reflecting over a line.Alternatively, since the parabola's definition relates distances to the focus and directrix, perhaps we can use that.Wait, here's an idea. Let's note that |MF| is equal to the distance from M to the directrix, which is x + 1.So, the problem becomes minimizing |MC| + (x + 1). So, |MC| + x + 1. But x is the x-coordinate of point M. Since M is on the parabola, x = y² / 4.Alternatively, maybe consider expressing the problem in terms of coordinates.Let me denote M as (x, y) on the parabola, so x = y² / 4. Then, the expression |MC| + |MF| is sqrt[(x - 4)^2 + (y - 1)^2] + (x + 1). Substituting x = y² / 4, we get sqrt[(y² / 4 - 4)^2 + (y - 1)^2] + (y² / 4 + 1).So, this is a function of y. To find its minimum, we could take the derivative with respect to y and set it to zero. However, this might be algebraically intensive, but let's try.Let me denote f(y) = sqrt[( (y² / 4 - 4)^2 + (y - 1)^2 )] + (y² / 4 + 1)We need to find the minimum of f(y). Let's compute the derivative f'(y).First, let's compute the derivative of the square root term, let's call that term S(y):S(y) = sqrt[(y² / 4 - 4)^2 + (y - 1)^2]So, S'(y) = [ (2(y² / 4 - 4)(y/2) ) + 2(y - 1)(1) ] / (2 * S(y)) )Simplify numerator:First term: 2*(y²/4 - 4)*(y/2) = (y²/4 - 4)*ySecond term: 2*(y - 1)*1 = 2(y - 1)Therefore, numerator is y*(y²/4 - 4) + 2(y - 1)Simplify:= y³/4 - 4y + 2y - 2= y³/4 - 2y - 2Therefore, S'(y) = (y³/4 - 2y - 2) / (2 * S(y))Then, the derivative of the second term, (y²/4 + 1), is y/2.Therefore, f'(y) = [ (y³/4 - 2y - 2) / (2 * S(y)) ] + (y/2)Set f'(y) = 0:[ (y³/4 - 2y - 2) / (2 * S(y)) ] + (y/2) = 0Multiply both sides by 2 * S(y):(y³/4 - 2y - 2) + y * S(y) = 0This equation looks very complicated. Solving this analytically might be difficult. Maybe there's a better approach.Alternatively, maybe parametrize M in terms of a parameter t. For the parabola y² = 4x, a parametrization can be x = t², y = 2t. Let's use that. So, let M be (t², 2t). Then, let's express |MC| + |MF|.First, F is (1,0), so |MF| is the distance from (t², 2t) to (1,0):sqrt[(t² - 1)^2 + (2t - 0)^2] = sqrt[(t² - 1)^2 + 4t²]But earlier we saw that |MF| = x + 1 = t² + 1. Wait, that's inconsistent. Wait, if x = t², then |MF| should be x + 1 = t² + 1, but the distance formula gives sqrt[(t² - 1)^2 + (2t)^2]. Let's check this.Compute sqrt[(t² - 1)^2 + (2t)^2] = sqrt[t^4 - 2t² + 1 + 4t²] = sqrt[t^4 + 2t² + 1] = sqrt[(t² + 1)^2] = t² + 1. Oh, right! So, the distance from M to F is indeed t² + 1, which matches the formula |MF| = x + 1. That's a good check.Therefore, |MF| = t² + 1. Then, |MC| is the distance from M(t², 2t) to C(4,1):sqrt[(t² - 4)^2 + (2t - 1)^2]So, the expression to minimize is sqrt[(t² - 4)^2 + (2t - 1)^2] + t² + 1Therefore, f(t) = sqrt[(t² - 4)^2 + (2t - 1)^2] + t² + 1Again, to find the minimum, take derivative f'(t) and set to zero.Let's compute f'(t):First term: d/dt [sqrt((t² - 4)^2 + (2t - 1)^2)]Let’s denote this as sqrt(Q(t)), where Q(t) = (t² - 4)^2 + (2t - 1)^2Then, derivative is (1/(2*sqrt(Q(t)))) * [2(t² - 4)(2t) + 2(2t - 1)(2)]Simplify numerator:First part: 2(t² - 4)(2t) = 4t(t² - 4)Second part: 2(2t - 1)(2) = 4(2t - 1)Total numerator: 4t(t² - 4) + 4(2t - 1) = 4t³ - 16t + 8t - 4 = 4t³ - 8t - 4Therefore, derivative of the sqrt term is (4t³ - 8t - 4) / (2*sqrt(Q(t))) = (2t³ - 4t - 2)/sqrt(Q(t))Derivative of the second term, t² + 1, is 2t.Therefore, total derivative f'(t) = (2t³ - 4t - 2)/sqrt(Q(t)) + 2tSet f'(t) = 0:(2t³ - 4t - 2)/sqrt(Q(t)) + 2t = 0Multiply both sides by sqrt(Q(t)):2t³ - 4t - 2 + 2t * sqrt(Q(t)) = 0This equation is still quite complicated. Maybe there is a value of t that makes this true. Let's test some integer values for t.First, let's try t = 1:Compute Q(1) = (1 - 4)^2 + (2 - 1)^2 = 9 + 1 = 10Left side: 2(1)^3 -4(1) -2 = 2 -4 -2 = -4Right side: -2t * sqrt(Q(t)) = -2*1*sqrt(10) ≈ -6.324Wait, equation is 2t³ -4t -2 + 2t*sqrt(Q(t)) = 0At t=1: -4 + 2*1*sqrt(10) ≈ -4 + 6.324 ≈ 2.324 ≠ 0Not zero.Try t = 2:Q(2) = (4 -4)^2 + (4 -1)^2 = 0 + 9 = 9Left side: 2*(8) - 8 -2 = 16 -8 -2 = 6Right side: 6 + 2*2*sqrt(9) = 6 + 4*3 = 6 +12 = 18 ≠0Wait, equation is 2t³ -4t -2 + 2t*sqrt(Q(t)) =0At t=2: 16 -8 -2 + 4*3 =6 +12=18≠0Not zero.t=0:Q(0)= (0 -4)^2 + (0 -1)^2=16 +1=17Left side:0 -0 -2= -2Right side: -2 +0= -2≠0Equation: -2 + 0= -2≠0Not zero.t= -1:Q(-1)=(1 -4)^2 + (-2 -1)^2=9 +9=18Left side: -2 - (-4) -2= -2 +4 -2=0Right side: 0 + 2*(-1)*sqrt(18)= -2*4.242≈-8.484≠0Equation: 0 -8.484≈-8.484≠0t=3:Q(3)= (9 -4)^2 + (6 -1)^2=25 +25=50Left side:2*27 -12 -2=54-12-2=40Right side:40 +2*3*sqrt(50)=40 +6*7.071≈40+42.426≈82.426≠0Not zero.Hmm, not getting anywhere. Maybe this approach is too calculation-heavy. Let's think differently.We need to minimize |MC| + |MF|. Since |MF| = x +1, and x = y²/4, which can be written as t² for parameter t. So, |MF| = t² +1, and |MC| is the distance from (t², 2t) to (4,1). So, sqrt( (t² -4)^2 + (2t -1)^2 )Alternatively, maybe we can consider the point M such that the line joining C and F reflects off the parabola at point M. Wait, that's a possible approach using reflection properties.Alternatively, since the problem involves points on a parabola and a circle, perhaps the minimal value is achieved when points M, A, and F are aligned in some way, or when MA is tangent to the circle.Wait, but since A is on the circle, maybe the minimal |MA| occurs when A is the point on the circle closest to M. So, if we can model the problem as |MA| ≥ |MC| -1, then |MA| + |MF| ≥ |MC| + |MF| -1. Therefore, if we can find the minimum of |MC| + |MF|, then subtract 1 to get the answer.So, first find the minimal |MC| + |MF|, then subtract 1.Let me try to find the minimal |MC| + |MF|.Given that F is (1,0), C is (4,1). So, points F and C are fixed. We need to find a point M on the parabola y²=4x such that |MC| + |MF| is minimized.This is similar to the problem of finding a point on a curve that minimizes the sum of distances to two fixed points. In general, such a problem can be approached by reflecting one of the points over the curve and then finding the straight line distance. However, since the curve here is a parabola, reflecting over a parabola is non-trivial.Alternatively, we can use calculus to minimize the expression. Let's parameterize M as (t², 2t) as before. Then, |MF| is t² +1, and |MC| is sqrt( (t² -4)^2 + (2t -1)^2 ). So, f(t) = sqrt( (t² -4)^2 + (2t -1)^2 ) + t² +1.To find the minimum, take the derivative f’(t) and set it to zero. But as before, solving f’(t)=0 is difficult. Maybe we can approximate or find critical points numerically.Alternatively, maybe the minimal value occurs when the derivative of |MC| is equal to the negative derivative of |MF|. But I need to think.Alternatively, using the reflection property. Let me recall that the shortest path from C to F via a point M on the parabola would reflect off the parabola. But the reflection property of parabola is about reflecting to or from the focus.Alternatively, consider the following approach: For any point M on the parabola, |MF| is equal to the distance from M to the directrix. So, |MF| = x +1.Therefore, the problem is to minimize |MC| + x +1, where x = y²/4.Alternatively, we can write |MC| as sqrt( (x -4)^2 + (y -1)^2 ). So, substituting x = y²/4, we have:sqrt( (y²/4 -4)^2 + (y -1)^2 ) + y²/4 +1Let me let y = 2t for easier substitution. Then x = t². Wait, same as before. Hmm.Alternatively, consider specific points. For example, if M is the vertex of the parabola, which is (0,0). Then |MF| = 0 +1 =1, |MC| is sqrt( (0 -4)^2 + (0 -1)^2 ) = sqrt(16 +1)=sqrt(17)≈4.123. So, |MC| + |MF| ≈4.123 +1≈5.123. Then subtract 1, we get≈4.123, which is higher than the options given (the options are 2,3,4,5). But this might not be the minimal value.If M is at the point where the parabola and the circle are closest? The circle is at (4,1) with radius 1. The closest point on the parabola to the circle's center (4,1) would perhaps be somewhere on the parabola. Let's compute the distance from (4,1) to a general point (t², 2t) on the parabola: sqrt( (t² -4)^2 + (2t -1)^2 ). To minimize this, we can set its derivative to zero. Let’s compute derivative:d/dt [ (t² -4)^2 + (2t -1)^2 ] = 2(t² -4)(2t) + 2(2t -1)(2) = 4t(t² -4) + 4(2t -1) =4t³ -16t +8t -4=4t³ -8t -4Set to zero: 4t³ -8t -4=0 => t³ -2t -1=0. Let's try rational roots: t=1: 1 -2 -1=-2≠0; t=-1: -1 +2 -1=0. So t=-1 is a root. Then factor: (t +1)(t² - t -1)=0. So other roots are t=(1±sqrt(5))/2≈1.618 and -0.618. So critical points at t=-1, t≈1.618, t≈-0.618.Compute the distances:For t=-1: point (1, -2). Distance to (4,1): sqrt(9 +9)=sqrt(18)≈4.242For t≈1.618: t=(1+√5)/2≈1.618, so point (t², 2t)≈(2.618,3.236). Distance to (4,1): sqrt( (2.618 -4)^2 + (3.236 -1)^2 )≈sqrt(1.906 + 4.999)≈sqrt(6.905)≈2.628For t≈-0.618: t=(1 -√5)/2≈-0.618, point (t², 2t)≈(0.618², -1.236)≈(0.382, -1.236). Distance to (4,1): sqrt( (0.382 -4)^2 + (-1.236 -1)^2 )≈sqrt(13.03 + 5.03)≈sqrt(18.06)≈4.25So the minimal distance from C to the parabola is≈2.628 at t≈1.618, which is the closest point. Then, the minimal |MC| is≈2.628, and |MF| would be x +1≈2.618 +1≈3.618, so |MC| + |MF|≈2.628 +3.618≈6.246. Then subtract 1 gives≈5.246. But the options include 5, which is close but maybe not exactly.But the options don't have 5.246, but they have 5. Wait, but perhaps the minimal value is actually 4. Let me think.Alternatively, maybe we need to find when MA + MF is minimized, taking into account that A is on the circle, so perhaps MA is minimized when A is along the line MC, so |MA| = |MC| -1. Then, |MA| + |MF| = |MC| + |MF| -1. So, the problem reduces to minimizing |MC| + |MF|, then subtracting 1.But how do we minimize |MC| + |MF|? Maybe using reflection.Wait, here's a better idea. Since |MF| = x +1, and x = y² /4, then |MF| = y² /4 +1. Therefore, |MC| + |MF| = sqrt( (x -4)^2 + (y -1)^2 ) + x +1. Let me denote this as f(x,y) = sqrt( (x -4)^2 + (y -1)^2 ) + x +1, with the constraint y² =4x.Substituting x = y² /4 into f(x,y):f(y) = sqrt( (y² /4 -4)^2 + (y -1)^2 ) + y² /4 +1We need to find the minimum of this function. Maybe completing the square or using calculus.Alternatively, let's consider using the method of Lagrange multipliers. Let me set up the Lagrangian.We want to minimize f(x,y) = sqrt( (x -4)^2 + (y -1)^2 ) + x +1, subject to g(x,y) = y² -4x =0.The Lagrangian is L(x,y,λ) = sqrt( (x -4)^2 + (y -1)^2 ) + x +1 + λ(y² -4x)Compute the partial derivatives:∂L/∂x = [ (x -4) / sqrt( (x -4)^2 + (y -1)^2 ) ] +1 -4λ =0∂L/∂y = [ (y -1) / sqrt( (x -4)^2 + (y -1)^2 ) ] + 2λ y =0∂L/∂λ = y² -4x =0So, we have three equations:1. (x -4)/D +1 -4λ =02. (y -1)/D + 2λ y =03. y² -4x =0Where D = sqrt( (x -4)^2 + (y -1)^2 )From equation 1: (x -4)/D =4λ -1From equation 2: (y -1)/D = -2λ yLet me denote equation 1 as A =4λ -1, equation 2 as B = -2λ y, where A = (x -4)/D and B=(y -1)/D.Therefore, we have:A =4λ -1B = -2λ yAlso, from A and B, since A^2 + B^2 = [ (x -4)^2 + (y -1)^2 ] / D^2 =1/D^2 * D^2=1. Therefore:A² + B² =1So, substituting:(4λ -1)^2 + (-2λ y)^2 =1Expand:16λ² -8λ +1 +4λ² y² =1Simplify:(16λ² +4λ² y² ) -8λ +1 -1=0Thus:4λ² (4 + y² ) -8λ =0Factor out 4λ:4λ [ λ(4 + y² ) -2 ]=0Therefore, either λ=0 or λ(4 + y² ) -2=0.Case 1: λ=0Then from equation 1: (x -4)/D +1=0 => (x -4)/D = -1 => x -4 = -D => x -4 = -sqrt( (x -4)^2 + (y -1)^2 )Square both sides:(x -4)^2 = (x -4)^2 + (y -1)^2 => 0 = (y -1)^2 => y=1From constraint y²=4x => x=1/4.So, point M=(1/4,1). Check if this is a minimum.Compute f(x,y)=sqrt( (1/4 -4)^2 + (1 -1)^2 ) +1/4 +1= sqrt( (-15/4)^2 ) +5/4=15/4 +5/4=20/4=5. So, value 5. But we need to subtract 1, so 5-1=4. But wait, this is under case λ=0. However, let's check other case.Case 2: λ(4 + y² ) -2=0 => λ =2/(4 + y² )Now, from equation 1: (x -4)/D =4λ -1=4*(2/(4 + y² )) -1=8/(4 + y² ) -1=(8 - (4 + y² ))/(4 + y² )=(4 - y² )/(4 + y² )From equation 2: (y -1)/D = -2λ y= -2*(2/(4 + y² ))* y= -4y/(4 + y² )Let’s denote ratio: [(x -4)/D] / [(y -1)/D] = (x -4)/(y -1)= [ (4 - y² )/(4 + y² ) ] / [ -4y/(4 + y² ) ]= (4 - y² )/( -4y )Therefore, (x -4)/(y -1)= (4 - y² )/( -4y )Cross-multiplied:(x -4)( -4y )= (4 - y² )(y -1 )Left side: -4y(x -4)Right side: (4 - y² )(y -1 )But x = y² /4, so substitute:Left side: -4y(y² /4 -4 )= -4y( (y² -16)/4 )= -y(y² -16 )= -y³ +16yRight side: (4 - y² )(y -1 )=4y -4 -y³ + y²Therefore, equate left and right:- y³ +16y =4y -4 -y³ + y²Simplify:Left: -y³ +16yRight: -y³ + y² +4y -4Subtract right from left:0 +0 +12y - y² +4=0 => -y² +12y +4=0 => y² -12y -4=0Solutions: y=(12 ±sqrt(144 +16))/2=(12±sqrt(160))/2=(12±4sqrt(10))/2=6±2sqrt(10)Approximately sqrt(10)=3.162, so y≈6±6.324. So y≈12.324 or y≈-0.324.Now, check if these y-values satisfy the original constraint y²=4x. For each y, x=y²/4.First, y≈12.324, then x=(12.324)^2 /4≈151.94/4≈37.98. Then point M≈(37.98,12.324). The distance |MC| would be sqrt( (37.98-4)^2 + (12.324 -1)^2 )≈sqrt(1155 +128)≈sqrt(1283)≈35.82, and |MF|=x +1≈38.98. So |MC| + |MF|≈74.8. Then subtract 1≈73.8, which is way larger than previous.For y≈-0.324, x≈(0.324)^2 /4≈0.105/4≈0.026. Then |MC|≈sqrt( (0.026 -4)^2 + (-0.324 -1)^2 )≈sqrt(15.68 +1.73)≈sqrt(17.41)≈4.17, and |MF|≈0.026 +1≈1.026. So |MC| + |MF|≈5.196. Subtract 1≈4.196, which is around 4.2. So this is better than the previous case but still higher than the case when λ=0.So, comparing the two cases: when λ=0, we get f(x,y)=5, and when case 2, we get approximately 5.196. Therefore, the minimal value in this scenario is 5, achieved at point M=(1/4,1). Then, the minimal |MA| + |MF| would be 5 -1=4. So, the answer is C:4.But wait, let me check the value at M=(1/4,1). Compute |MA| + |MF|. Since A is on the circle (x-4)^2 + (y -1)^2=1, the closest point on the circle to M=(1/4,1) is along the line connecting the center of the circle (4,1) to M. The distance from M to the center is sqrt( (4 -1/4)^2 +0 )=sqrt( (15/4)^2 )=15/4=3.75. Therefore, the minimal |MA| is 15/4 -1=11/4=2.75. Then |MF| is x +1=1/4 +1=5/4=1.25. So |MA| + |MF|=2.75 +1.25=4. So, that's exactly 4. Hence, the minimal value is 4, which is option C.Therefore, the answer is C:4.Final Answerboxed{C}</think>Given the problem of finding the minimum value of ( |MA| + |MF| ) where ( M ) is on the parabola ( y^2 = 4x ), ( F ) is the focus of the parabola, and ( A ) is on the circle ( (x-4)^2 + (y-1)^2 = 1 ):1. Identify Key Points and Properties: - The parabola ( y^2 = 4x ) has its focus at ( F(1, 0) ). - The circle ( (x-4)^2 + (y-1)^2 = 1 ) has center ( C(4, 1) ) and radius 1. - For any point ( M ) on the parabola, the distance ( |MF| ) is equal to ( x + 1 ) where ( x ) is the x-coordinate of ( M ).2. Transform the Problem: - The minimum value of ( |MA| + |MF| ) can be transformed by considering the minimum distance from ( M ) to the circle center ( C ) and subtracting the radius. This gives ( |MA| geq |MC| - 1 ). - Therefore, ( |MA| + |MF| geq |MC| + |MF| - 1 ).3. Minimize ( |MC| + |MF| ): - Using the reflection property of the parabola and Lagrange multipliers, we find that the minimum value of ( |MC| + |MF| ) occurs at the point ( M left( frac{1}{4}, 1 right) ).4. Calculate the Distances: - The distance ( |MC| ) from ( M left( frac{1}{4}, 1 right) ) to ( C(4, 1) ) is ( frac{15}{4} ). - The minimum ( |MA| ) is ( frac{15}{4} - 1 = frac{11}{4} ). - The distance ( |MF| ) is ( frac{1}{4} + 1 = frac{5}{4} ).5. Sum the Distances: - The minimum value of ( |MA| + |MF| ) is ( frac{11}{4} + frac{5}{4} = 4 ).Thus, the minimum value of ( |MA| + |MF| ) is (boxed{C}).,<think>Okay, let me try to figure out how to solve this problem. So, we have a parabola y² = 4x, which I remember is a right-opening parabola. The focus F of this parabola should be at (1, 0) because the standard form of a parabola that opens to the right is y² = 4ax, where the focus is at (a, 0). Comparing that to our equation, 4a = 4, so a = 1. Therefore, the focus F is indeed (1, 0). Then, there's a point M on the parabola, so M must satisfy y² = 4x. The point A is on the circle C: (x - 4)² + (y - 1)² = 1. The circle has center at (4, 1) and radius 1. We need to find the minimum value of |MA| + |MF|. Hmm, so we need to minimize the sum of the distances from M to A and from M to F. Since both M and A are variables here—M is on the parabola, and A is on the circle—this seems a bit complicated. But maybe there's a way to approach this using reflection properties of parabolas or circles? Let me recall that in optimization problems involving distances, reflection can sometimes be useful. For instance, in the classic problem where you have to find the shortest path that reflects off a line, you reflect a point across the line and then find the straight line distance. Maybe something similar applies here?Since the problem involves a parabola and a circle, perhaps reflecting the focus across the circle's center or something like that? Wait, the parabola's focus is F at (1, 0). The circle is centered at (4, 1) with radius 1. Maybe we can use the reflection property of parabolas here. Parabolas have the property that any ray coming from the focus reflects off the parabola and travels parallel to the axis of the parabola. Conversely, any incoming ray parallel to the axis is reflected through the focus. But I'm not sure how this helps directly here.Alternatively, maybe we can use the fact that for any point on the parabola, the distance from that point to the focus is equal to the distance from that point to the directrix. The directrix of the parabola y² = 4x is x = -1. So, for any point M on the parabola, |MF| equals the distance from M to the directrix x = -1, which is just the horizontal distance since the directrix is vertical. So, if M is (x, y), then distance to directrix is x - (-1) = x + 1. But since M is on the parabola, x = y²/4. So, |MF| = x + 1 = (y²)/4 + 1. But how does that help? Maybe not directly. Let me think again about the problem. We need to minimize |MA| + |MF| where M is on the parabola and A is on the circle. Since A is on the circle, for a fixed M, the minimum |MA| would be the distance from M to the center of the circle minus the radius, right? Because the closest point on the circle to M would be along the line connecting M to the center of the circle, and the minimum distance is |MC| - 1, where C is (4,1). Wait, is that correct? Let me verify. If you have a circle with center C and radius r, then the minimum distance from a point M to any point A on the circle is |MC| - r, provided that M is outside the circle. If M is inside the circle, then the minimum distance is r - |MC|. But in this case, since the parabola is y² = 4x, which is a right-opening parabola, points on the parabola can be anywhere from x = 0 to infinity. The circle is centered at (4,1) with radius 1. So, the distance from the focus F (1,0) to the center C (4,1) is sqrt((4-1)² + (1-0)²) = sqrt(9 + 1) = sqrt(10) ≈ 3.16, which is larger than the radius 1, so the focus is outside the circle. Similarly, points on the parabola: the vertex of the parabola is at (0,0). The distance from (0,0) to center (4,1) is sqrt(16 + 1) = sqrt(17) ≈ 4.12, which is also greater than 1, so all points on the parabola are outside the circle? Wait, not necessarily. For example, a point M on the parabola could be (x, y) where x is close to 4. Let me check. The circle is centered at (4,1) with radius 1, so it spans from x=3 to x=5. The parabola at x=4 would have y²=16, so y=±4. So the point (4,4) is on the parabola. The distance from (4,4) to the center (4,1) is 3, which is greater than the radius 1, so that point is outside the circle. Similarly, any point on the parabola at x ≥ 0. The closest point on the parabola to the center (4,1) would be when the distance is minimized. Let's compute that distance squared: (x - 4)^2 + (y - 1)^2. Since y² = 4x, substitute x = y²/4. So distance squared is (y²/4 - 4)^2 + (y - 1)^2. To minimize this, take derivative with respect to y and set to zero. But maybe that's complicated. Alternatively, maybe the minimum distance is more than 1, so all points on the parabola are outside the circle. Therefore, for any M on the parabola, |MA| ≥ |MC| - 1, where C is (4,1). Therefore, the minimum of |MA| + |MF| would be greater than or equal to (|MC| - 1) + |MF|. But since we need to minimize this sum over M on the parabola. So perhaps we can write the expression as (|MC| - 1) + |MF| and then try to minimize that. But how?Wait, maybe we can switch the order. For each M, the minimal |MA| is |MC| - 1. So the total expression is |MC| - 1 + |MF|. Therefore, the problem reduces to minimizing |MC| + |MF| - 1. So if we can find the minimum of |MC| + |MF| over M on the parabola, then subtract 1 to get the answer.So perhaps the key is to minimize |MC| + |MF| where C is (4,1), F is (1,0), and M is on the parabola y²=4x.Hmm, how to minimize |MC| + |MF|. This is similar to finding a point M on the parabola such that the sum of distances from M to C and M to F is minimized. In optimization, the minimal sum of distances from a moving point to two fixed points is a classic problem, but here the moving point is constrained to lie on a parabola. Usually, without constraints, the minimal path would be the straight line between the two points, but here M has to be on the parabola. Alternatively, maybe we can use reflection properties. For instance, in the Fermat-Toricelli problem, but that's for three points. Maybe reflecting one of the points across the parabola's axis or directrix?Wait, the parabola's axis is the x-axis here since it's a right-opening parabola. The directrix is x = -1. If we reflect point F over the directrix, we get a point on the other side. Wait, but reflecting F over the directrix would be symmetric with respect to the directrix. Since F is at (1,0) and the directrix is x = -1, the reflection of F over the directrix would be (-3, 0). Because the distance from F to directrix is 1 - (-1) = 2 units, so reflecting across the directrix would subtract twice that distance from the x-coordinate. Wait, actually, reflection over the vertical line x = -1. To reflect a point (a, b) over the vertical line x = c, the reflection is (2c - a, b). So here, reflecting F (1,0) over x = -1 gives 2*(-1) - 1 = -2 -1 = -3, so (-3, 0). But how does that help? Wait, in the reflection property of a parabola, the path from F to M to a point on the directrix is such that the path reflects off the parabola. But perhaps that's not directly useful here. Alternatively, since for any point on the parabola, the distance from M to F is equal to the distance from M to the directrix. But I don't see a direct connection here.Wait, let's think of |MC| + |MF|. Maybe we can use some geometric transformations or optimization techniques. Let me consider parametrizing point M on the parabola. Let’s let M be (t², 2t) since for parabola y²=4x, parametric equations can be written as x = t², y = 2t. Then, coordinates of M are (t², 2t). Then, coordinates of C are (4,1), and coordinates of F are (1,0). So, distance MC is sqrt[(t² - 4)^2 + (2t - 1)^2], and distance MF is sqrt[(t² - 1)^2 + (2t - 0)^2]. So, the expression to minimize is sqrt[(t² - 4)^2 + (2t - 1)^2] + sqrt[(t² - 1)^2 + (2t)^2] - 1. Hmm, this seems quite complicated. Alternatively, maybe we can use calculus here. Let’s denote f(t) = |MC| + |MF| = sqrt[(t² - 4)^2 + (2t - 1)^2] + sqrt[(t² - 1)^2 + (2t)^2]. Then, find the derivative f’(t), set it to zero, and solve for t. But this might be messy. Let me see.First, compute the derivatives. Let’s denote:Let’s compute f(t) = sqrt[(t² - 4)^2 + (2t - 1)^2] + sqrt[(t² - 1)^2 + (2t)^2]Let’s denote:A = t² - 4B = 2t - 1C = t² - 1D = 2tThen, f(t) = sqrt(A² + B²) + sqrt(C² + D²)Compute derivative f’(t):First term derivative: [2A*(2t) + 2B*(2)] / (2 sqrt(A² + B²)) ) Wait, wait. Let me recall that derivative of sqrt(u) is (1/(2 sqrt(u))) * du/dt.So, derivative of sqrt[(t² - 4)^2 + (2t - 1)^2] is [2(t² - 4)(2t) + 2(2t - 1)(2)] / [2 sqrt((t² - 4)^2 + (2t - 1)^2)]Simplify numerator: 2*(t² - 4)*2t + 2*(2t -1)*2 = 4t(t² - 4) + 4(2t -1)Similarly, derivative of sqrt[(t² - 1)^2 + (2t)^2] is [2(t² -1)(2t) + 2*(2t)*2 ] / [2 sqrt((t² -1)^2 + (2t)^2)]Simplify numerator: 2*(t² -1)*2t + 2*(2t)*2 = 4t(t² -1) + 8tTherefore, putting it all together:f’(t) = [4t(t² -4) + 4(2t -1)] / [2 sqrt((t² -4)^2 + (2t -1)^2)] + [4t(t² -1) + 8t] / [2 sqrt((t² -1)^2 + (2t)^2)]Simplify each term:First term numerator: 4t(t² -4) + 4(2t -1) = 4t³ -16t +8t -4 = 4t³ -8t -4Divide by denominator 2 sqrt(...):[4t³ -8t -4] / [2 sqrt((t² -4)^2 + (2t -1)^2)] = [2t³ -4t -2]/sqrt((t² -4)^2 + (2t -1)^2)Second term numerator: 4t(t² -1) +8t = 4t³ -4t +8t =4t³ +4tDivide by denominator 2 sqrt(...):[4t³ +4t]/[2 sqrt(...)] = [2t³ +2t]/sqrt((t² -1)^2 + (2t)^2)So f’(t) = [2t³ -4t -2]/sqrt((t² -4)^2 + (2t -1)^2) + [2t³ +2t]/sqrt((t² -1)^2 + (2t)^2)Set this derivative to zero:[2t³ -4t -2]/sqrt((t² -4)^2 + (2t -1)^2) + [2t³ +2t]/sqrt((t² -1)^2 + (2t)^2) = 0This equation looks quite complicated. Solving this analytically might be challenging. Maybe there is a smarter approach.Alternatively, let's think geometrically. The problem is to minimize |MA| + |MF|, where A is on the circle C. Since A is on the circle, perhaps the minimal value occurs when A is chosen such that MA + MF is minimized. But since A is variable, maybe we can think of this as first choosing M on the parabola, then choosing A on the circle to minimize |MA|, which gives |MC| - 1, then we need to minimize |MC| -1 + |MF|. So, as before, minimize |MC| + |MF|, then subtract 1. Alternatively, maybe we can use the triangle inequality. Let me see.We want |MA| + |MF|. Let's consider |MA| + |MF| ≥ |FA|, by triangle inequality. But since A is on the circle, the minimal |FA| would be |FC| -1, where FC is the distance from F to C. Wait, FC is the distance from (1,0) to (4,1), which is sqrt((4-1)^2 + (1-0)^2) = sqrt(9 +1) = sqrt(10) ≈3.16. So |FA| ≥ |FC| - |CA|, but |CA| is the radius 1. Wait, actually, A is on the circle centered at C with radius 1, so |FA| ≥ |FC| - |CA| = sqrt(10) -1 ≈2.16. But the answer choices are integers: 2,3,4,5. So maybe 3? But this is just a rough estimation.But the triangle inequality approach might not directly give the minimal value because M is involved. The path is M to A and M to F, so maybe another approach is needed.Alternatively, since M is on the parabola, and F is the focus, perhaps we can use the definition of the parabola. For any point M on the parabola, |MF| is equal to the distance from M to the directrix. The directrix is x = -1. So |MF| = x +1 where x is the x-coordinate of M. So |MF| = x +1. But |MA| is the distance from M to A, where A is on the circle. So, to minimize |MA| + |MF|, we can write it as |MA| + (x +1). Since x is the x-coordinate of M, which is on the parabola, so x = y² /4. So, maybe substituting that in? But not sure. Alternatively, perhaps fix point A and find the minimal |MA| + |MF|, then minimize over all A on the circle. But this seems complex. Wait, maybe reflection can help here. Let's consider reflecting point F over the circle C. Wait, reflection over a circle is a concept in inversive geometry. The reflection of a point over a circle is another point such that the product of their distances to the center is equal to the square of the radius. But I'm not sure if that's helpful here. Alternatively, reflecting across the center? Wait, another thought. If we can find a point Q such that for any point A on the circle, |MA| + |MF| is equivalent to |MQ| + |MF|, then maybe we can reduce the problem. Alternatively, think of the problem as optimizing |MA| + |MF|. Since A is on the circle, perhaps we can use some form of Minkowski sum or consider the locus of points A. But I'm not sure. Wait, let's consider |MA| + |MF|. Since A is on the circle, we can think of this as |MA| + |MF| = |MF| + |MA|. For each M, |MA| is minimized when A is the closest point on the circle to M. So as before, |MA| ≥ |MC| -1. So, |MA| + |MF| ≥ |MC| -1 + |MF|. Therefore, the minimal value of |MA| + |MF| is at least the minimal value of |MC| + |MF| minus 1. Therefore, if we can find the minimal value of |MC| + |MF| for M on the parabola, then subtract 1 to get the answer. So, how to find the minimal |MC| + |MF|. Let's see, points C is (4,1), F is (1,0). So, we need to find a point M on the parabola y²=4x such that the sum |MC| + |MF| is minimized.This seems like a problem that can be approached using calculus, as we tried before, but perhaps there's a geometric interpretation. Alternatively, since F is the focus of the parabola, maybe there's a connection between the reflection property and the problem. For instance, if we consider reflecting point C over the focus or something. Wait, perhaps reflecting point C across the directrix?Wait, the directrix is x = -1. Reflecting C (4,1) across x = -1 would be a point (-6,1). Because the reflection over x = -1 would invert the x-coordinate across that line. The distance from C to the directrix is 4 - (-1) = 5, so the reflection would be 5 units on the other side: x = -1 -5 = -6. So the reflection point is (-6,1). Alternatively, maybe reflecting point F over the circle's center. Wait, F is (1,0), center C is (4,1). The reflection of F over C would be (4 + (4 -1), 1 + (1 -0)) = (7, 2). Not sure if that's helpful.Alternatively, using the method of images in optimization. Let me recall that sometimes to minimize the sum of distances, you can reflect one point and then find the distance. For example, in the shortest path problems involving reflections. Suppose we want to minimize |MC| + |MF|. If we can find a reflection such that the path MC + MF becomes a straight line in the reflected space. Alternatively, if we reflect point C over the parabola, then the minimal path from F to C via a point M on the parabola would correspond to the straight line from F to the reflection of C. But reflecting over a parabola is complicated. Wait, perhaps using the focus-directrix property. Since for any M on the parabola, |MF| = distance from M to directrix x = -1. So, |MF| = x +1. So, the sum |MC| + |MF| is |MC| + x +1. Since x is the x-coordinate of M, which is y²/4. So, |MC| + x +1. But |MC| is sqrt((x -4)^2 + (y -1)^2). So, the expression becomes sqrt((x -4)^2 + (y -1)^2) + x +1. Since x = y²/4, substitute that in:sqrt((y²/4 -4)^2 + (y -1)^2) + y²/4 +1.Hmm, still messy. Maybe take derivative with respect to y. Let me attempt that.Let’s denote f(y) = sqrt(( (y²)/4 -4 )² + (y -1)^2 ) + (y²)/4 +1.Simplify inside the sqrt:(y²/4 -4) = (y² -16)/4. So, ( (y² -16)/4 )² = (y² -16)² / 16(y -1)^2 = y² -2y +1So inside the sqrt: [ (y² -16)² /16 + y² -2y +1 ]Combine terms:= [ (y^4 -32y² +256)/16 + y² -2y +1 ]= [ y^4/16 - 2y² + 16 + y² -2y +1 ]= y^4/16 - y² -2y +17So f(y) = sqrt( y^4/16 - y² -2y +17 ) + y²/4 +1This seems very complicated to differentiate. Maybe this approach is not feasible.Alternatively, maybe parametrize M as (t², 2t) and use numerical methods to approximate the minimum. Let me pick some values of t and compute |MC| + |MF|.Let’s try t =0: M is (0,0). |MC| = sqrt(16 +1) = sqrt(17) ≈4.123, |MF| = sqrt(1 +0) =1, so total ≈5.123. Subtract 1 gives ≈4.123, but the answer choices are integers.t=1: M is (1,2). |MC| = sqrt((1-4)^2 + (2-1)^2)=sqrt(9 +1)=sqrt(10)≈3.16, |MF| = sqrt((1-1)^2 + (2-0)^2)=2, so total≈5.16. Subtract 1:≈4.16t=2: M is (4,4). |MC|=sqrt((4-4)^2 + (4-1)^2)=3, |MF|=sqrt((4-1)^2 + (4-0)^2)=5, total=8. Subtract1=7.t= -1: M is (1, -2). |MC|=sqrt((1-4)^2 + (-2 -1)^2)=sqrt(9 +9)=sqrt(18)≈4.24, |MF|=sqrt(0 +4)=2, total≈6.24. Subtract1≈5.24t=0.5: M=(0.25,1). |MC|=sqrt((0.25-4)^2 + (1-1)^2)=sqrt(14.0625)=3.75, |MF|=sqrt((0.25-1)^2 + (1-0)^2)=sqrt(0.5625 +1)=sqrt(1.5625)=1.25, total≈3.75+1.25=5, subtract1=4.t=0.6: M=(0.36,1.2). |MC|=sqrt((0.36-4)^2 + (1.2-1)^2)=sqrt(13.03 +0.04)=sqrt(13.07)≈3.616, |MF|=sqrt((0.36-1)^2 + (1.2)^2)=sqrt(0.4096 +1.44)=sqrt(1.8496)=1.36, total≈3.616+1.36≈4.976, subtract1≈3.976≈4. So that's close to 4.t=0.7: M=(0.49,1.4). |MC|=sqrt((0.49-4)^2 + (1.4-1)^2)=sqrt(12.35 +0.16)=sqrt(12.51)≈3.537, |MF|=sqrt((0.49-1)^2 +1.96)=sqrt(0.26 +1.96)=sqrt(2.22)=1.49, total≈3.537+1.49≈5.027, subtract1≈4.027.Wait, but earlier when t=0.5, the total was 5, subtract1=4.Wait, maybe even lower. Let's try t=0.4:M=(0.16,0.8). |MC|=sqrt((0.16-4)^2 + (0.8-1)^2)=sqrt(14.7456 +0.04)=sqrt(14.7856)=3.845, |MF|=sqrt((0.16-1)^2 +0.64)=sqrt(0.7056 +0.64)=sqrt(1.3456)=1.16, total≈3.845+1.16≈5.005, subtract1≈4.005.Wait, so even at t=0.4, it's about 4.005. At t=0.6, it's 3.976, which is less than 4. Hmm, but these are approximate. Maybe the minimal value is 4? The options include 4 as option C. So perhaps the answer is C:4.But need to confirm. Let me try t=0.7:Wait, earlier calculation for t=0.6, total before subtracting 1 is≈4.976, so subtract1≈3.976. But this is inconsistent with previous. Wait, wait, actually, when I tried t=0.6, |MC| + |MF| was≈3.616 +1.36≈4.976, so subtract1 gives≈3.976. But answer choices are integers, so 3.976 is close to 4, but the problem says the minimal value. However, maybe the exact minimal value is 4. Alternatively, perhaps there's a smarter way. Let's consider that the minimal value of |MA| + |MF| is equivalent to |MA| + |MF| ≥ |FA|, by triangle inequality. But A is on the circle. So the minimal |FA| is |FC| -1 = sqrt(10) -1 ≈2.16. But this is not an integer. However, since M is involved, perhaps this approach is not valid.Wait, another thought. Suppose we consider the problem as a reflection. If we can reflect the focus F over the circle, then the minimal path from the reflection to M would correspond to the sum |MA| + |MF|. Wait, reflecting F over the circle C: (x-4)^2 + (y-1)^2=1. The reflection of a point over a circle can be calculated, but it's a bit involved. The formula for inversion over a circle. The inversion of a point (a,b) over the circle (x - h)^2 + (y - k)^2 = r² is given by:(x', y') = (h + r²(a - h)/d², k + r²(b - k)/d²), where d is the distance from (a,b) to (h,k).So, reflecting F(1,0) over circle C: center (4,1), radius 1.First, compute the distance d from F(1,0) to center C(4,1):d = sqrt((4-1)^2 + (1-0)^2) = sqrt(9 +1) = sqrt(10).Then, the inversion (reflection) of F over circle C would be:x' = 4 + (1²)*(1 -4)/d² = 4 + (1)*( -3)/10 = 4 - 3/10 = 3.7y' =1 + (1²)*(0 -1)/d² =1 + (-1)/10 = 0.9So, the reflection point F' is (3.7, 0.9). But I'm not sure if this inversion helps here. Alternatively, maybe considering the reflection of the circle C over the parabola? That seems complicated. Alternatively, recall that in some problems, the minimal distance from a point to a curve plus another distance can be found by considering the intersection of certain lines or curves.Wait, let's consider that |MA| + |MF| is the sum we need to minimize. Since A is on the circle, maybe for the minimal value, the point A is such that MA and MF are aligned in a specific way. For example, A, M, and F colinear? Or A, M, and some other point colinear. Alternatively, maybe the minimal value is achieved when M is such that the derivative of |MC| + |MF| is zero, which we tried earlier but it was too complicated. Alternatively, maybe we can set up parametric equations and find critical points.Alternatively, since the minimal value is among the options given, and the approximate calculation gave us something close to 4, maybe the answer is 4. But let's think of another way.Let me recall that the minimal value of |MA| + |MF| can be seen as the shortest path from A to F via a point M on the parabola. This is similar to reflection. If we reflect F over the parabola and find the shortest distance from A to the reflection. Wait, but reflecting over a parabola is not straightforward.Alternatively, since the parabola is the set of points equidistant from F and the directrix. So, maybe if we consider the directrix x = -1, then reflecting A over the directrix and connecting to F might intersect the parabola at the optimal M. Let me try this. Reflect point A over the directrix x = -1. If A is on the circle (x-4)^2 + (y-1)^2=1, then reflecting A over x=-1 would be (x', y') where x' = -1 - (a +1), y' = y. Wait, no. To reflect a point (a,b) over the line x = -1, the reflection is (-2 -a, b). So for a general point (a,b), the reflection over x=-1 is (-2 -a, b).So, the reflection of center C (4,1) over x=-1 is (-2 -4,1) = (-6,1). Then, the reflection of the circle C over x=-1 is the circle with center (-6,1) and radius 1. Therefore, the reflection of any point A on circle C over x=-1 is a point A' on the circle centered at (-6,1) with radius 1. Now, the distance from A' to F is the same as the distance from A to the reflection of F over x=-1. Wait, but maybe this reflection helps in terms of the path. Since for any point M on the parabola, |MF| is equal to the distance from M to the directrix. So, if we reflect F over the directrix, we get a point F'. Then, the distance from M to F is equal to the distance from M to F'. Wait, no. Wait, reflection over directrix: since for any M on the parabola, |MF| equals distance to directrix. So, if we reflect F over the directrix, which would be (-3,0), then for any M on the parabola, |MF| = distance from M to directrix, which is equal to distance from M to F'. But I think this is a standard property. Wait, yes, in the definition of the parabola, the distance from any point M on the parabola to the focus F is equal to the distance from M to the directrix. So, if we reflect F over the directrix, we get a point F', and for any M on the parabola, |MF| = |MF'|. But wait, no. If you reflect F over directrix x=-1, you get F' at (-3,0). Then, the distance from M to directrix is |MF|, and since the directrix is vertical, the distance from M to directrix is horizontal distance, which is x +1. But |MF| is sqrt((x -1)^2 + y^2). For parabola y²=4x, x = y²/4. So, sqrt((y²/4 -1)^2 + y²) should equal x +1. Let's check:Compute sqrt((y²/4 -1)^2 + y²):= sqrt( ( (y² -4)/4 )² + y² )= sqrt( (y⁴ -8y² +16)/16 + y² )= sqrt( (y⁴ -8y² +16 +16y²)/16 )= sqrt( (y⁴ +8y² +16)/16 )= sqrt( (y² +4)^2 /16 )= (y² +4)/4But x +1 = y²/4 +1 = (y² +4)/4. So yes, it's equal. Therefore, |MF| = x +1 = (y² +4)/4.But how does this help with the problem?Wait, perhaps using this property, the expression |MA| + |MF| becomes |MA| + x +1. Since x is the x-coordinate of M, which is on the parabola. So, x = y²/4. Therefore, the expression is |MA| + y²/4 +1. But I'm not sure if that's helpful.Alternatively, think of |MA| + |MF| = |MA| + distance from M to directrix. But the directrix is x=-1. So, the problem is to minimize |MA| + distance from M to the line x=-1. Hmm, but this seems like a different problem.Wait, another idea. Maybe use the fact that for any point M, |MA| + |MF| ≥ |FA|, but equality holds only when M is on the segment FA. But M has to be on the parabola. So, if FA intersects the parabola, then the minimal value is |FA|. But FA might not intersect the parabola. But A is on the circle. So, the minimal |FA| for A on the circle is |FC| -1 ≈sqrt(10) -1 ≈2.16, but answer choices are integers. Maybe the minimal |FA| is 3? But 2.16 is less than 3. However, since M must lie on the parabola, maybe the actual minimal sum is higher.Alternatively, since |MA| + |MF| is being minimized over both M and A, perhaps consider the following: Fix A, then the minimal |MA| + |MF| is the distance from A to F via a point M on the parabola. Then, the minimal over all A on the circle would be the minimal such distance.But this is still vague. Maybe use Lagrange multipliers with constraints.Alternatively, since the problem gives answer choices, and through approximate calculation we saw that the minimal value before subtracting 1 is around 4.976 -1 ≈3.976, which is close to 4. Since the answer choices are integers, 4 is an option, so likely the answer is C:4.Alternatively, think of the minimal |MC| + |MF|. If we can find a point M where the line connecting C and F reflects off the parabola, then that might be the minimal path. But I’m not sure.Wait, let's consider points C(4,1), F(1,0). The line connecting C and F has slope (1-0)/(4-1)=1/3. Equation: y = (1/3)(x -1). Let's see if this line intersects the parabola y²=4x. Substitute y=(1/3)(x -1) into y²=4x:[(1/3)(x -1)]² =4x(1/9)(x -1)^2 =4xMultiply both sides by 9:(x -1)^2 =36xx² -2x +1 =36xx² -38x +1=0Solutions x=(38 ±sqrt(1444 -4))/2=(38 ±sqrt(1440))/2=(38 ±12sqrt(10))/2=19 ±6sqrt(10)Since sqrt(10)≈3.162, 6sqrt(10)≈18.97. So, x≈19 +18.97≈37.97 or x≈19 -18.97≈0.03. So, there are two intersection points. One at x≈0.03, y≈(1/3)(0.03 -1)=≈-0.323. So M≈(0.03, -0.323). The other intersection is at x≈37.97, which is far away. So the line CF intersects the parabola at two points, one near the vertex and one far away. The point near the vertex is M≈(0.03, -0.323). Let's compute |MC| + |MF| for this M. |MC|≈sqrt((0.03 -4)^2 + (-0.323 -1)^2)=sqrt(15.52 +1.78)=sqrt(17.3)≈4.16|MF|≈sqrt((0.03 -1)^2 + (-0.323 -0)^2)=sqrt(0.94 +0.104)=sqrt(1.044)≈1.022So total≈4.16 +1.022≈5.182, subtract 1 gives≈4.182, which is still about 4.18. Not lower than our previous approximation. So not helpful.Alternatively, think of the minimal value being achieved when M is such that the angles to C and F satisfy some condition. Maybe using calculus of variations. But this is too advanced.Alternatively, think of this as an optimization problem with two variables: parameterizing M on the parabola and A on the circle, then setting up equations. But this would be very involved.Wait, but since the answer choices are integers, maybe there's a trick to get the exact value. Let's consider that the minimal value is 4. If we can find points M and A such that |MA| + |MF| =4.Let’s suppose |MA| + |MF| =4. Since |MA| ≥ |MC| -1, then |MC| -1 + |MF| ≤4. Therefore, |MC| + |MF| ≤5. So, we need to find M such that |MC| + |MF|=5. Is there such a point M on the parabola where |MC| + |MF|=5?Let’s check if this is possible. For example, take M at (0,0). |MC| + |MF|=sqrt(16 +1) + sqrt(1 +0)=sqrt(17)+1≈4.123+1≈5.123>5. If M is at (1,2): |MC|=sqrt(9 +1)=sqrt(10)≈3.16, |MF|=2, so total≈5.16>5. If M is at (0.25,1): |MC|=sqrt(3.75² +0)=3.75, |MF|=sqrt(0.75² +1²)=sqrt(0.5625 +1)=sqrt(1.5625)=1.25, total=5. So here, |MC| + |MF|=5. Therefore, subtract 1 gives 4. So the minimal value is 4. Hence, answer is C.So, when M is at (0.25,1), which is (1/4,1), the sum |MC| + |MF|=3.75 +1.25=5, then subtract 1 gives 4. So this is achievable. Hence, the minimal value is 4. Therefore, the answer is C.Final Answerboxed{C}</think>Given the problem of finding the minimum value of ( |MA| + |MF| ) where ( M ) is on the parabola ( y^2 = 4x ), ( F ) is the focus of the parabola, and ( A ) is on the circle ( (x-4)^2 + (y-1)^2 = 1 ):1. Identify Key Points and Properties: - The parabola ( y^2 = 4x ) has its focus at ( F(1, 0) ). - The circle is centered at ( C(4, 1) ) with radius 1. - For any point ( M ) on the parabola, the distance ( |MF| ) is equal to the distance from ( M ) to the directrix ( x = -1 ).2. Reflection and Optimization: - The minimum distance from a point ( M ) on the parabola to a point ( A ) on the circle is ( |MC| - 1 ) (since ( A ) is on the circle). - The expression to minimize becomes ( |MC| - 1 + |MF| ).3. Key Insight: - Reflect the focus ( F ) over the directrix ( x = -1 ) to get the point ( (-3, 0) ). - Use the property that the distance from ( M ) to ( F ) is equal to the distance from ( M ) to the directrix.4. Critical Point Calculation: - For a point ( M ) on the parabola, the minimum value of ( |MC| + |MF| ) was found by considering specific points and using geometric reflections. - The point ( M ) at ( (0.25, 1) ) (or ( (1/4, 1) )) was found to satisfy ( |MC| + |MF| = 5 ).5. Result: - Subtracting the radius of the circle (1) from the total distance gives the minimum value ( 5 - 1 = 4 ).Thus, the minimum value of ( |MA| + |MF| ) is (boxed{C}).]

🔑:C Key Points: Integration of circles and conic sections; simple properties of parabolas. Topics: Comprehensive problems; challenging problems. Analysis: First, determine the equation of the directrix based on the equation of the parabola. Draw a perpendicular line MN from point M to the directrix, with N being the foot of the perpendicular. According to the definition of a parabola, |MN|=|MF|. The problem is then transformed into finding the minimum value of |MA|+|MN|. Given that A is on circle C, it is deduced that when points N, M, and C are collinear, |MA|+|MN| has its minimum value, which leads to the solution. Solution: The equation of the directrix of the parabola y^2=4x is x=-1. Draw a perpendicular line MN from point M to the directrix, with N being the foot of the perpendicular. Since point M is on the parabola y^2=4x and F is the focus of the parabola, it follows that |MN|=|MF|. Therefore, |MA|+|MF|=|MA|+|MN|. Given that A is on the circle C: (x-4)^2+(y-1)^2=1, with the center of the circle C at (4, 1) and radius r=1, it is deduced that when points N, M, and C are collinear, |MA|+|MF| is minimized. Therefore, the minimum value of (|MA|+|MF|)_{min}=(|MA|+|MN|)_{min}=|CN|-r=5-1=4. Thus, the minimum value of (|MA|+|MF|)_{min}=4. Hence, the correct choice is boxed{C}. Review: This problem focuses on the integration of circles and conic sections, examining the simple properties of parabolas and the minimum sum of distances. The key to solving the problem is to use the idea of transformation and reduction, converting the problem into finding the minimum of |MA|+|MF| when points N, M, and C are collinear.

🧠:[<think>Okay, let's try to figure out this problem step by step. So, Masha was supposed to survey 50 customers, but there were fewer customers that day. We need to find the maximum number of customers she actually surveyed. The data she collected includes three categories: impulse purchases, advertising influence, and sales consultant advice. Let me break down the information given.First, it says 7 of the respondents made an impulse purchase. Then, 75% of the remaining respondents bought the product under the influence of advertising. And the number of customers who chose based on the sales consultant's advice is one-third of those influenced by advertising. We need to model this into equations.Let me denote the total number of customers surveyed as N. Since she was supposed to survey 50 but surveyed fewer, N is less than or equal to 50, but we need the maximum possible N that satisfies the given conditions.From the problem, 7 respondents made an impulse purchase. So, that leaves N - 7 respondents who didn't make an impulse purchase. Of these remaining N - 7, 75% were influenced by advertising. Let me write that as:Number influenced by advertising = 0.75 * (N - 7)Then, the number influenced by sales consultant advice is one-third of those influenced by advertising. So:Number influenced by sales consultant = (1/3) * [0.75 * (N - 7)] = 0.25 * (N - 7)Now, all these categories should add up to the total number surveyed, which is N. So, let's set up the equation:Impulse purchases + Advertising influence + Sales consultant advice = Total surveyed7 + 0.75(N - 7) + 0.25(N - 7) = NHmm, let's simplify that. Combine the terms for advertising and sales consultant:0.75(N - 7) + 0.25(N - 7) = (0.75 + 0.25)(N - 7) = 1.00(N - 7) = N - 7So the equation becomes:7 + (N - 7) = NBut 7 + N - 7 = N, which simplifies to N = N. Wait, that's an identity, meaning it's always true. That seems odd. Did I model this correctly?Let me check again. The total surveyed is N. 7 are impulse. The remaining N - 7 are split into advertising and sales consultant. 75% of N - 7 are advertising, and the remaining 25% are sales consultant. So, 0.75(N - 7) + 0.25(N - 7) = N -7. Then, total surveyed is 7 + (N -7) = N, which checks out. So according to this, the equation is always satisfied as long as the categories are split that way. So why is the problem asking for the maximum number of customers surveyed? There must be some constraints I'm missing.Wait, maybe the problem is that the numbers of people in each category must be whole numbers. Since you can't have a fraction of a person. So 0.75*(N - 7) and 0.25*(N -7) must both be integers. Therefore, N -7 must be a number such that when multiplied by 0.75 and 0.25, both results are integers. Let me verify that.So, N -7 must be divisible by 4. Because 75% is 3/4, so if N -7 is divisible by 4, then 3/4*(N -7) is an integer, and 1/4*(N -7) is also an integer. Therefore, N -7 must be a multiple of 4. So, N -7 = 4k, where k is an integer. Therefore, N = 4k +7. Since N must be less than or equal to 50, the maximum possible N is the largest number of the form 4k +7 that is less than or equal to 50.Let's calculate that. Let's see, starting from 50, subtract 7: 50 -7 =43. Is 43 divisible by 4? 43 divided by 4 is 10.75, so no. The previous multiple would be 4*10=40, so 40 +7=47. Check if 47 is less than or equal to 50. Yes. Let's confirm. 47 -7=40, which is divisible by 4. Then 0.75*40=30, which is an integer, and 0.25*40=10, also an integer. So 30 +10=40, and 7 +40=47. That works.But wait, could there be a higher N? If N=47, but maybe N=49? Let's check. 49 -7=42. 42 divided by 4 is 10.5, which is not an integer. So 42 isn't divisible by 4. 46-7=39, 39/4=9.75, nope. 45-7=38, 38/4=9.5, nope. 44-7=37, nope. 43-7=36. Wait, 43-7=36? Wait, no. 43 is N? Wait, no. Wait, N -7 must be divisible by 4. So, if N=43, then N-7=36, which is divisible by 4. Wait, 36 divided by 4 is 9. So 36 is divisible by 4. Then, why did I think 43 was N?Wait, earlier steps: if N-7 must be divisible by 4, then N =4k +7. So possible Ns are 7,11,15,...,47,51,... but since N must be ≤50, the maximum N is 47. But wait, if N=47, then N-7=40. 40/4=10. So 0.75*40=30, 0.25*40=10. So 30+10+7=47. That works. But if N=51, that's over 50. But wait, if N=43, then N-7=36. 36 is divisible by 4, right? 36/4=9. Then 0.75*36=27, 0.25*36=9. Then 27+9+7=43. So that's also valid. But 43 is less than 47, so 47 is a higher number.Wait, but 47 is the maximum. Let me check N=47. 47-7=40. 40 is divisible by 4. So that works. What's the next possible N after 47? 47 +4=51, which is over 50, so 51 is too big. Therefore, 47 is the maximum possible N. So the answer is 47.But let me confirm once more. Let's check if N=47 satisfies all conditions. 7 impulse buyers. Remaining 40. 75% of 40 is 30 (advertising), 25% is 10 (sales consultant). 30 +10=40. Then total is 7+30+10=47. Perfect. All integers, all conditions met. And since 47 is less than 50, which is the original target, but there were fewer customers, so 47 is acceptable.Therefore, the maximum number of customers Masha could have surveyed is 47.Final AnswerThe maximum number of customers Masha could have surveyed is boxed{47}.</think>Given that Masha was supposed to survey 50 customers but surveyed fewer, we need to determine the maximum number of customers she actually surveyed, denoted as ( N ). The data collected includes three categories: impulse purchases, advertising influence, and sales consultant advice.1. Impulse Purchases: 7 respondents made an impulse purchase.2. Advertising Influence: 75% of the remaining respondents (after removing the impulse purchases) bought the product under the influence of advertising.3. Sales Consultant Advice: The number of customers influenced by the sales consultant is one-third of those influenced by advertising.We denote the total number of customers surveyed as ( N ). The remaining respondents after accounting for impulse purchases are ( N - 7 ). - The number influenced by advertising is ( 0.75(N - 7) ).- The number influenced by the sales consultant is ( frac{1}{3} times 0.75(N - 7) = 0.25(N - 7) ).The total number of surveyed customers must be an integer, and both ( 0.75(N - 7) ) and ( 0.25(N - 7) ) must also be integers. This implies ( N - 7 ) must be divisible by 4 (since 75% is equivalent to ( frac{3}{4} ), requiring ( N - 7 ) to be a multiple of 4).Thus, ( N - 7 = 4k ) where ( k ) is an integer. Therefore, ( N = 4k + 7 ). To find the maximum ( N ) less than or equal to 50, we solve for the largest ( k ):[4k + 7 leq 50 implies 4k leq 43 implies k leq 10.75]The largest integer ( k ) is 10, giving:[N = 4 times 10 + 7 = 47]Verification:- ( 47 - 7 = 40 ) (which is divisible by 4)- ( 0.75 times 40 = 30 )- ( 0.25 times 40 = 10 )- Total: ( 7 + 30 + 10 = 47 )Thus, the maximum number of customers Masha could have surveyed is (boxed{47}).,<think>Okay, let's try to tackle this problem step by step. So, Masha was supposed to survey 50 customers but ended up surveying fewer. We need to find the maximum number she could have surveyed based on the given data. First, let me parse the information given. According to her data:1. 7 respondents made an impulse purchase.2. 75% of the remaining respondents bought the product under the influence of advertising.3. The number of customers who chose the product based on the advice of a sales consultant is one-third of the number who chose the product under the influence of advertising.We need to figure out how these numbers relate to each other and find the maximum possible number of customers surveyed, which must be less than or equal to 50. Since we want the maximum, we need to see what constraints are limiting the number and how to satisfy them with integer values, since you can't survey a fraction of a person.Let me denote the total number of customers surveyed as N. According to the problem, N is less than 50, but as close to 50 as possible given the constraints.From the data:- 7 made impulse purchases.- The remaining respondents are N - 7.Of these remaining N - 7 respondents:- 75% bought under the influence of advertising. So that's 0.75*(N - 7).- The rest, which would be 25%, bought based on the sales consultant's advice. But the problem states that the number who chose based on the sales consultant is one-third of those influenced by advertising. Wait, let me check that again.Wait, hold on. The third point says: "the number of customers who chose the product based on the advice of a sales consultant is one-third of the number who chose the product under the influence of advertising."Hmm. So let's denote:Let A = number influenced by advertising.Let S = number influenced by sales consultant.Given that S = (1/3) * A.But also, from the second point, A is 75% of the remaining respondents (after the 7 impulse buyers). But wait, perhaps these are overlapping?Wait, perhaps the remaining respondents (N - 7) are split into two categories: those influenced by advertising (A) and those influenced by sales consultant (S). But according to the problem statement, 75% of the remaining bought under advertising, and the number influenced by sales consultant is one-third of those influenced by advertising. So, let's check if these two statements are consistent.First interpretation: 75% of (N - 7) are influenced by advertising, so A = 0.75*(N - 7). Then S is one-third of A, so S = (1/3)*A = (1/3)*0.75*(N - 7) = 0.25*(N - 7). But 0.25*(N - 7) is exactly 25% of the remaining respondents. So that would mean that the remaining respondents are split into 75% advertising and 25% sales consultant, which is consistent with S being one-third of A. Because 25% is one-third of 75% (since 75% divided by 3 is 25%). So that works out.Therefore, total remaining respondents: N - 7 = A + S = 0.75*(N - 7) + 0.25*(N - 7) = (0.75 + 0.25)*(N - 7) = N - 7. So that checks out.Therefore, these divisions are consistent. So, given that, then N must be such that all these numbers are integers because you can't have a fraction of a person.So, the key here is that 0.75*(N - 7) must be an integer, and 0.25*(N - 7) must also be an integer. Because both A and S need to be whole numbers.Alternatively, since S = (1/3)*A, and A = 0.75*(N - 7), then S = 0.25*(N - 7). Therefore, 0.25*(N - 7) must be an integer. Which is the same as (N - 7) must be divisible by 4, since 0.25 is 1/4. Similarly, 0.75*(N - 7) is equivalent to (3/4)*(N - 7), so (N - 7) must be divisible by 4 for both A and S to be integers.Therefore, (N - 7) must be a multiple of 4. Let me write that as:N - 7 = 4k, where k is a positive integer.Therefore, N = 4k + 7.Since N has to be less than or equal to 50, but as close to 50 as possible. So, we need the maximum integer N = 4k + 7 <= 50.Solving for k:4k + 7 <= 504k <= 43k <= 43/4k <= 10.75Since k must be an integer, the maximum k is 10. Therefore, N = 4*10 + 7 = 47.But wait, let's check if N = 47 is possible.If N = 47,then N - 7 = 40So A = 0.75*40 = 30S = 0.25*40 = 10Check if S is one-third of A: 10 is one-third of 30? Yes, because 30 / 3 = 10. So that works.So N = 47 is valid. But let's check if N can be higher. The next possible N would be N = 47 + 4 = 51, but 51 exceeds 50, which is the original target. So 47 is the maximum possible N less than 50 that satisfies all the conditions.Wait, but hold on. The problem says "there were fewer customers in the store that day," so Masha surveyed fewer than 50. But the question is "What is the maximum number of customers Masha could have surveyed, given that..." So N must be less than 50, but as high as possible. So 47 is the answer. But let me confirm again.Wait, the initial total she was supposed to survey is 50. But there were fewer customers, so she surveyed fewer. The problem is asking for the maximum number she could have surveyed, given the data conditions. So, the answer is 47. But let me check once more.Wait, another way. Let me think. If N must be such that N - 7 is divisible by 4. So possible N values are 7 + 4k. The maximum N less than 50 is when k is as large as possible. So k = 10 gives N = 47, k=11 gives N=51, which is over 50. So yes, 47 is the maximum.But wait, maybe there's a different way to interpret the problem? Let me check again.The problem states:"7 of the respondents made an impulse purchase, 75% of the remaining respondents bought the product under the influence of advertising, and the number of customers who chose the product based on the advice of a sales consultant is one-third of the number who chose the product under the influence of advertising."So, first group: 7 impulse buyers.Remaining respondents: N - 7.Of these, 75% are advertising-influenced: A = 0.75*(N -7).Then, sales consultant influenced: S = (1/3)*A.But S is part of the remaining respondents. So total remaining respondents should be A + S.But A + S = 0.75*(N -7) + 0.25*(N -7) = (N -7). So that's correct. Therefore, the entire remaining respondents are accounted for by A and S. So the structure is correct.Therefore, the key is that N -7 must be divisible by 4, because 0.75*(N -7) must be integer (since 0.75 is 3/4, so N -7 must be divisible by 4). Therefore, N must be 7 + 4k.So the maximum N less than 50 is 47. Because 47 -7 =40, which is divisible by 4. 40/4 =10, so k=10.If N=47:Impulse:7Advertising: 30Sales consultant:10Total:7+30+10=47.Yes, that adds up. And 10 is one-third of 30. All conditions satisfied.If N=46:N-7=39. 39 is not divisible by 4. 39 divided by 4 is 9.75. So 0.75*39=29.25, which is not an integer. So invalid.Similarly, N=45: N-7=38. 38/4=9.5. Not integer.N=44: N-7=37. Not divisible by 4.N=43: N-7=36. 36 is divisible by 4? 36 divided by 4 is 9. Yes. Wait, 36 is divisible by 4? Wait, 4*9=36? No, 4*9=36? Wait, 4*9=36? 4*9 is 36? Wait, no. 4*9 is 36? 4*9=36. Wait, 4*9=36. But 36 divided by 4 is 9. So 36 is divisible by 4. Wait, but 36/4=9. So yes, 36 is divisible by 4. Wait, 4*9=36, yes. Therefore, N=43 would give N-7=36. Then A=0.75*36=27, S=0.25*36=9. Then 27 +9=36, which is correct. Then total surveyed is 7+27+9=43. But 43 is less than 47, so 47 is higher. So 47 is better.Wait, but why did I think N-7 must be divisible by 4? Because A=0.75*(N-7)=3/4*(N-7) must be integer. Therefore, (N-7) must be divisible by 4. Therefore, (N-7)/4 must be integer. Therefore, N-7 is multiple of 4. Therefore, N=7+4k. So N can be 7,11,15,19,23,27,31,35,39,43,47,51,...But since N must be less than 50, the maximum N is 47.Wait, but earlier when I checked N=43, which is 7 + 4*9=43, that also works. But 47 is higher. So 47 is the maximum possible.Therefore, the answer should be 47.But let me check one more thing. Suppose N=47:Impulse:7Advertising:30Sales:10Total:47. All integers. 10 is 1/3 of 30. Correct.N=51 would be 7 + 44=51, but 51 exceeds 50, so not allowed. So 47 is the answer.Yes, I think that's solid.Final AnswerThe maximum number of customers Masha could have surveyed is boxed{47}.</think>Given that Masha was supposed to survey 50 customers but surveyed fewer, we need to determine the maximum number of customers she could have surveyed based on the given data. The key constraints are:1. 7 respondents made an impulse purchase.2. 75% of the remaining respondents bought the product under the influence of advertising.3. The number of customers who chose the product based on the advice of a sales consultant is one-third of the number who chose the product under the influence of advertising.Let ( N ) be the total number of customers surveyed. The remaining respondents after the 7 impulse purchases are ( N - 7 ). According to the problem, 75% of these remaining respondents were influenced by advertising, and the number influenced by the sales consultant is one-third of those influenced by advertising.This implies:- The number influenced by advertising is ( 0.75(N - 7) ).- The number influenced by the sales consultant is ( 0.25(N - 7) ).For these numbers to be integers, ( N - 7 ) must be divisible by 4. Therefore, we can write ( N - 7 = 4k ) where ( k ) is an integer. Hence, ( N = 4k + 7 ).To find the maximum ( N ) less than 50:[4k + 7 leq 50 implies 4k leq 43 implies k leq 10.75]The largest integer ( k ) is 10. Substituting ( k = 10 ):[N = 4(10) + 7 = 47]Verification:- Impulse purchases: 7- Remaining respondents: ( 47 - 7 = 40 )- Influenced by advertising: ( 0.75 times 40 = 30 )- Influenced by sales consultant: ( 0.25 times 40 = 10 )Check if 10 is one-third of 30: ( 10 = frac{1}{3} times 30 ), which is true.Thus, the maximum number of customers Masha could have surveyed is (boxed{47}).]

🔑:Answer: 47.Solution: Let the number of customers surveyed be x. Then, the number of customers who made a purchase under the influence of advertising is (x-7) cdot 3 / 4, and the number of customers who made a purchase on the advice of a sales consultant is (x-7)/4. Since the number of customers can only be an integer, x-7 must be divisible by 4. The maximum suitable number x, less than 50, is 47.

🧠:[<think>Okay, let's see. I have this arithmetic sequence {a_n}, and the sum of the first 13 terms is 39. I need to find the value of a_6 + a_7 + a_8. The options are A:6, B:12, C:18, D:9. Hmm. Alright, arithmetic sequences. Let me recall what I know about them.First, an arithmetic sequence is a sequence where each term after the first is obtained by adding a constant difference. So, if the first term is a_1 and the common difference is d, then the nth term is a_n = a_1 + (n-1)d. Got that.The sum of the first n terms of an arithmetic sequence is given by the formula S_n = n/2 * (2a_1 + (n-1)d) or also S_n = n*(a_1 + a_n)/2. Either formula should work. Since we're given the sum of the first 13 terms is 39, I can use that to set up an equation.Let me write that down:S_13 = 39 = 13/2 * [2a_1 + 12d]Alternatively, S_13 = 13*(a_1 + a_13)/2 = 39.Either way, maybe I can express a_1 in terms of d or vice versa. Let's try the first formula:39 = (13/2) * [2a_1 + 12d]Multiply both sides by 2 to eliminate the denominator:78 = 13*(2a_1 + 12d)Divide both sides by 13:78/13 = 2a_1 + 12d6 = 2a_1 + 12dSimplify:Divide both sides by 2:3 = a_1 + 6dSo, a_1 + 6d = 3. Hmm. Okay, that's one equation. But we have two variables here, a_1 and d. So maybe we need another equation, but the problem doesn't give more information. Wait, but the question is asking for a_6 + a_7 + a_8. Maybe I can express those terms in terms of a_1 and d and then use the equation we have.Let's compute each term:a_6 = a_1 + 5da_7 = a_1 + 6da_8 = a_1 + 7dSo, adding them together:a_6 + a_7 + a_8 = (a_1 + 5d) + (a_1 + 6d) + (a_1 + 7d) = 3a_1 + 18dHmm. So, the sum is 3a_1 + 18d. Wait, from the earlier equation, we have a_1 + 6d = 3. Let's see if we can relate this.Let me write 3a_1 + 18d as 3*(a_1 + 6d). Oh! That's exactly 3 times (a_1 + 6d). But we already know that a_1 + 6d is equal to 3. Therefore, 3*(3) = 9. So, a_6 + a_7 + a_8 = 9. So, the answer should be D:9.Wait, but let me verify to make sure I didn't make a mistake. Let me check each step again.First, sum of first 13 terms is 39. The formula gives S_13 = 13/2*(2a1 + 12d) = 39. Multiply both sides by 2: 78 = 13*(2a1 + 12d). Divide by 13: 6 = 2a1 + 12d. Divide by 2: 3 = a1 + 6d. Correct.Then, a6 + a7 + a8 = (a1 +5d)+(a1 +6d)+(a1 +7d) = 3a1 + 18d. Factor out 3: 3(a1 +6d). Since a1 +6d =3, then 3*3=9. So, 9. Answer is D. That seems correct.Alternatively, maybe another approach. In an arithmetic sequence, the sum of the first n terms is also equal to the average of the first and last term multiplied by the number of terms. So, S_13 = (a1 + a13)/2 *13 =39. Then, (a1 + a13)/2 =39/13=3. Therefore, (a1 + a13)=6.But a13 =a1 +12d. Therefore, a1 +a1 +12d=6 => 2a1 +12d=6. Then, same as before, a1 +6d=3. So, same result.Alternatively, maybe think of the middle term. In an arithmetic sequence, the average of the terms is equal to the average of the first and last term, which is also equal to the middle term. For an odd number of terms, the middle term is the (n+1)/2 th term. So, for 13 terms, the middle term is the 7th term, a7. So, the sum S_13 =13*a7=39. Therefore, a7=39/13=3. Then, a7=3.But the question is asking for a6 +a7 +a8. Since the sequence is arithmetic, a6, a7, a8 are three consecutive terms. So, a6 =a7 -d, a8 =a7 +d. Therefore, a6 +a7 +a8= (a7 -d) +a7 + (a7 +d)= 3a7. Since a7=3, then 3*3=9. Therefore, the answer is 9, which is D.Oh, that's another way to think about it! That's clever. Since the sum of the first 13 terms is 13 times the average term, which is the 7th term (since 13 is odd), so the 7th term is 3. Then, adding the 6th, 7th, and 8th terms is three times the 7th term. Therefore, 9. That's a quicker way.So both methods lead to the same answer. That gives me more confidence that D:9 is correct. Let me check if any of the other options make sense. If the answer were 6, 12, or 18, how would that happen? But according to both approaches, it's 9. So, D.I think that's solid. The answer is D.Final Answerboxed{D}</think>Given an arithmetic sequence ({a_n}), the sum of the first 13 terms is 39. We need to find the value of (a_6 + a_7 + a_8).First, recall the formula for the sum of the first (n) terms of an arithmetic sequence:[S_n = frac{n}{2} left(2a_1 + (n-1)dright)]For the given problem, the sum of the first 13 terms is 39:[S_{13} = frac{13}{2} left(2a_1 + 12dright) = 39]Multiplying both sides by 2:[78 = 13 left(2a_1 + 12dright)]Dividing both sides by 13:[6 = 2a_1 + 12d]Simplifying:[3 = a_1 + 6d]Next, we need to find (a_6 + a_7 + a_8). Using the formula for the (n)-th term of an arithmetic sequence (a_n = a_1 + (n-1)d):[a_6 = a_1 + 5d, quad a_7 = a_1 + 6d, quad a_8 = a_1 + 7d]Adding these terms together:[a_6 + a_7 + a_8 = (a_1 + 5d) + (a_1 + 6d) + (a_1 + 7d) = 3a_1 + 18d]Factoring out 3:[3(a_1 + 6d)]We already know from earlier that (a_1 + 6d = 3), so:[3 times 3 = 9]Thus, the value of (a_6 + a_7 + a_8) is (boxed{D}).,<think>Okay, so I need to solve this arithmetic sequence problem. Let me start by recalling what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference, usually denoted as ( d ). The ( n )-th term of an arithmetic sequence can be expressed as:[a_n = a_1 + (n - 1)d]where ( a_1 ) is the first term.The problem says that the sum of the first 13 terms of an arithmetic sequence ( {a_n} ) is 39. Then, we need to find the value of ( a_6 + a_7 + a_8 ).The options given are A: 6, B: 12, C: 18, D: 9. Hmm, let's break this down step by step.First, let's recall the formula for the sum of the first ( n ) terms of an arithmetic sequence. The sum ( S_n ) can be calculated by:[S_n = frac{n}{2} times (2a_1 + (n - 1)d)]Alternatively, this can also be written as:[S_n = frac{n}{2} times (a_1 + a_n)]Since the sum of the first 13 terms is 39, we can write:[S_{13} = frac{13}{2} times (2a_1 + 12d) = 39]Alternatively, using the second formula, since the 13th term ( a_{13} = a_1 + 12d ), then:[S_{13} = frac{13}{2} times (a_1 + a_{13}) = 39]Either way, let's solve for the terms. Let me use the first formula for clarity.So,[frac{13}{2} times (2a_1 + 12d) = 39]Multiply both sides by 2 to eliminate the denominator:[13 times (2a_1 + 12d) = 78]Now divide both sides by 13:[2a_1 + 12d = 6]Simplify this equation by dividing all terms by 2:[a_1 + 6d = 3]Hmm, interesting. So we have ( a_1 + 6d = 3 ). Wait a second, ( a_1 + 6d ) is actually the 7th term of the sequence. Because in an arithmetic sequence, the ( n )-th term is ( a_1 + (n - 1)d ). So, when ( n = 7 ):[a_7 = a_1 + (7 - 1)d = a_1 + 6d]Which is exactly the left-hand side of our equation. So this tells us that ( a_7 = 3 ).But the problem is asking for ( a_6 + a_7 + a_8 ). If ( a_7 = 3 ), maybe we can express ( a_6 ) and ( a_8 ) in terms of ( a_7 )?In an arithmetic sequence, each term is the average of the terms around it. Specifically, the term before and after a given term are equidistant from it. So:[a_6 = a_7 - d][a_8 = a_7 + d]Therefore, the sum ( a_6 + a_7 + a_8 ) would be:[(a_7 - d) + a_7 + (a_7 + d) = a_7 - d + a_7 + a_7 + d = 3a_7]Since ( a_7 = 3 ), then:[3a_7 = 3 times 3 = 9]Wait, that would be 9. But 9 is option D. But hold on, let me verify this again because the answer seems straightforward but the options are given, so maybe I made a mistake somewhere.Alternatively, maybe I can approach this differently. Let's consider that in an arithmetic sequence, the sum of terms equidistant from the ends is constant. For example, in the first 13 terms, the sum of the first and the 13th term is equal to the sum of the second and the 12th term, and so on. Since there are 13 terms (which is odd), the middle term is the 7th term. So the sum of the first 13 terms can also be expressed as 13 times the middle term (7th term). That is:[S_{13} = 13 times a_7 = 39]Therefore:[13a_7 = 39 implies a_7 = 3]So again, ( a_7 = 3 ). Then, as before, ( a_6 + a_7 + a_8 = 3a_7 = 9 ), which is 9, option D. But wait, I thought maybe 9 is the answer. But the options include 6, 12, 18, 9. So D is 9. Hmm, but I need to check my reasoning again.Alternatively, perhaps I can compute ( a_6 + a_7 + a_8 ) directly using the formula for the sum.Let me see. Let's express ( a_6 ), ( a_7 ), ( a_8 ) in terms of ( a_1 ) and ( d ):[a_6 = a_1 + 5d][a_7 = a_1 + 6d][a_8 = a_1 + 7d]Therefore, ( a_6 + a_7 + a_8 = (a_1 + 5d) + (a_1 + 6d) + (a_1 + 7d) = 3a_1 + 18d ).But from earlier, we found that ( a_1 + 6d = 3 ). So:[3a_1 + 18d = 3(a_1 + 6d) = 3 times 3 = 9]Yes, that's consistent. So that's 9. Therefore, the answer is D: 9.But wait, let me check if there's another way someone might approach this problem and possibly get a different answer. For example, sometimes mistakes happen when people confuse the number of terms or miscalculate the common difference.Suppose someone tried to compute ( a_1 ) and ( d ) individually. Let's see if that's possible.From the equation ( a_1 + 6d = 3 ), we only have one equation with two variables. So we can't uniquely determine ( a_1 ) and ( d ). However, the problem asks for ( a_6 + a_7 + a_8 ), which we already found to be 9 regardless of the values of ( a_1 ) and ( d ). So even without knowing ( a_1 ) and ( d ), the answer is fixed.Alternatively, maybe another approach. The sum of the first 13 terms is 39, which averages to 3 per term. Since it's an arithmetic sequence, the average term is the middle term, which is the 7th term. So again, ( a_7 = 3 ). Then, as before, the three terms around ( a_7 ) would average to ( a_7 ), so their sum is 3 times that.Therefore, the answer is D: 9. I think that's correct. Let me check if the answer is in the options. Yes, D is 9. So that's the answer.But wait, let me check again. If someone thought that the sum of three terms around the middle term is equal to three times the middle term, but maybe that's only for symmetric terms? Wait, in this case, the three terms are consecutive: 6th, 7th, 8th. So they are not symmetric around the 7th term unless we have more terms. However, in an arithmetic sequence, the average of any three consecutive terms would still be the middle term. So, ( a_6 + a_7 + a_8 = 3a_7 ). Since ( a_7 = 3 ), this sum is 9. Therefore, the answer is D.Alternatively, maybe someone could think that since the total sum is 39 over 13 terms, each term averages 3, so the middle term is 3. Then, the three terms around it would also average 3, so their sum is 9. That's the same conclusion.Alternatively, perhaps if someone tried to use the formula for the sum between terms 6 to 8. The sum from term 6 to term 8 is three terms: 6,7,8. The sum of these can be calculated as:Number of terms: 3First term: ( a_6 = a_1 + 5d )Last term: ( a_8 = a_1 + 7d )Sum ( S = frac{3}{2}(a_6 + a_8) )But since ( a_6 + a_8 = 2a_7 ), because ( a_6 = a_7 - d ), ( a_8 = a_7 + d ), so:[S = frac{3}{2}(2a_7) = 3a_7 = 9]Same result.Therefore, I'm confident that the answer is D: 9. The other options don't fit. For example, if someone miscalculates the middle term as 39 divided by 6 (if they thought 13 terms but divided by 6), but that would be 6.5, which isn't an option. Alternatively, maybe if someone thought the sum of 13 terms is 39, so each term is 3, and then multiplied by 3 to get 9. That's another way to see it. So yeah, 9 is the correct answer.Final Answerboxed{D}</think>Given an arithmetic sequence ({a_n}), the sum of the first 13 terms is 39. We need to find the value of (a_6 + a_7 + a_8).First, recall the formula for the sum of the first (n) terms of an arithmetic sequence:[S_n = frac{n}{2} times (2a_1 + (n - 1)d)]For the given problem, the sum of the first 13 terms is 39:[S_{13} = frac{13}{2} times (2a_1 + 12d) = 39]Multiplying both sides by 2:[13 times (2a_1 + 12d) = 78]Dividing both sides by 13:[2a_1 + 12d = 6]Simplifying:[a_1 + 6d = 3]This equation shows that the 7th term (a_7 = a_1 + 6d = 3).Next, we need to find (a_6 + a_7 + a_8). In an arithmetic sequence, the terms (a_6), (a_7), and (a_8) can be expressed as:[a_6 = a_7 - d, quad a_7 = 3, quad a_8 = a_7 + d]Thus, the sum is:[a_6 + a_7 + a_8 = (a_7 - d) + a_7 + (a_7 + d) = 3a_7]Since (a_7 = 3):[3a_7 = 3 times 3 = 9]Therefore, the answer is (boxed{D}).]

🔑:AnalysisThis question tests the student's mastery of the general formula for an arithmetic sequence and the formula for the sum of the first n terms. According to the formula for the sum of the first n terms of an arithmetic sequence, we have S_{13}=39. Simplifying this yields a relationship, and then using the general formula for an arithmetic sequence to express the desired expression, the value can be obtained by substitution. The key is the application of the idea of overall substitution.SolutionAccording to the formula for the sum of an arithmetic sequence, we have:therefore S_{13}=13a_1+ frac{13×12}{2} d=39,therefore Simplifying gives: a_1+6d=3, therefore a_6+a_7+a_8=a_1+5d+a_1+6d+a_1+7d=3a_1+18d=3(a_1+6d)=3×3=9. Therefore, the correct answer is boxed{D}.

🧠:[<think>Okay, so I have this problem where there's a unit square, and at each of its four vertices, they've erected perpendiculars to the plane of the square. On these perpendiculars, on one side of the square's plane, there are points at distances 3, 4, 6, and 5 from the plane, given in the order of traversal. I need to find the volume of the polyhedron formed by these points and the original square's vertices.Hmm. Let me visualize this first. A unit square, so all sides are 1 unit long. The perpendiculars are like poles sticking out from each corner, all on the same side of the square. Each pole has a point at a certain height: 3, 4, 6, 5. Since they're in the order of traversal, I assume that means going around the square, like clockwise or counter-clockwise. So the heights correspond to each vertex in order.The polyhedron's vertices are both the original square's vertices and these four points on the perpendiculars. So the polyhedron is like a combination of the square and these four points connected in some way. Maybe it's a kind of pyramid or a prism, but since the heights are different, it's probably a more complex shape.I remember that volumes of such polyhedrons can sometimes be calculated by dividing them into simpler shapes like pyramids or using formulas like the scalar triple product. But since the heights are different, maybe it's a frustum-like shape or something with a non-uniform top.Wait, maybe this is a type of prismatoid? Because a prismatoid is a polyhedron with all vertices lying in two parallel planes. In this case, the original square is in one plane, and the four points are in another plane (since they're all on perpendiculars to the original plane). Wait, but actually, they are on different perpendiculars, so they might not lie on a single plane. Hmm, if the four points are not coplanar, then the top is a quadrilateral in 3D space, not necessarily planar. So the polyhedron isn't a prismatoid. Then how?Alternatively, maybe think of it as a combination of pyramids with the square as the base. But since each vertex has a different height, that complicates things. Wait, but if we connect each of the top points to the square, maybe we can decompose the polyhedron into four tetrahedrons, each with three edges from the square and one edge going up to the top point. Wait, but maybe not. Let me think.Alternatively, perhaps the volume can be calculated by some formula related to the heights. I vaguely recall that if you have a base polygon and points elevated perpendicularly from each vertex, the volume might be related to the area of the base times the average height? But I need to check if that's accurate.Wait, for a prism, where all heights are equal, the volume is indeed base area times height. For a pyramid, it's (base area × height)/3. But here, the heights are different. So maybe there's a generalized formula.Wait, here's a thought. If you have a base polygon and lift each vertex to a different height, forming a polyhedron, the volume can be computed as (1/3) times the area of the base times the average height? Wait, not sure. Let me think. If all heights are the same, that would give the volume of a prism, but (1/3) base area times height is for a pyramid. So maybe not.Alternatively, maybe the volume is (1/3) times the sum over each edge of something? Hmm, I might need to recall the formula for a non-uniform extrusion.Alternatively, perhaps use the divergence theorem or integrate over the area. But that might be complicated.Alternatively, since all the vertical edges (the perpendiculars) are orthogonal to the base square, maybe we can decompose the polyhedron into simpler components. Let's try.Suppose we consider the original square in the plane z=0, and the four points at (x_i, y_i, h_i), where (x_i, y_i) are the square's vertices, and h_i are 3,4,6,5.So, the polyhedron is bounded by the original square (z=0) and the four vertical edges, and the top quadrilateral (though the top points might not be coplanar). Then, the sides of the polyhedron are quadrilaterals connecting each vertical edge.Alternatively, perhaps the volume can be found by integrating the height over the area. Wait, if you imagine that at each point (x,y) on the base, the height is some interpolated value between the heights at the corners, then the volume would be the integral of the height function over the base. But since the heights are only given at the corners, and the interpolation isn't specified, this might not be straightforward. But maybe it's a linear interpolation, forming a ruled surface.Alternatively, if we model the top surface as a bilinear surface (a hyperbolic paraboloid) interpolating the four heights, then the volume can be computed using some formula. I think in that case, the volume is equal to the average of the heights multiplied by the base area. Wait, is that the case?Wait, suppose we have four heights at the corners. If the top surface is a bilinear interpolation, then the volume under that surface over the unit square would be the average of the four corner heights times the area. So if the base area is 1, then the volume would be (3 + 4 + 6 + 5)/4 = 18/4 = 4.5. Then, since the original problem includes both the base and the top, is that the volume?Wait, but in the problem, the polyhedron's vertices are both the original square's vertices and the four elevated points. So the polyhedron is the space between the square and the four points. If the top is a bilinear surface connecting the four elevated points, then integrating the height would give the volume. But if the top is instead made of triangles or something else, the volume could be different.But without more information about how the sides are connected, it's ambiguous. However, in standard problems like this, it's usually assumed that the top is a ruled surface, connecting corresponding edges, so forming a kind of prismoid. In which case, there's a formula called the prismoidal formula, which is Volume = (h1 + h2 + h3 + h4)/4 * Area, but I need to check.Wait, no, the prismoidal formula is more specific. For a prismoid, which is a polyhedron with two parallel polygonal bases and lateral faces that are trapezoids or triangles. The formula for the volume is (Height/6)*(Area1 + Area2 + 4*MiddleArea), but in this case, the top isn't a square, so maybe not.Alternatively, since the problem mentions the polyhedron's vertices are the original square's vertices and the four elevated points, the polyhedron is likely composed of the original square, the four elevated points, and the lateral faces connecting them. So, each edge of the square is connected to the corresponding elevated point, and the top edges connect the elevated points in order.Thus, the polyhedron is a type of prismatoid with the base a square and the top a quadrilateral (possibly non-planar), connected by four trapezoidal faces. If the top is a planar quadrilateral, then it's a frustum of a pyramid, but given the heights are different, the top is probably not planar.Hmm, tricky. Maybe we can use the general formula for the volume of a prismatoid. Wait, the formula for a prismatoid is Volume = (1/6) * h * (B1 + B2 + 4*M), where h is the distance between the two planes, B1 and B2 are the areas of the two bases, and M is the area of the midsection. But in our case, the two bases are the original square and the top quadrilateral. However, since the top quadrilateral is not necessarily planar, the prismatoid formula might not apply.Alternatively, perhaps we can divide the polyhedron into tetrahedrons. Since any convex polyhedron can be divided into tetrahedrons, and then sum their volumes. Let's try that approach.First, let's assign coordinates to the square. Let's assume the square is in the xy-plane with vertices at (0,0,0), (1,0,0), (1,1,0), (0,1,0). Then, the four elevated points are at (0,0,3), (1,0,4), (1,1,6), (0,1,5). So we have the original square at z=0 and the four points above each vertex with the given heights.So the polyhedron is bounded by the square and these four points. The faces would be:1. The original square (z=0).2. Four vertical edges connecting each square vertex to its elevated point.3. The four lateral faces: each face connects two adjacent elevated points and their corresponding square edges.4. The top "roof" formed by connecting the elevated points in order.But since the top is a quadrilateral in 3D space (the four elevated points), which might not be planar, the polyhedron is a type of a non-convex or twisted prism. However, given the order of traversal, it's probably a convex polyhedron.To compute the volume, maybe we can use the following method: the volume can be found by subtracting the space below the polyhedron from some larger volume. But perhaps a better way is to use the divergence theorem or coordinates.Alternatively, use the shoelace formula in 3D. But I don't recall such a formula. Alternatively, use coordinates and compute the volume via determinants.Yes, since we can represent the polyhedron as a collection of tetrahedrons. For example, we can pick a common vertex and divide the polyhedron into tetrahedrons from that vertex.Let's choose the origin (0,0,0) as the common vertex. Then, we can decompose the polyhedron into tetrahedrons each having (0,0,0) as a vertex. Wait, but I need to check how.Alternatively, another approach is to use the convex hull of the eight points: the four original square vertices and the four elevated points. But actually, the problem states that the polyhedron's vertices are the specified points (the elevated ones) and the original square's vertices. So it's their convex hull? Or is it the polyhedron formed by connecting each edge of the square to the elevated points and forming the lateral faces.Wait, perhaps it's better to model the polyhedron as follows:The base is the original square. The top is the quadrilateral connecting the four elevated points. The sides are quadrilaterals connecting each edge of the base to the corresponding edge of the top. However, since the top edges are not necessarily parallel to the base edges, these side faces might not be planar.But in 3D, the volume can be calculated by integrating over the base. If we can express the height at each point (x,y) as a function, then the volume would be the integral of that function over the unit square.But since the heights are given only at the four corners, maybe the height function is interpolated bilinearly. If that's the case, the volume can be computed as the average of the four heights multiplied by the area of the base. Since the base area is 1, the volume would be (3 + 4 + 6 + 5)/4 = 18/4 = 4.5. However, this is assuming a bilinear interpolation, which forms a hyperbolic paraboloid. But is this the correct interpretation?Alternatively, if we use linear interpolation along the edges and then across, maybe the volume is different. Let me check.Alternatively, perhaps the formula for the volume in such a case is indeed the average height multiplied by the base area. For instance, in a similar way to how the area under a piecewise linear function over a grid is the average of the corner values times the area. If we have a unit square with heights at the corners, the volume under a bilinear interpolation is indeed (h1 + h2 + h3 + h4)/4. But I need to verify this.Let me recall: for a bilinear interpolation over a unit square, the function is f(x,y) = a + bx + cy + dxy. The integral over the unit square would be the average value, which is the integral from 0 to 1 in x and y of f(x,y) dx dy. Let's compute that:∫₀¹ ∫₀¹ (a + bx + cy + dxy) dx dyFirst integrate with respect to x:∫₀¹ [a x + (b/2)x² + c y x + (d/2)x² y] from 0 to 1= [a + b/2 + c y + d/2 y] dyNow integrate with respect to y:∫₀¹ (a + b/2) dy + ∫₀¹ c y dy + ∫₀¹ (d/2) y dy= (a + b/2)(1) + (c/2)(1²) + (d/4)(1²)= a + b/2 + c/2 + d/4Now, the coefficients a, b, c, d can be determined by the four corner values. Let's denote the heights at the corners as follows:At (0,0): h1 = f(0,0) = aAt (1,0): h2 = f(1,0) = a + bAt (1,1): h3 = f(1,1) = a + b + c + dAt (0,1): h4 = f(0,1) = a + cSo we can solve for a, b, c, d:From h1: a = h1From h2: h1 + b = h2 => b = h2 - h1From h4: h1 + c = h4 => c = h4 - h1From h3: h1 + (h2 - h1) + (h4 - h1) + d = h3 => h2 + h4 - h1 + d = h3 => d = h3 - h2 - h4 + h1Thus, the integral (average value) is:a + b/2 + c/2 + d/4Substituting the values:= h1 + (h2 - h1)/2 + (h4 - h1)/2 + (h3 - h2 - h4 + h1)/4Simplify term by term:First term: h1Second term: (h2 - h1)/2Third term: (h4 - h1)/2Fourth term: (h3 - h2 - h4 + h1)/4Combine all terms:= h1 + (h2/2 - h1/2) + (h4/2 - h1/2) + (h3/4 - h2/4 - h4/4 + h1/4)Combine like terms:h1 - h1/2 - h1/2 + h1/4 + h2/2 - h2/4 + h4/2 - h4/4 + h3/4Calculate coefficients for each h:h1: 1 - 1/2 -1/2 + 1/4 = 0 + 1/4 = 1/4h2: 1/2 - 1/4 = 1/4h3: 1/4h4: 1/2 - 1/4 = 1/4So total integral = (h1 + h2 + h3 + h4)/4Thus, the average height is indeed (h1 + h2 + h3 + h4)/4, so the volume under the bilinear interpolation is equal to the average height times the base area. Since the base area is 1, the volume is just 18/4 = 4.5.But wait, in our problem, is the polyhedron exactly the volume under this bilinear surface? Because the problem says the polyhedron's vertices are the specified points and the square's vertices. So, if we connect the elevated points with a bilinear surface, then the polyhedron is bounded by the original square, the four vertical edges, and the bilinear surface. Then, the volume would indeed be 4.5.But maybe the problem assumes that the top is a planar quadrilateral? If that's the case, then the volume would be different. Let's check if the four elevated points are coplanar.To check coplanarity, let's see if the volume of the tetrahedron formed by any three points and another is zero. Let's take the four points: A'(0,0,3), B'(1,0,4), C'(1,1,6), D'(0,1,5). Compute the volume of the tetrahedron A'B'C'D'.Wait, but four points in 3D space are coplanar if the scalar triple product of vectors formed by them is zero. Let's pick three vectors from point A' and check if they are linearly dependent.Vectors:A'B' = (1,0,1)A'C' = (1,1,3)A'D' = (0,1,2)Compute the scalar triple product [A'B' , A'C' , A'D'] = determinant of the matrix formed by these vectors as columns.The determinant is:|1 1 0||0 1 1||1 3 2|Calculating this:1*(1*2 - 1*3) - 1*(0*2 - 1*1) + 0*(0*3 - 1*1)= 1*(2 - 3) - 1*(0 - 1) + 0*(0 - 1)= 1*(-1) - 1*(-1) + 0= -1 + 1 = 0So the scalar triple product is zero, which means the four points A', B', C', D' are coplanar. Therefore, the top quadrilateral is planar. Therefore, the polyhedron is a frustum-like shape with the top face being a planar quadrilateral. Therefore, the volume can be calculated as the average of the heights times the base area? Wait, but if it's a frustum of a pyramid, then the formula is different.Wait, if both the base and the top are planar and parallel, then the volume is (h/3)(B1 + B2 + sqrt(B1*B2)), but in this case, the top is not parallel to the base. Since the base is a square in the z=0 plane, and the top is a quadrilateral in some plane. However, since the top is planar but not parallel to the base, the volume would require a different approach.Alternatively, since we found that the four elevated points are coplanar, we can compute the volume of the polyhedron as the volume of the prismatoid with the two bases being the original square and the top quadrilateral. The formula for the volume of a prismatoid is:Volume = (1/6) * h * (B1 + B2 + 4*M)where h is the distance between the planes, B1 and B2 are the areas of the two bases, and M is the area of the midsection. However, in our case, the two bases are not parallel, so this formula doesn't apply directly.Alternatively, since the top is planar, we can consider the polyhedron as a type of wedge. Maybe we can split the polyhedron into two tetrahedrons. Let's see.Alternatively, use coordinates and compute the volume via the divergence theorem or by coordinates.Since we have coordinates for all eight vertices (the four original and four elevated), we can use the convex hull of these points. But since the original square is in z=0 and the elevated points are above, the polyhedron is the convex hull of these points. However, since the elevated points are coplanar, the convex hull consists of the original square, the top quadrilateral, and the four trapezoidal faces connecting them.But perhaps we can compute the volume by decomposing the polyhedron into pyramids. For example, choosing a point as the apex and computing the volume.Alternatively, use the formula for the volume of a prismatoid when the top and bottom are polygons (not necessarily parallel or similar) but connected in a certain way. However, I'm not sure about the exact formula here.Wait, here's an idea. If the top and bottom faces are both quadrilaterals, and they are connected by their corresponding edges, forming a prismatoid, then the volume can be computed using the following formula:Volume = (1/3) * | (a × b + c × d) · (h1 + h2 + h3 + h4)/4 | ?Wait, maybe not. Alternatively, use the scalar triple product.Alternatively, use the shoelace formula extended to 3D. But I need to think step by step.Alternatively, since we know the coordinates of all eight vertices, perhaps we can divide the polyhedron into tetrahedrons whose volumes we can compute and sum them up.Let me try that. Let's list all the vertices:Original square (z=0):A: (0,0,0)B: (1,0,0)C: (1,1,0)D: (0,1,0)Elevated points:A': (0,0,3)B': (1,0,4)C': (1,1,6)D': (0,1,5)Now, the polyhedron consists of these eight points. But actually, in the problem statement, the polyhedron's vertices are the specified points (i.e., A', B', C', D') and the original square's vertices (A, B, C, D). So the polyhedron is the convex hull of these eight points? Wait, but the original square is part of the polyhedron. Wait, no, the original square's vertices are included, but the edges and faces need to be connected properly.But given that the problem states "the polyhedron whose vertices are the specified points and the vertices of the square," so all eight points are vertices. Thus, the polyhedron is the convex hull of these eight points. However, since the elevated points are coplanar (as we found earlier), the convex hull would consist of two squares (the original and the elevated) connected by four trapezoidal faces, but since the elevated square is not parallel to the original, the shape is a type of prismatoid.But given that the top quadrilateral is planar and the bottom is the original square, perhaps we can compute the volume by splitting the polyhedron into two parts: the original square-based pyramid with apex at some point, but since the top is a quadrilateral, maybe split it into two tetrahedrons and a prism?Alternatively, split the polyhedron into pyramids with the original square as the base. Wait, but there are four elevated points.Alternatively, use the coordinates and apply the divergence theorem. Since the polyhedron is a closed 3D figure, but this might be complex.Alternatively, use the following formula for the volume of a polyhedron given its vertices: divide the polyhedron into tetrahedrons, compute each volume using determinants, and sum them up.To do this, we need to choose a decomposition. Let's pick a common vertex and decompose the polyhedron into tetrahedrons from that vertex. Let's choose vertex A (0,0,0). Then, we can decompose the polyhedron into tetrahedrons that all include A.But the polyhedron has vertices A, B, C, D, A', B', C', D'. Let's see:First, consider the tetrahedrons:1. A, B, C, A'But wait, need to ensure that the tetrahedrons lie entirely within the polyhedron. Alternatively, perhaps decompose the polyhedron into simpler parts.Alternatively, split the polyhedron into the original square and the four "walls" connecting each edge to the elevated points, and the top quadrilateral.But maybe a better approach is to split the entire polyhedron into tetrahedrons such that each tetrahedron is uniquely defined and their volumes can be summed. For example, connect A to all elevated points and decompose accordingly.But this might be complicated. Let me try:First, consider the square base ABCD. Connect each elevated point A', B', C', D' to the square. The polyhedron can be divided into the following parts:1. The square base ABCD.2. The top quadrilateral A'B'C'D'.3. Four quadrilaterals connecting AB to A'B', BC to B'C', CD to C'D', DA to D'A'.But this is the structure. To compute the volume, perhaps split the polyhedron into pyramids. For example, the volume can be calculated as the sum of the volumes of four pyramids, each with a triangular base on the original square and apex at the elevated point.Wait, no, because each elevated point is only connected to one vertex. Alternatively, think of each edge of the square connected to the elevated points, forming vertical edges.Alternatively, perhaps split the polyhedron into five tetrahedrons: for example, A-B-B'-A', B-C-C'-B', C-D-D'-C', D-A-A'-D', and the middle part. But this might not cover the entire volume.Alternatively, use the following decomposition:1. The lower part, which is the original square.2. The upper part, which is the top quadrilateral.But how?Alternatively, perhaps split the polyhedron into four tetrahedrons:1. A-B-B'-A'2. B-C-C'-B'3. C-D-D'-C'4. D-A-A'-D'But this leaves out the central part. Alternatively, also include the tetrahedrons A'-B'-C'-D', but since that's a quadrilateral, not a tetrahedron.Alternatively, since the top face is planar (we checked coplanar), maybe split the top quadrilateral A'B'C'D' into two triangles, say A'B'C' and A'C'D', then compute the volume of the polyhedron as the sum of two prism-like shapes.But prisms have parallel bases. Alternatively, decompose the entire polyhedron into tetrahedrons.Let me try this approach. Let's split the polyhedron into tetrahedrons such that all the vertices are included. Let's start by selecting tetrahedron A-B-C-A'. Wait, but A' is above A. Wait, maybe not. Let me see.Alternatively, here's a systematic way. The polyhedron can be divided into five tetrahedrons:1. A-B-C-D (the original square, but this is a square, not a tetrahedron). So, need to split the square into two triangles first. For example, split the square into triangles A-B-C and A-C-D.Then, for each triangle, connect to the elevated points. But this might complicate things.Alternatively, consider the following five tetrahedrons:1. A-B-C-A'2. A-C-D-A'3. A'-B'-C'-D'4. A'-B'-B-A5. C'-D'-D-CBut this is getting too vague. Let's use coordinates and compute the volume.The coordinates are:A: (0,0,0)B: (1,0,0)C: (1,1,0)D: (0,1,0)A': (0,0,3)B': (1,0,4)C': (1,1,6)D': (0,1,5)We need to compute the volume of the convex hull of these points.Alternatively, use the Cayley-Menger determinant. But that might be complicated.Alternatively, consider the following decomposition:The polyhedron can be divided into the original square ABCD and four "walls" which are quadrilaterals between each edge of the square and the corresponding edge of the top quadrilateral A'B'C'D'. Since A', B', C', D' are coplanar, and the original square is in another plane, the volume can be calculated as follows.Since both the base and the top are planar, even though they are not parallel, the volume can be calculated using the formula for a non-parallel prismatoid. Wait, but I don't recall such a formula.Alternatively, use the general formula for the volume between two polygons in 3D space connected by a non-planar lateral surface. I think this is complex.Alternatively, use the following approach: the volume can be found by subtracting the volume below the polyhedron from some reference volume. But I'm not sure.Wait, here's another idea. Since the four elevated points are coplanar, we can find the equation of that plane and then compute the volume between the original square and this plane, bounded by the vertical edges.The top plane equation can be found using the four points A', B', C', D'. Let's compute the equation of the plane.We have four points:A' (0,0,3)B' (1,0,4)C' (1,1,6)D' (0,1,5)We already confirmed they are coplanar. To find the equation of the plane, let's compute the normal vector.Take vectors A'B' = (1,0,1), A'C' = (1,1,3). The cross product of these vectors will give the normal vector.Compute A'B' × A'C':|i   j   k ||1   0   1||1   1   3|= i*(0*3 - 1*1) - j*(1*3 - 1*1) + k*(1*1 - 0*1)= i*(0 - 1) - j*(3 - 1) + k*(1 - 0)= -i - 2j + kSo the normal vector is (-1, -2, 1). The equation of the plane can be written as:-1(x - 0) - 2(y - 0) + 1(z - 3) = 0Simplify:- x - 2y + z - 3 = 0=> z = x + 2y + 3So the equation of the top plane is z = x + 2y + 3.Now, the volume between the original square (z=0) and this plane (z = x + 2y + 3) over the unit square can be computed as the double integral over the unit square of (x + 2y + 3) dx dy.Compute this integral:∫₀¹ ∫₀¹ (x + 2y + 3) dx dyFirst integrate with respect to x:∫₀¹ [ (1/2)x² + 2y x + 3x ] from 0 to 1= [ (1/2)(1)² + 2y(1) + 3(1) ] - [0]= 1/2 + 2y + 3= 2y + 3.5Now integrate with respect to y:∫₀¹ (2y + 3.5) dy= [ y² + 3.5y ] from 0 to 1= (1 + 3.5) - 0= 4.5So the volume under the plane z = x + 2y + 3 over the unit square is 4.5, which is the same as the average height times the area. Hence, this confirms the earlier result.Therefore, the volume of the polyhedron is 4.5, which is 9/2. So, in boxed form, boxed{dfrac{9}{2}}.Final AnswerThe volume of the polyhedron is boxed{dfrac{9}{2}}.</think>To find the volume of the polyhedron formed by the vertices of a unit square and the points erected perpendicularly from each vertex at distances 3, 4, 6, and 5, we start by noting that these points are coplanar. 1. Coordinates of the vertices: - Original square vertices: (A(0,0,0)), (B(1,0,0)), (C(1,1,0)), (D(0,1,0)) - Elevated points: (A'(0,0,3)), (B'(1,0,4)), (C'(1,1,6)), (D'(0,1,5))2. Checking coplanarity: - The scalar triple product of vectors formed by these points confirms they are coplanar.3. Equation of the plane: - The plane equation is found using the normal vector from the cross product of vectors ( overrightarrow{A'B'} ) and ( overrightarrow{A'C'} ), resulting in the plane equation ( z = x + 2y + 3 ).4. Volume calculation: - The volume under the plane ( z = x + 2y + 3 ) over the unit square is computed by integrating the height function over the unit square: [ int_0^1 int_0^1 (x + 2y + 3) , dx , dy ] - Evaluating the integral: [ int_0^1 left[ frac{1}{2}x^2 + 2yx + 3x right]_0^1 dy = int_0^1 (2y + 3.5) , dy = left[ y^2 + 3.5y right]_0^1 = 4.5 ]Thus, the volume of the polyhedron is (boxed{dfrac{9}{2}}).,<think>Okay, so I have this problem here where there's a unit square, and at each of its four vertices, they've erected perpendiculars to the plane of the square. On each of these perpendiculars, on the same side of the square's plane, there are points at distances 3, 4, 6, and 5 from the plane. The order is given as the order of traversal, which I assume means going around the square, like clockwise or something. The question is to find the volume of the polyhedron formed by these four points and the original four vertices of the square. Hmm, okay.First, let me visualize this. A unit square, so each side is length 1. Let me label the square's vertices as A, B, C, D going around clockwise. Then from each of these points, there's a perpendicular line sticking out of the plane. On each of these lines, they've placed a point above the square at heights 3, 4, 6, and 5 respectively. So, for example, if A is (0,0,0), then the point above A is (0,0,3), then above B is (1,0,4), above C is (1,1,6), above D is (0,1,5). The original square is in the z=0 plane, and these new points are at z=3,4,6,5. So the polyhedron has vertices at the original square's corners (z=0) and these elevated points (z=3,4,6,5). So the polyhedron is like a sort of prismatoid, connecting the square base to the top quadrilateral formed by these four points. Wait, but a prismatoid is a polyhedron with all vertices lying in two parallel planes. In this case, the original square is in z=0, and the top points are all in different z's. Wait, no, they are in different heights. So the top isn't a plane, but a skew quadrilateral? Hmm. So maybe the polyhedron isn't a prismatoid. Hmm. So maybe the volume can be computed by breaking it down into simpler parts.Alternatively, since all the edges connecting the base square to the top points are perpendicular to the base, maybe there's a formula for such a shape. Wait, but the top isn't a translated version of the base, because the heights are different. If all the heights were the same, it would be a prism, and the volume would be base area times height. But here, heights vary. So maybe it's a type of frustum, but not with similar top and bottom bases. Alternatively, maybe we can use the formula for the volume of a non-uniform prismatoid. Let me recall. For a prismatoid, the volume is (h/6)(A1 + A2 + 2Am), where h is the distance between the planes, A1 and A2 are the areas of the two bases, and Am is the area of the midsection. But in this case, the top isn't a plane, so the prismatoid formula might not apply directly.Wait, but maybe I can model this as a combination of pyramids or frustums. Each vertical edge from the base to the top has a different height. Alternatively, maybe decompose the polyhedron into tetrahedrons. If I can split the polyhedron into tetrahedrons whose volumes I can compute and sum them up.Alternatively, use the shoelace formula extended to 3D? Wait, not sure. Alternatively, maybe consider the volume as the integral over the area of the square of the height function. Hmm, since each point on the square is connected to a point above it with varying heights. Wait, but the heights are only given at the four corners. So maybe we can model this as a linear interpolation between the heights? Like a tent function over the square. Then the volume would be the integral of the height function over the square.Yes, that might work. If the heights at the four corners are given, and we assume that the top surface is a bilinear interpolation (a hyperbolic paraboloid), then the volume under that surface would be the average of the heights times the area of the base. Wait, but actually, for a linear interpolation over a square grid, the volume under the surface (i.e., the integral over the square) is equal to the average of the corner heights multiplied by the area of the base. Wait, is that true?Let me check. Suppose the square is divided into four quadrants, each with a different height. But actually, if you have a square with corners at (0,0), (1,0), (1,1), (0,1) and heights z1, z2, z3, z4 at the respective corners. If we model the surface as a bilinear patch, then the volume under it can be found by integrating over the square. The formula for the integral of a bilinear function over a unit square is (z1 + z2 + z3 + z4)/4. Since the area is 1, the average height times area would give the same result. Wait, but that's only if the surface is bilinear? Hmm.Wait, no, actually, the integral over the unit square of a bilinear function f(x,y) = a + bx + cy + dxy is equal to the average of the four corner values. Let's test this. Suppose we have a unit square from (0,0) to (1,1). The integral of f(x,y) over the square would be ∫ from 0 to1 ∫ from 0 to1 (a + bx + cy + dxy) dx dy.Integrate with respect to x first:∫0^1 [a + bx + cy + dxy] dx = [a x + (b/2)x² + cy x + (d/2)x² y] from 0 to1= a + b/2 + cy + d/2 yThen integrate with respect to y:∫0^1 [a + b/2 + cy + d/2 y] dy = [a y + (b/2)y + (c/2)y² + (d/4)y²] from 0 to1= a + b/2 + c/2 + d/4But the average of the four corner values is [f(0,0) + f(1,0) + f(1,1) + f(0,1)] /4f(0,0) = af(1,0) = a + bf(1,1) = a + b + c + df(0,1) = a + cSo average is [a + (a + b) + (a + b + c + d) + (a + c)] /4= [4a + 2b + 2c + d]/4 = a + (b/2) + (c/2) + (d/4)Which is exactly the result of the integral. Therefore, the volume under the bilinear surface is equal to the average of the four corner heights times the area (which is 1 here). So if the area is 1, the volume is (z1 + z2 + z3 + z4)/4. Therefore, in this case, since the original square is unit area, the volume between the square and the top surface would be (3 + 4 + 6 + 5)/4 = 18/4 = 4.5. Then, the total volume of the polyhedron would be this 4.5 plus the volume of the original square? Wait, no, because the original square is part of the polyhedron. Wait, but actually, the polyhedron is bounded by the original square and the four vertical edges and the top surface. So the volume is just the volume under the top surface (the bilinear interpolated one) above the square. But since the square is in z=0, and the top surface is in z=3,4,6,5, then yes, the volume would be the integral of the height over the square. Therefore, 4.5. Wait, but 3+4+6+5=18, divided by 4 is 4.5, and since the area is 1, 4.5*1=4.5. So the volume is 4.5. But let me check if this is correct.But wait, the problem says "the polyhedron whose vertices are the specified points and the vertices of the square." So the polyhedron includes the original square and the four elevated points. So this polyhedron is like a frustum-like shape but with a non-planar top. So how exactly is the volume calculated? If the top is a bilinear surface, then integrating over the square would give the volume between the square and the top surface. But is that the actual volume of the polyhedron?Wait, maybe. Because the polyhedron is formed by connecting the original square's vertices to the elevated points. So each edge of the square is connected to two elevated points, forming quadrilaterals. Each of these quadrilaterals is a side face. Then the top is the quadrilateral formed by the four elevated points. So the polyhedron is a kind of a "twisted prism," but with a non-planar top. The volume in such a case can be calculated using the average height formula? Wait, maybe. Because if you think of it as a linear extrusion with varying height, the volume would be the integral of the height over the base. Which would be the average height times area. Which is 4.5. So that's 9/2. So 4.5. But the problem states the square is a unit square, so area is 1. So 4.5*1=4.5. Therefore, 9/2. So the answer should be 9/2, which is 4.5, or in boxed form, boxed{dfrac{9}{2}}.But wait, let me confirm. Let's think of another way. Let's divide the polyhedron into pyramids. Each pyramid has a base as the original square and a apex at one of the elevated points. But wait, no, because all four elevated points are connected to the square. Alternatively, maybe decompose the polyhedron into four tetrahedrons. Let's see.The polyhedron has 8 vertices: the four original square vertices (A, B, C, D) and the four elevated points (A', B', C', D'). The edges are the original square edges, the vertical edges from each original vertex to its elevated point, and the edges connecting the elevated points in order (A'B'C'D'). The faces are the original square, the top quadrilateral A'B'C'D', and four side quadrilaterals (AA'B'B, BB'C'C, CC'D'D, DD'A'A). So the polyhedron is a prismatoid? Wait, the definition of a prismatoid is a polyhedron with all vertices lying in two parallel planes. The original square is in z=0, and the elevated points are in z=3,4,6,5. Since these z's are not the same, the top vertices are not coplanar. Therefore, the polyhedron is not a prismatoid. So the prismatoid formula does not apply here. Therefore, my initial thought about the volume being the average height times area may not be correct. Wait, but maybe it's still valid? Wait, if the top is a ruled surface but not planar, is there a formula?Alternatively, maybe the volume can be calculated using the divergence theorem or something, but that seems complicated. Alternatively, I can use coordinates. Let's assign coordinates to the square. Let me place the square in the xy-plane with vertices at A(0,0,0), B(1,0,0), C(1,1,0), D(0,1,0). The elevated points are A'(0,0,3), B'(1,0,4), C'(1,1,6), D'(0,1,5). Now, the polyhedron is formed by these 8 points. To compute its volume, perhaps we can use the convex hull of these points. But since the square is part of the polyhedron, the volume is the space between the square and the "roof" formed by A', B', C', D'. So maybe decomposing the polyhedron into tetrahedrons whose volumes can be summed.Alternatively, use the formula for the volume of a non-planar top. Let me recall that the volume can be computed as 1/3 times the sum over each face of the base area times height. Wait, no. Alternatively, maybe use the following approach: the volume can be considered as the sum of the volumes of the four "columns" plus some correction terms. Wait, not sure.Alternatively, think of the polyhedron as a combination of four square pyramids, but with different heights. Wait, but overlapping. Alternatively, use the shoelace formula in 3D. There is a formula called the Cayley-Menger determinant, which can compute the volume of a tetrahedron given its edge lengths. But since this is a more complex polyhedron, maybe not.Alternatively, perhaps split the polyhedron into tetrahedrons. For example, choose a diagonal of the base square and connect it to the elevated points. Let me try this.First, split the base square into two triangles, say ABC and ACD. Then, connect each triangle to the elevated points. But how?Alternatively, for each edge of the square, connect it to the corresponding elevated edges. Hmm, perhaps another approach. Let's consider the polyhedron as a hexahedron (8 vertices, 12 edges). To compute its volume, one way is to divide it into five tetrahedrons (since a convex polyhedron can often be divided into tetrahedrons). But this requires choosing appropriate diagonals.Alternatively, use the following formula for a prismatoid (even though the top is not planar). Wait, prismatoid formula is (h/6)(B1 + B2 + 4M), where M is the area of the midsection. But since the top isn't planar, the midsection isn't defined. Alternatively, there's a formula called the volume integral. Since we know the coordinates of all eight vertices, maybe use the divergence theorem.Alternatively, compute the volume using coordinates. Let me try that.Given the coordinates:A(0,0,0), B(1,0,0), C(1,1,0), D(0,1,0)A'(0,0,3), B'(1,0,4), C'(1,1,6), D'(0,1,5)We can use the formula for the volume of a polyhedron given its vertices. This can be done by dividing the polyhedron into tetrahedrons, each of which has a known volume. The key is to partition the polyhedron into non-overlapping tetrahedrons whose volumes can be summed.One way to do this is to pick a common vertex and form tetrahedrons with that vertex and each face. For example, choose vertex A(0,0,0) and form tetrahedrons with A and each of the five other faces. Wait, but the polyhedron is complex. Let's see.Alternatively, split the polyhedron along the diagonal from A to C, creating two halves. Each half can be split into tetrahedrons.Alternatively, here's a systematic approach:The polyhedron has two quadrilaterals (the base and the top) connected by four trapezoidal faces. To compute the volume, perhaps use the formula for the volume between two non-parallel planes. Wait, but the top isn't a plane.Alternatively, use the following method: the volume can be calculated as the average of the heights at each corner multiplied by the area of the base. Wait, but earlier I thought that gives 4.5. But does that apply here?Wait, let's think. If the top were a flat surface, then the volume would indeed be the area of the base times the average height. However, in this case, the top is not flat. But maybe the formula still works? Wait, actually, the formula for the volume under any surface (flat or not) over a base is the integral of the height over the base. If we approximate the height as varying bilinearly, then the integral is equal to the average of the four corner heights. But is the actual surface of the polyhedron a bilinear surface?Wait, the polyhedron's upper faces are the four trapezoids and the top quadrilateral. The top quadrilateral is a skew quadrilateral. The bilinear surface is a different surface. So if we model the upper surface as the bilinear interpolation, then the volume would be the average height times area, but in reality, the actual volume of the polyhedron might be different.But in the problem statement, the polyhedron is defined as having vertices at the specified points and the vertices of the square. Therefore, the upper surface is the quadrilateral A'B'C'D', which is a skew quadrilateral. So the polyhedron is a kind of a "twisted prism," and its volume is not simply the average height times the area.Therefore, I need to compute the volume correctly. Let's try splitting the polyhedron into tetrahedrons.First, note that the polyhedron can be split into two parts: the original square pyramid with apex at some point, but since all the elevated points are different, it's more complicated.Alternatively, pick a point inside the polyhedron and connect it to all faces. But that might be difficult.Alternatively, consider that the polyhedron can be split into four tetrahedrons by connecting each elevated point to the base square appropriately.Wait, here's an idea. Let's connect A' to B, C, D; B' to C, D, A; etc. But not sure.Alternatively, here's a better approach. The polyhedron is a hexahedron with eight vertices. One way to compute its volume is to use the following formula for a hexahedron:Divide the hexahedron into five tetrahedrons. For example, select a diagonal of the base square, say from A(0,0,0) to C(1,1,0). Then, we can split the base into two triangles: ABC and ADC.Then, connect these triangles to the top quadrilateral. However, the top is a skew quadrilateral, so we need to find a way to divide it into tetrahedrons.Alternatively, split the top quadrilateral A'B'C'D' into two triangles, say A'B'C' and A'C'D'. Then, the entire polyhedron can be split into the following tetrahedrons:1. A B C A'2. A C D A'3. A' B' C' B4. A' C' D' DWait, maybe not. Alternatively, perhaps we need a more systematic approach.Alternatively, use the scalar triple product for each tetrahedron. Let's list all possible tetrahedrons within the polyhedron.But this might be time-consuming. Alternatively, use coordinates and the divergence theorem. The volume can be computed by integrating 1 over the polyhedron. To do this, we can use the coordinates of the vertices and apply the formula for the volume of a non-convex polyhedron. But this seems complex.Wait, another approach. Since the sides are vertical (i.e., the edges AA', BB', CC', DD' are perpendicular to the base), this shape is called a prismoid with non-uniform heights. Wait, in engineering, there's a formula for the volume of a prismoid when the top is a plane. The formula is (h1 + h2 + h3 + h4)/4 * area. Wait, but that's only when the top is planar. But here, the top is not planar, so the formula might not apply. However, if we use the average of the heights regardless of the top being planar or not, is that still valid?Wait, let's compute the volume using coordinates. Let me model the shape in 3D coordinates.The base square has coordinates:A(0,0,0)B(1,0,0)C(1,1,0)D(0,1,0)The top points:A'(0,0,3)B'(1,0,4)C'(1,1,6)D'(0,1,5)Now, the polyhedron is composed of the base square, the top quadrilateral A'B'C'D', and the four vertical quadrilaterals AA'B'B, BB'C'C, CC'D'D, DD'A'A.To compute the volume, we can use the following method: for each vertical column over the base square, integrate the height over the base. But since the heights are only given at the corners, and the height in between is determined by the bilinear interpolation, then the volume would indeed be the average height times the area, as we thought earlier. So even though the top is a hyperbolic paraboloid (due to bilinear interpolation), the integral over the square is equal to the average of the corner heights. Therefore, the volume would be (3 + 4 + 6 + 5)/4 = 18/4 = 4.5. Hence, 9/2. Therefore, the answer is boxed{dfrac{9}{2}}.But wait, let me verify this with a different method. Let's split the polyhedron into two parts: the original square and the four "walls" connecting to the top quadrilateral. But integrating the height over the base gives the volume between the base and the top surface. Since the walls are vertical, this integral should equal the volume. Therefore, even if the top is not flat, the volume is still the integral of the height function over the base. Since we only have the heights at the corners, assuming linear interpolation along the edges and bilinear over the square, the integral is the average height times area. Therefore, the answer is 9/2.Alternatively, perhaps think of the polyhedron as a frustum of a pyramid but with different heights at each corner. There's a formula for such a volume. Wait, actually, in the case of a truncated pyramid (frustum), the volume is (h/3)(A1 + A2 + sqrt(A1 A2)). But in this case, the top isn't a scaled version of the base, so that formula doesn't apply. However, if we consider the general formula for the volume between two parallel polygons, but here they aren't parallel. The top is a skew quadrilateral, so that complicates things.But going back to the integration approach. If we parametrize the square as (x,y,0) where 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, and the height at each point (x,y) is given by some function z(x,y). Since we have the heights at the four corners, we can model z(x,y) as a bilinear function:z(x,y) = a + bx + cy + dxyWe can solve for a, b, c, d using the given heights.At (0,0): z = 3 = aAt (1,0): z = 4 = a + b => 4 = 3 + b => b = 1At (1,1): z = 6 = a + b + c + d => 6 = 3 + 1 + c + d => c + d = 2At (0,1): z = 5 = a + c => 5 = 3 + c => c = 2Then from c + d = 2, and c = 2, so d = 0.Therefore, the bilinear function is z(x,y) = 3 + x + 2y + 0*xy = 3 + x + 2y.Therefore, the height at any point (x,y) is 3 + x + 2y.Therefore, the volume under this surface over the unit square is the double integral of (3 + x + 2y) dx dy over [0,1]x[0,1].Compute the integral:Integral from 0 to1 Integral from 0 to1 (3 + x + 2y) dx dyFirst integrate with respect to x:∫0^1 (3 + x + 2y) dx = [3x + (1/2)x² + 2y x] from 0 to1 = 3(1) + (1/2)(1) + 2y(1) - 0 = 3 + 0.5 + 2y = 3.5 + 2yThen integrate with respect to y:∫0^1 (3.5 + 2y) dy = [3.5 y + y²] from 0 to1 = 3.5(1) + (1)^2 - 0 = 3.5 + 1 = 4.5Therefore, the volume is indeed 4.5, which is 9/2. Therefore, the answer is boxed{dfrac{9}{2}}.This makes sense. So even though the top is a ruled surface (since it's a bilinear function), the volume can be calculated by integrating the height function over the base. The key insight was that the heights at the corners define a bilinear interpolation over the square, and integrating that gives the volume. Therefore, the volume is indeed the average of the four heights times the area of the base. Since the area is 1, it's just (3 + 4 + 6 + 5)/4 = 18/4 = 9/2.Final Answerboxed{dfrac{9}{2}}</think>Given a unit square with vertices (A(0,0,0)), (B(1,0,0)), (C(1,1,0)), and (D(0,1,0)), perpendiculars are erected at each vertex, and points are placed on these perpendiculars at heights 3, 4, 6, and 5 respectively. We need to find the volume of the polyhedron formed by these points and the vertices of the square.1. Coordinates of Elevated Points: - (A'(0,0,3)) - (B'(1,0,4)) - (C'(1,1,6)) - (D'(0,1,5))2. Bilinear Interpolation: - The height function (z(x,y)) over the unit square can be modeled as a bilinear function. - Solving for the coefficients using the given heights: - At ((0,0)): (z = 3) - At ((1,0)): (z = 4) - At ((1,1)): (z = 6) - At ((0,1)): (z = 5) - The bilinear function is found to be (z(x,y) = 3 + x + 2y).3. Integral of the Height Function: - The volume under the bilinear surface over the unit square is calculated by integrating (z(x,y)) over the square: [ int_0^1 int_0^1 (3 + x + 2y) , dx , dy ] - First integrate with respect to (x): [ int_0^1 (3 + x + 2y) , dx = left[3x + frac{1}{2}x^2 + 2yxright]_0^1 = 3.5 + 2y ] - Then integrate with respect to (y): [ int_0^1 (3.5 + 2y) , dy = left[3.5y + y^2right]_0^1 = 4.5 ]4. Result: - The volume of the polyhedron is the result of the integral, which is (frac{9}{2}).[boxed{dfrac{9}{2}}]]

🔑:Let A B C D be a unit square, and A_{1}, B_{1}, C_{1}, and D_{1} be the specified points on the perpendiculars to its plane, such that A A_{1}=3, B B_{1}=4, C C_{1}=6, D D_{1}=5.Extend the perpendiculars to segments A A_{2}=B B_{2}=C C_{2}=D D_{2}=9. Note that A_{1} B_{1} C_{1} D_{1} is a parallelogram (for example, because the midpoints of segments A_{1} C_{1} and B_{1} D_{1} coincide). Therefore, the points A_{1}, B_{1}, C_{1}, D_{1} lie in the same plane, which divides the parallelepiped A B C D A_{2} B_{2} C_{2} D_{2} into two equal figures.Since the volume of the specified parallelepiped is 9, the volume of each of the halves is 9 / 2.## Answer4.5 .

🧠:[<think>Okay, so I need to prove that if a and b are coprime positive integers, then the sum [a/b] + [2a/b] + ... + [(b-1)a/b] equals (a-1)(b-1)/2. Hmm, let me start by understanding the problem better.First, remember that [x] denotes the floor function, right? Which is the greatest integer less than or equal to x. So for each term [ka/b], where k goes from 1 to b-1, we're taking the floor of ka divided by b.Since a and b are coprime, their greatest common divisor is 1. That fact must be important here because otherwise, if they had a common divisor, maybe some terms would repeat or something. But since they're coprime, each ka mod b should be distinct for k from 1 to b-1. Wait, is that true? Let me recall that if a and b are coprime, then multiplication by a modulo b is a bijection. So the residues ka mod b for k=1,2,...,b-1 should be a permutation of 1,2,...,b-1. That seems like a key point.So, for each k, ka = q_k*b + r_k, where q_k is the quotient and r_k is the remainder when ka is divided by b. Then [ka/b] would be q_k. So the sum we're looking at is the sum of all these q_k's from k=1 to k=b-1.Alternatively, since ka = q_k*b + r_k, then summing over k from 1 to b-1:Sum_{k=1}^{b-1} ka = Sum_{k=1}^{b-1} (q_k*b + r_k)Which means:a * Sum_{k=1}^{b-1} k = b * Sum_{k=1}^{b-1} q_k + Sum_{k=1}^{b-1} r_kSo, rearranging,Sum_{k=1}^{b-1} q_k = (a * Sum_{k=1}^{b-1} k - Sum_{k=1}^{b-1} r_k)/bBut Sum_{k=1}^{b-1} k is (b-1)b/2. So,Sum q_k = (a*(b-1)b/2 - Sum r_k)/bWhich simplifies to:Sum q_k = (a(b-1)/2 - (Sum r_k)/b)So, if I can compute Sum r_k, then I can find Sum q_k. But what is Sum r_k?Earlier, we noted that since a and b are coprime, the residues r_k when ka is divided by b are just a permutation of 1, 2, ..., b-1. Because multiplying by a modulo b is a bijection. Therefore, Sum r_k = Sum_{k=1}^{b-1} k = (b-1)b/2.Wait, that's a crucial point. So if the residues r_k are a permutation of 1 to b-1, their sum is the same as the sum from 1 to b-1, which is (b-1)b/2. Therefore:Sum q_k = (a(b-1)/2 - ( (b-1)b/2 ) / b )Simplify that:= (a(b-1)/2 - ( (b-1)/2 ))= ( (a(b-1) - (b-1)) / 2 )= ( (a - 1)(b - 1) ) / 2Which is exactly the desired result. Therefore, the sum of the floors [ka/b] from k=1 to b-1 is (a-1)(b-1)/2.Wait, let me check if this makes sense. Let's take a small example where a and b are coprime. Let's say a=3, b=4. Then the sum is [3/4] + [6/4] + [9/4] = [0.75] + [1.5] + [2.25] = 0 + 1 + 2 = 3. On the other hand, (3-1)(4-1)/2 = 2*3/2 = 3. That works.Another example: a=5, b=7. Then the sum would be [5/7] + [10/7] + [15/7] + [20/7] + [25/7] + [30/7] = 0 + 1 + 2 + 2 + 3 + 4 = 0+1=1, 1+2=3, 3+2=5, 5+3=8, 8+4=12. Wait, (5-1)(7-1)/2 =4*6/2=12. So that works too.Another example: a=2, b=3. Then the sum is [2/3] + [4/3] = 0 + 1 = 1. The formula gives (2-1)(3-1)/2=1*2/2=1. Perfect.So the logic seems to hold. By expressing each ka as q_k*b + r_k, summing over k, and using the fact that the residues r_k are a permutation of 1 to b-1 when a and b are coprime, we can relate the sum of the floors (which are the q_k's) to the sum of the residues. Then, substituting the known sum of residues, the terms cancel out appropriately to give the desired result. That's a neat argument. I didn't know this identity before, but breaking it down using division algorithm and properties of coprime numbers makes sense.I should check one more example where a and b are not coprime, just to see if the formula fails. Let's take a=2, b=4. They are not coprime. Then the sum is [2/4] + [4/4] + [6/4] = 0 + 1 + 1 = 2. The formula would give (2-1)(4-1)/2 =1*3/2=1.5, which is not an integer, and the sum is 2. So indeed, the formula doesn't hold when a and b are not coprime, which matches the problem's requirement. So the coprimality is essential here for the residues to cover all numbers 1 to b-1 exactly once, ensuring that their sum is (b-1)b/2. If they weren't coprime, the residues would repeat, and their sum would be different, hence the formula would not hold.Therefore, the key steps are:1. Recognize that for each k, ka = q_k*b + r_k, with 0 < r_k < b (since a and b are coprime, r_k can't be zero for k < b).2. Sum over all k from 1 to b-1: sum(ka) = sum(q_k*b + r_k).3. Rearranged, sum(q_k) = [a*sum(k) - sum(r_k)] / b.4. Since a and b are coprime, {r_k} is a permutation of {1, 2, ..., b-1}, so sum(r_k) = sum(k) = (b-1)b/2.5. Substitute back into the equation for sum(q_k), simplify, and get (a-1)(b-1)/2.Therefore, the proof is complete.Final Answerboxed{frac{1}{2}(a-1)(b-1)}</think>Given coprime positive integers (a) and (b), we need to prove that:[leftlfloor frac{a}{b} rightrfloor + leftlfloor frac{2a}{b} rightrfloor + cdots + leftlfloor frac{(b-1)a}{b} rightrfloor = frac{1}{2}(a-1)(b-1)]Key Steps:1. Division Algorithm: For each (k) from 1 to (b-1), write (ka = q_k b + r_k) where (q_k) is the quotient and (r_k) is the remainder when (ka) is divided by (b). Here, (leftlfloor frac{ka}{b} rightrfloor = q_k).2. Summing Both Sides: Summing over (k) from 1 to (b-1): [ sum_{k=1}^{b-1} ka = sum_{k=1}^{b-1} (q_k b + r_k) ] This simplifies to: [ a sum_{k=1}^{b-1} k = b sum_{k=1}^{b-1} q_k + sum_{k=1}^{b-1} r_k ]3. Sum of (k): The sum (sum_{k=1}^{b-1} k) is (frac{(b-1)b}{2}).4. Residues (r_k): Since (a) and (b) are coprime, the residues (r_k) are a permutation of (1, 2, ldots, b-1). Thus, (sum_{k=1}^{b-1} r_k = frac{(b-1)b}{2}).5. Solving for (sum q_k): [ a cdot frac{(b-1)b}{2} = b sum_{k=1}^{b-1} q_k + frac{(b-1)b}{2} ] Rearranging gives: [ sum_{k=1}^{b-1} q_k = frac{a cdot frac{(b-1)b}{2} - frac{(b-1)b}{2}}{b} = frac{(a - 1)(b - 1)}{2} ]Conclusion:The sum (leftlfloor frac{a}{b} rightrfloor + leftlfloor frac{2a}{b} rightrfloor + cdots + leftlfloor frac{(b-1)a}{b} rightrfloor) simplifies to (frac{1}{2}(a-1)(b-1)).[boxed{frac{1}{2}(a-1)(b-1)}],<think>Okay, so I need to prove that if a and b are coprime positive integers, then the sum [a/b] + [2a/b] + ... + [(b-1)a/b] equals (a-1)(b-1)/2. Hmm, let's start by understanding what each term in the sum represents. The notation [x] usually denotes the floor function, which gives the greatest integer less than or equal to x. So each term [ka/b] is the integer part of ka divided by b, right?Since a and b are coprime, that means their greatest common divisor is 1. That might be important because when a and b are coprime, the numbers ka mod b for k from 1 to b-1 should cycle through all residues modulo b except 0. Wait, is that true? Let me think. Since a and b are coprime, multiplying by a modulo b is a bijection. So the residues of ka modulo b for k=1,2,...,b-1 should be a permutation of 1,2,...,b-1. That seems right. So each ka can be written as b*q_k + r_k where r_k is between 1 and b-1, and all r_k are distinct. Then [ka/b] would be q_k. So the sum we need is the sum of all q_k for k from 1 to b-1.But how does that relate to (a-1)(b-1)/2? Maybe there's a relationship between the quotients and the remainders. Let's write each ka as q_k * b + r_k. Then if we sum over all k from 1 to b-1, we have:Sum_{k=1}^{b-1} ka = Sum_{k=1}^{b-1} (q_k * b + r_k)Which simplifies to:a * Sum_{k=1}^{b-1} k = b * Sum_{k=1}^{b-1} q_k + Sum_{k=1}^{b-1} r_kThe left side is a*( (b-1)b)/2 ). The right side is b*(sum of q_k) + sum of r_k. Since the remainders r_k are a permutation of 1 to b-1, their sum is (b-1)b/2. So plugging these in:a*( (b-1)b/2 ) = b*(sum of q_k) + (b-1)b/2Subtracting (b-1)b/2 from both sides:a*(b-1)b/2 - (b-1)b/2 = b*(sum of q_k)Factor out (b-1)b/2:(b-1)b/2 * (a - 1) = b*(sum of q_k)Divide both sides by b:(b-1)/2 * (a - 1) = sum of q_kWhich is exactly the desired result: sum of [ka/b] from k=1 to b-1 equals (a-1)(b-1)/2. Oh, so that works out! Wait, let me check if I made any mistakes here.So, key steps:1. Express each ka as q_k * b + r_k, where r_k is the remainder when ka is divided by b, so 1 ≤ r_k ≤ b-1.2. Since a and b are coprime, the r_k are all distinct, so they are a permutation of 1, 2, ..., b-1. Therefore, sum of r_k = sum from 1 to b-1 = (b-1)b/2.3. Sum both sides over k=1 to b-1: sum(ka) = sum(q_k * b) + sum(r_k).4. Left side: a * sum(k) from 1 to b-1 = a*(b-1)b/2.5. Right side: b*sum(q_k) + (b-1)b/2.6. Equate the two: a*(b-1)b/2 = b*sum(q_k) + (b-1)b/2.7. Subtract (b-1)b/2 from both sides: (a - 1)*(b-1)b/2 = b*sum(q_k).8. Divide both sides by b: (a - 1)(b - 1)/2 = sum(q_k).Therefore, the sum of the floor functions [ka/b] (which are q_k) is indeed (a - 1)(b - 1)/2. That seems to check out. Let me test this with an example to make sure.Let's take a=3, b=4, which are coprime. Then the sum is [3/4] + [6/4] + [9/4] = [0.75] + [1.5] + [2.25] = 0 + 1 + 2 = 3. The formula gives (3-1)(4-1)/2 = 2*3/2 = 3. Correct.Another example: a=5, b=7. The sum is [5/7] + [10/7] + [15/7] + [20/7] + [25/7] + [30/7] = 0 + 1 + 2 + 2 + 3 + 4 = 0+1=1, 1+2=3, 3+2=5, 5+3=8, 8+4=12. Formula: (5-1)(7-1)/2 = 4*6/2 = 12. Correct again.Wait, let's check the remainders for a=5, b=7:5*1 = 5, remainder 55*2 = 10, remainder 35*3 = 15, remainder 15*4 = 20, remainder 65*5 = 25, remainder 45*6 = 30, remainder 2So remainders are 5,3,1,6,4,2 which are indeed all residues from 1 to 6. Their sum is 5+3+1+6+4+2=21, which is 6*7/2=21. Correct. Then using the equation:Left side: a*(sum k) = 5*(6*7/2) = 5*21 = 105Right side: b*sum(q_k) + sum(r_k) = 7*sum(q_k) + 21But sum(q_k) is [5/7] + [10/7] + [15/7] + [20/7] + [25/7] + [30/7] = 0 + 1 + 2 + 2 + 3 + 4 = 12So 7*12 + 21 = 84 +21=105. Which equals left side. So that works. Therefore, the proof seems solid.Therefore, the reasoning is correct. The key idea was to relate the sum of the floor terms to the sum of the remainders, leveraging the fact that the remainders form a complete set of residues modulo b when a and b are coprime. Then, by expressing the sum of ka in two different ways, we can solve for the sum of the floor terms. That gives the desired identity. I think that's the proof.Final AnswerThe given identity is proven, and the result is boxed{dfrac{(a-1)(b-1)}{2}}.</think>Given coprime positive integers (a) and (b), we need to prove:leftlfloor frac{a}{b} rightrfloor + leftlfloor frac{2a}{b} rightrfloor + cdots + leftlfloor frac{(b-1)a}{b} rightrfloor = frac{1}{2}(a-1)(b-1).Key Steps:1. Express each term (ka) as (q_k b + r_k): For each (k) from 1 to (b-1), write (ka = q_k b + r_k), where (q_k = leftlfloor frac{ka}{b} rightrfloor) and (1 leq r_k leq b-1). Since (a) and (b) are coprime, the remainders (r_k) are distinct and form a permutation of (1, 2, ldots, b-1).2. Sum both sides over (k = 1) to (b-1): [ sum_{k=1}^{b-1} ka = sum_{k=1}^{b-1} (q_k b + r_k) ] This simplifies to: [ a sum_{k=1}^{b-1} k = b sum_{k=1}^{b-1} q_k + sum_{k=1}^{b-1} r_k ]3. Calculate the sums: - The left side: (a sum_{k=1}^{b-1} k = a frac{(b-1)b}{2}) - The sum of remainders: (sum_{k=1}^{b-1} r_k = frac{(b-1)b}{2}) (since they are a permutation of (1, 2, ldots, b-1))4. Substitute and solve for (sum q_k): [ a frac{(b-1)b}{2} = b sum_{k=1}^{b-1} q_k + frac{(b-1)b}{2} ] Subtract (frac{(b-1)b}{2}) from both sides: [ (a - 1) frac{(b-1)b}{2} = b sum_{k=1}^{b-1} q_k ] Divide by (b): [ frac{(a - 1)(b - 1)}{2} = sum_{k=1}^{b-1} q_k ]Thus, the sum of the floor terms (leftlfloor frac{ka}{b} rightrfloor) for (k = 1, 2, ldots, b-1) is indeed (frac{(a-1)(b-1)}{2}).[boxed{dfrac{(a-1)(b-1)}{2}}]]

🔑:Prove that, as shown in Figure 7-6, the four vertices of rectangle O A B C are O(0,0), A(b, 0), B(b, a), and C(0, a). All integer points (x, y) inside this rectangle satisfy 1 leqslant x leqslant b-1, 1 leqslant y leqslant a-1, and there are a total of (a-1) cdot (b-1) such integer points.The equation of the diagonal O B of the rectangle is y = frac{a}{b} x. Since (a, b) = 1, when x = 1, 2, cdots, b-1, y = frac{a}{b} x cannot be an integer, meaning there are no integer points on O B. Thus, the number of integer points inside the triangle below O B is equal to the number of integer points inside the triangle above O B, both equal to frac{1}{2}(b-1)(a-1). On the other hand, sum_{k=1}^{b-1}left[frac{k a}{b}right] represents the number of integer points inside triangle O A B. Therefore, sum_{k=1}^{b-1}left[frac{k a}{b}right] = frac{1}{2}(a-1)(b-1).