🧠:[<think>Okay, so I need to find the negation of the proposition "There exists an x in the real numbers such that x squared is less than or equal to the absolute value of x." Hmm, let's break this down. First, I remember that the negation of an existential quantifier is a universal quantifier with the negated statement. So, if the original statement is ∃x ∈ R, P(x), then the negation should be ∀x ∈ R, ¬P(x). Right? So in this case, P(x) is "x² ≤ |x|", so the negation would be "For all x in R, x² > |x|". Looking at the options, option B is "For all x in R, x² > |x|". That seems to match. But let me check the other options to be sure. Option A says "For all x in R, x² ≥ |x|". Wait, the negation of ≤ is actually >, not ≥. Because if it's not less than or equal, it must be greater than. So if we have ¬(x² ≤ |x|), that's equivalent to x² > |x|. Therefore, the correct negation should be that for all x, x² is greater than |x|. So option B.But let me think again. Maybe I made a mistake. Let's recall De Morgan's laws for quantifiers. The negation of ∃x P(x) is ∀x ¬P(x). So yes, in this case, the original statement is ∃x∈ℝ (x² ≤ |x|). The negation is ∀x∈ℝ, ¬(x² ≤ |x|), which is ∀x∈ℝ, x² > |x|. So that's definitely option B.But wait, the answer might be option A? Because sometimes people might confuse the strict inequality. Let me test with some numbers. For example, take x = 0. Then x² = 0 and |x| = 0, so x² ≤ |x| is true. But in the negation, if it's ∀x∈ℝ, x² > |x|, then x=0 would make x² = 0, which is not greater than |x|=0, so that's false. Therefore, the negation would be false because there exists x (like 0) where x² ≤ |x|. But actually, the original statement is true because such an x exists (like 0). So the negation should be false, but if the answer were option B, then since there exists x=0 where x² is not greater than |x|, the statement ∀x∈ℝ, x² > |x| is false. Which is correct because the negation of a true statement should be false. So that's consistent. Alternatively, if the negation were option A: ∀x∈ℝ, x² ≥ |x|. Let's check x=0.5. Then x² = 0.25, |x| = 0.5. So 0.25 is not ≥ 0.5, so the statement would be false. But the original statement is ∃x∈ℝ, x² ≤ |x|. If we take x=0.5, then x²=0.25 ≤ 0.5, so that's true. So the original statement is true. Its negation should be false. Option A is ∀x∈ℝ, x² ≥ |x|, which is false because x=0.5 is a counterexample. Option B is ∀x∈ℝ, x² > |x|, which is also false because x=1, x²=1 which is not greater than |x|=1. Wait, x=1: x²=1 and |x|=1, so 1 > 1 is false. So both A and B are false? But the original statement is true, so its negation must be false, which both A and B are. That's confusing. Wait, maybe my reasoning is flawed here. Let me check again. Let's consider the original statement: ∃x∈ℝ, x² ≤ |x|. Is this true? Let's test x=0: 0 ≤ 0, true. x=0.5: 0.25 ≤ 0.5, true. x=1: 1 ≤ 1, true. So the original statement is definitely true. Therefore, its negation must be false. Both options A and B are universal statements. If they are negations, they should be false. But how can both be false? That can't be. So maybe I made a mistake in the negation.Wait, hold on. Let's carefully do the negation step by step. The original statement is ∃x∈ℝ (x² ≤ |x|). The negation is ¬∃x∈ℝ (x² ≤ |x|) which is equivalent to ∀x∈ℝ ¬(x² ≤ |x|). Now, ¬(x² ≤ |x|) is equivalent to x² > |x|. Therefore, the negation is ∀x∈ℝ, x² > |x|, which is option B. But if we check option B, for x=1, x²=1 and |x|=1, so 1 > 1 is false. Therefore, the statement ∀x∈ℝ, x² > |x| is false, which is correct because the original statement was true.However, option A is ∀x∈ℝ, x² ≥ |x|. For x=0.5, x²=0.25 < 0.5, so 0.25 ≥ 0.5 is false. Therefore, option A is also false. So both A and B are false? But the original statement's negation is only supposed to be one of these. Hmm. Wait, maybe there's a problem with the options. Let me check again.The original question is: The negation of "∃x∈ℝ, x² ≤ |x|" is which option. The options are A: ∀x∈ℝ, x² ≥ |x|; B: ∀x∈ℝ, x² > |x|; C: ∃x∈ℝ, x² > |x|; D: ∃x∈ℝ, x² ≥ |x|.So, according to logic, the negation should be ∀x∈ℝ, x² > |x|, which is option B. However, when checking with x=1, the statement x² > |x| becomes 1 > 1, which is false, making the entire universal statement false. That's correct because the original statement is true, so its negation should be false. But let's check option A. If the negation were option A, which is ∀x∈ℝ, x² ≥ |x|, then this is a weaker statement than option B. If option B is false, option A could still be false. For example, if there exists an x where x² < |x|, then both A and B are false. Since there are x's (like 0.5) where x² < |x|, option A is false. So the only correct negation is option B, even though it's false. The question is about the correct form of the negation, not whether the negation is true or false.Therefore, regardless of the truth value, the correct logical negation is option B. So the answer is B.Final Answerboxed{B}</think>To find the negation of the proposition "exists x in mathbb{R}, x^2 leqslant |x|", we start by recalling that the negation of an existential quantifier is a universal quantifier with the negated statement. The original statement is exists x in mathbb{R}, P(x) where P(x) is x^2 leqslant |x|. The negation of this statement is forall x in mathbb{R}, neg P(x). The negation of P(x), which is neg (x^2 leqslant |x|), translates to x^2 > |x|. Therefore, the negation of the original statement is forall x in mathbb{R}, x^2 > |x|, which corresponds to option B.To verify, we check the options:- Option A: forall x in mathbb{R}, x^2 geqslant |x| is incorrect because the negation of leqslant is >, not geqslant.- Option B: forall x in mathbb{R}, x^2 > |x| is correct as it directly negates the original statement.- Options C and D are existential quantifiers, which are incorrect because the negation of an existential quantifier must be a universal quantifier.Thus, the correct answer is boxed{B}.,<think>Okay, so I need to find the negation of the proposition "There exists an x in the real numbers such that x squared is less than or equal to the absolute value of x." The options are A, B, C, D. Let me think through this step by step.First, I remember that the negation of an existential quantifier (∃) is a universal quantifier (∀) with the negation of the inner statement. So, if the original statement is ∃x ∈ R, P(x), then its negation should be ∀x ∈ R, ¬P(x). In this problem, the original proposition is ∃x ∈ R, x² ≤ |x|. So, the negation would be ∀x ∈ R, ¬(x² ≤ |x|). Now, I need to figure out what ¬(x² ≤ |x|) is equivalent to. I know that the negation of "less than or equal to" is "greater than." So, ¬(x² ≤ |x|) is the same as x² > |x|. Therefore, the negation of the original statement should be "For all x in real numbers, x squared is greater than the absolute value of x." Looking at the options:A: For all x in R, x² ≥ |x|B: For all x in R, x² > |x|C: There exists x in R, x² > |x|D: There exists x in R, x² ≥ |x|So, according to my reasoning, the correct negation is option B: ∀x ∈ R, x² > |x|. But wait, let me double-check. Is the negation of "there exists x such that P(x)" indeed "for all x, not P(x)"? Yes, that's correct in predicate logic. So, if there does not exist an x with x² ≤ |x|, that means for every x, x² is not ≤ |x|, which is the same as x² > |x|. But wait, could it be option A instead? If the original statement was ∃x, x² < |x|, then the negation would be ∀x, x² ≥ |x|. But here, the original statement is ∃x, x² ≤ |x|. So the negation should be ∀x, x² > |x|. Therefore, option B. But let me test with some numbers to see if the original statement is true or false. For example, if x = 0, then x² = 0 and |x| = 0, so 0 ≤ 0 is true. So the original statement is actually true because there exists at least one x (x=0) where x² ≤ |x|. Therefore, its negation should be false. So, if the original statement is true, then the negation (option B) is false. Let me check if option B is indeed false. For example, take x=0 again. If option B is ∀x, x² > |x|, then at x=0, x² = 0 and |x|=0, so 0 > 0 is false. So option B is false, which is correct because the negation of a true statement should be false. Alternatively, option A says ∀x, x² ≥ |x|. Let's test x=0.5. Then x² = 0.25 and |x|=0.5. So 0.25 ≥ 0.5? No, that's false. Therefore, option A is also false. Hmm, but the original statement is true, so its negation must be false, which both A and B are. Wait, but how does that work?Wait, maybe I made a mistake here. Let me re-examine. The original statement is ∃x, x² ≤ |x|. Since x=0 satisfies it, the original is true, so the negation must be false. Therefore, both A and B are universal statements which are false because there exist some x where they fail. For example, in option B, x=1: x²=1, |x|=1, so 1>1 is false. Wait, x=1: x²=1, |x|=1, so 1>1 is false. Therefore, for x=1, x² > |x| is false. So ∀x, x² > |x| is false because when x=1, it's not true. Similarly, for option A, x=0.5: x²=0.25, |x|=0.5, so 0.25 ≥ 0.5 is false. Therefore, both A and B are false. But the negation of the original statement should be false. Wait, but according to logic, the negation of ∃x P(x) is ∀x ¬P(x). So if the original is true, the negation is false. Therefore, the answer should be B, but how come B is false? Wait, no, the answer is not about which is true or false, but which is the correct logical negation. Wait, perhaps I confused the truth value with the structure. Let me get back. The problem is not asking which is true or false, but what is the correct form of the negation. Original statement: ∃x ∈ R, x² ≤ |x|. Negation: ∀x ∈ R, x² > |x|. So that's option B. Even though in reality, this negation is false, because there are x where x² ≤ |x|, like x=0, but the question is about the correct form of the negation, not its truth value. Therefore, the answer is B. But let me check again. Maybe there's a mistake in the negation. The original is ∃x, P(x). The negation is ∀x, ¬P(x). Here, P(x) is x² ≤ |x|, so ¬P(x) is x² > |x|. Therefore, the negation is ∀x, x² > |x|, which is option B. Therefore, the correct answer is B.But wait, in the options, A is ∀x, x² ≥ |x|. Could it be that the negation of ≤ is ≥? Wait, no. The negation of a ≤ b is a > b. Because if it's not less than or equal, then it must be greater. Similarly, the negation of a ≥ b is a < b. Therefore, yes, ¬(x² ≤ |x|) is x² > |x|. So the correct answer is B.Alternatively, maybe someone might think that ¬(a ≤ b) is a ≥ b, but that's not correct. Because if a is not less than or equal to b, then a must be greater than b. For example, if a=5 and b=3, 5 ≤ 3 is false, so 5 > 3 is true. Similarly, if a=3 and b=5, 3 ≤5 is true, so its negation is 3>5, which is false. So yes, ¬(a ≤ b) is equivalent to a > b.Therefore, the negation is ∀x ∈ R, x² > |x|, which is option B. So the answer should be B.Final Answerboxed{B}</think>To find the negation of the proposition "exists x in mathbb{R}, x^2 leqslant |x|", we start by understanding that the negation of an existential quantifier is a universal quantifier with the negation of the inner statement. The original statement is exists x in mathbb{R}, x^2 leqslant |x|. The negation of this statement is forall x in mathbb{R}, neg (x^2 leqslant |x|). The negation of x^2 leqslant |x| is x^2 > |x|. Therefore, the negation of the original statement is forall x in mathbb{R}, x^2 > |x|.We verify this by considering the logical negation principles and testing specific values. The negation of "there exists" is "for all", and the negation of "less than or equal to" is "greater than". This confirms that the correct negation is forall x in mathbb{R}, x^2 > |x|, which corresponds to option B.Thus, the final answer is:boxed{B}]

🔑:To negate the existential quantifier proposition "exists xin R, x^{2}leqslant |x|," we follow the logical negation rules. The negation of an existential quantifier (exists) is a universal quantifier (forall), and the inequality also flips from leqslant to >.1. Original statement: exists xin R, x^{2}leqslant |x|2. Negation of exists is forall: forall xin R3. Inequality negation: x^{2}leqslant |x| becomes x^{2} > |x|Combining these steps, the negation of the original statement is: forall xin R, x^{2} > |x|. Therefore, the correct answer is boxed{B}.

🧠:[<think>Okay, so the problem is: Among 5 students A, B, C, D, E, we need to select 3 students to participate in math, physics, and chemistry competitions. The question is asking for the number of ways in which A and B are selected at most one time. Hmm. Let me try to parse this.First, the total number of ways to select 3 students out of 5 without any restrictions. That would be combinations, right? Since the order of selection might matter if they are assigned to different subjects. Wait, wait, actually, the problem says "3 students are randomly selected to participate in the math, physics, and chemistry competitions." So each selected student is assigned to a specific subject. That means that the order matters here. So instead of combinations, this is actually a permutation problem because assigning Student X to math vs. physics is a different arrangement.So, total number of ways without restrictions would be P(5,3) which is 5*4*3 = 60. Let me confirm that: for the first subject (say math), we can choose any of the 5 students, then for physics, 4 remaining, and for chemistry, 3 remaining. Yep, 5*4*3=60.But the problem is asking for the number of ways in which A and B are selected at most one time. Wait, "selected at most one time." So that means either A is selected or B is selected, but not both? Or does it mean that each of A and B can be selected at most once? Wait, but each selection is choosing 3 distinct students, so A and B can each be selected only once in each selection. Wait, maybe I'm misunderstanding the question.Wait, the wording says: "the number of ways in which A and B are selected at most one time." Maybe it's a translation issue. Maybe it's supposed to mean that A and B are selected at most once each? But since we're selecting 3 students, each student can be selected only once. Wait, perhaps "at most one time" refers to the total number of times A and B are selected across all three competitions? Wait, no, the competitions are math, physics, and chemistry, and each selected student is assigned to one competition. So in each selection of 3 students, each student is assigned to exactly one subject. Therefore, in each arrangement, each of the 3 selected students is participating in one subject. So A and B can each be selected at most once in each such arrangement. So "A and B are selected at most one time" probably means that in the group of 3 selected students, either A is selected or B is selected, but not both. So "at most one of A or B is selected." So the problem is asking for the number of ways where either A is not selected, B is not selected, or neither is selected. So the total number of such arrangements where we don't have both A and B selected together.So the approach would be: total number of arrangements without restrictions (which is 60) minus the number of arrangements where both A and B are selected. So 60 minus the number of permutations where both A and B are selected. So let's compute the number of permutations where both A and B are selected.If we have 3 students selected, and two of them are A and B, then the third student is one of the remaining 3: C, D, or E. So there are 3 choices for the third student. Then, for each such trio (A, B, and one other), we need to assign them to the three subjects: math, physics, chemistry. Since there are 3 students, the number of permutations is 3! = 6. So for each third student, we have 6 permutations. Therefore, total number of permutations where both A and B are selected is 3 * 6 = 18.Therefore, subtracting that from the total, we get 60 - 18 = 42. So the answer should be 42.Wait, but let me verify that again. Another way: compute directly the number of permutations where at most one of A or B is selected. So this includes permutations where neither A nor B is selected, and permutations where exactly one of A or B is selected.First, permutations where neither A nor B is selected: we have to choose 3 students from C, D, E. But there are only 3 students: C, D, E. So the number of permutations is P(3,3) = 3*2*1 = 6.Second, permutations where exactly one of A or B is selected. So two cases: either A is selected and B is not, or B is selected and A is not.Case 1: A is selected, B is not. Then we need to choose 2 more students from C, D, E (since B is excluded). There are C(3,2) combinations for the other two students, which is 3. For each such combination, we have 3 students (A and the two others), and the number of permutations is 3! = 6. So total permutations for this case: 3 * 6 = 18.Case 2: B is selected, A is not. Similarly, choose 2 students from C, D, E. Again 3 combinations, each leading to 6 permutations, so 18.Therefore, total permutations where exactly one of A or B is selected is 18 + 18 = 36. Then, permutations where neither is selected is 6. So total permutations where at most one of A or B is selected is 36 + 6 = 42. Which matches the previous result. So that's good.Therefore, the answer should be 42. So boxed{42}Final Answerboxed{42}</think>To determine the number of ways in which students (A) and (B) are selected at most one time when 3 students are randomly selected from 5 students (A, B, C, D, E) to participate in math, physics, and chemistry competitions, we need to consider the total number of permutations and subtract the permutations where both (A) and (B) are selected.1. Total number of permutations without restrictions: [ P(5, 3) = 5 times 4 times 3 = 60 ]2. Number of permutations where both (A) and (B) are selected: - Choose the third student from (C, D, E) (3 choices). - Permute the three selected students (3! ways): [ 3 times 3! = 3 times 6 = 18 ]3. Subtract the permutations where both (A) and (B) are selected from the total: [ 60 - 18 = 42 ]4. Verification by direct calculation: - Permutations where neither (A) nor (B) is selected: [ P(3, 3) = 3! = 6 ] - Permutations where exactly one of (A) or (B) is selected: - Case 1: (A) is selected, (B) is not: [ binom{3}{2} times 3! = 3 times 6 = 18 ] - Case 2: (B) is selected, (A) is not: [ binom{3}{2} times 3! = 3 times 6 = 18 ] - Total for exactly one of (A) or (B): [ 18 + 18 = 36 ] - Total permutations where at most one of (A) or (B) is selected: [ 36 + 6 = 42 ]Thus, the number of ways in which (A) and (B) are selected at most one time is (boxed{42}).,<think>Okay, let me try to figure out this problem step by step. So, we have 5 students: A, B, C, D, and E. We need to randomly select 3 of them to participate in three different competitions: math, physics, and chemistry. The question is asking for the number of ways in which A and B are selected at most one time. Hmm, that wording is a bit tricky. Let me parse it again."At most one time" probably means that either A or B is selected, but not both. So, the total number of ways where both A and B are not selected together. Wait, actually, "at most one time" might mean that each of A and B is selected at most once. But since we're selecting 3 students, each student can only be selected once, right? Because each competition is different, so maybe the selection is about assigning the students to the competitions. Wait, maybe I need to clarify the problem.Let me read the problem again: "Among 5 students A, B, etc., 3 students are randomly selected to participate in the math, physics, and chemistry competitions. The number of ways in which A and B are selected at most one time is ______."Wait, maybe the problem is that when selecting 3 students, each student can be selected for one of the three competitions. So, it's possible that a student is selected for more than one competition? But the problem says "3 students are randomly selected", so maybe it's 3 different students, each assigned to one competition. So, it's permutations of 5 students taken 3 at a time, where each student is assigned to a specific competition. So, the total number of ways is 5*4*3 = 60.But the question is about A and B being selected "at most one time". So, does that mean that in the selection of the 3 students, A and B can each be selected at most once? But since we are selecting 3 distinct students, each student can only be selected once. So, maybe the problem is that initially, you can select a student multiple times, but here, since it's 3 different competitions, you need different students? Wait, the wording is a bit unclear.Alternatively, maybe "selected at most one time" refers to being selected in the competitions. Wait, maybe the original Chinese problem uses a term that translates to "at most one time," but actually means that either A or B is selected, but not both. Because in Chinese, sometimes "at most one" can mean "no more than one of them." So, maybe the problem is asking for the number of ways where A and B are not both selected. So, either only A is selected, only B is selected, or neither is selected. So, that's the interpretation.But let's confirm. If we have 5 students, selecting 3, and we want the number of selections where A and B are not both selected. Then, the total number of ways is C(5,3) = 10. The number of ways where both A and B are selected is C(3,1) = 3, because if we select both A and B, we need to choose 1 more student from the remaining 3 (C, D, E). Therefore, the number of ways where A and B are not both selected is total minus both selected, which is 10 - 3 = 7. But wait, the answer here would be 7. However, the problem mentions "the math, physics, and chemistry competitions," which implies that the order matters because each competition is different. So, maybe the initial total isn't combinations but permutations.So, if we have to assign 3 different students to 3 different competitions, the total number of ways would be permutations: 5P3 = 5*4*3 = 60. Then, the number of ways where both A and B are selected. If both A and B are selected, we need to assign them to two of the three competitions, and then assign the third competition to one of the remaining 3 students. So, first, choose the two competitions out of three for A and B: C(3,2) = 3. Then, assign A and B to those two competitions: 2! ways. Then, assign the third competition to one of the remaining 3 students: 3 choices. So, total ways with both A and B selected is 3 * 2! * 3 = 3*2*3=18. Therefore, the number of ways where A and B are not both selected is total permutations minus those 18: 60 - 18 = 42. So, the answer is 42.But wait, let me check again. If we consider that selecting 3 students where A and B are not both selected, then first, the number of such combinations is C(3,3) + C(3,2). If we don't select both A and B, then either we select neither A nor B, which is C(3,3)=1 way, or we select exactly one of A or B, which is C(2,1)*C(3,2)=2*3=6. So total combinations are 1 + 6 =7. Then, for each combination, the number of permutations is 3! =6. So, total permutations would be 7*6=42. Yes, that matches. So, 42 is the answer.Alternatively, the problem might be interpreted as the number of ways where A and B are each selected at most once. But since each competition is assigned to a different student, each student can only be selected once. So, this interpretation is redundant. So, the initial thought that "at most one time" refers to not having both A and B in the selected group is more plausible. Therefore, the answer is 42.Wait, but let me check once more. Let me think of the problem in terms of permutations.Total permutations: 5P3=60.Number of permutations where both A and B are selected: First, select A and B, and then one more student from C, D, E. Then assign the three students to the three competitions. The number of ways is C(3,1) * 3! = 3*6=18. Therefore, subtracting that from 60 gives 60-18=42. So, yes, 42.Alternatively, think of it as:Number of permutations where A and B are not both selected: So, either A is selected without B, B is selected without A, or neither is selected.Case 1: Neither A nor B is selected. Then, we have to choose 3 students from C, D, E. But there are only 3 students, so C(3,3)=1 combination, and each combination can be permuted in 3! ways, so 1*6=6.Case 2: A is selected but not B. Then, we need to choose 2 more students from C, D, E. So, C(3,2)=3. Each such combination can be permuted in 3! ways, so 3*6=18.Case 3: B is selected but not A. Similarly, C(3,2)=3. Each combination permuted in 3! ways, so 3*6=18.Total permutations: 6 + 18 + 18 = 42. Yep, same result.Therefore, the answer is 42.Final Answerboxed{42}</think>To solve the problem of finding the number of ways in which students (A) and (B) are selected at most one time among 5 students (A), (B), (C), (D), and (E) for the math, physics, and chemistry competitions, we need to consider the permutations of the students where (A) and (B) are not both selected.1. Total permutations: The total number of ways to select and assign 3 students out of 5 to three different competitions is given by permutations: [ P(5, 3) = 5 times 4 times 3 = 60 ]2. Permutations where both (A) and (B) are selected: - Choose one more student from the remaining 3 students ((C), (D), (E)): (binom{3}{1} = 3) - Assign the three selected students to the three competitions: (3! = 6) - Total permutations for this case: (3 times 6 = 18)3. Subtract the unwanted permutations: Subtract the permutations where both (A) and (B) are selected from the total permutations: [ 60 - 18 = 42 ]4. Verification by considering cases: - Case 1: Neither (A) nor (B) is selected. Permutations of 3 students from (C), (D), (E): (3! = 6) - Case 2: (A) is selected but not (B). Choose 2 more students from (C), (D), (E): (binom{3}{2} = 3). Each combination permuted: (3 times 6 = 18) - Case 3: (B) is selected but not (A). Similarly: (3 times 6 = 18) - Total permutations: (6 + 18 + 18 = 42)Thus, the number of ways in which (A) and (B) are selected at most one time is (boxed{42}).]

🔑:To solve this problem, we consider the cases where A and B are selected at most one time. This can be broken down into three distinct scenarios:1. Case 1: A is selected, but B is not. - Since A is selected, we need to choose 2 more students from the remaining 3 (excluding B) to participate in the competitions. This can be done in {C}_{3}^{2} ways. - The 3 selected students can be assigned to the math, physics, and chemistry competitions in {A}_{3}^{3} ways. - Therefore, the total number of ways for this case is {C}_{3}^{2}cdot {A}_{3}^{3} = 3 cdot 6 = 18.2. Case 2: B is selected, but A is not. - Similar to Case 1, we choose 2 more students from the remaining 3 (excluding A) to participate. This can be done in {C}_{3}^{2} ways. - Again, the 3 selected students can be assigned to the competitions in {A}_{3}^{3} ways. - Thus, the total number of ways for this case is also {C}_{3}^{2}cdot {A}_{3}^{3} = 3 cdot 6 = 18.3. Case 3: Neither A nor B is selected. - In this scenario, we select all 3 students from the remaining 3 students. This can only be done in one way, which is equivalent to {A}_{3}^{3} ways. - The 3 selected students are then assigned to the competitions, which can be done in {A}_{3}^{3} = 6 ways.Adding up the total number of ways from all three cases, we get:[18 + 18 + 6 = 42]Therefore, the total number of ways in which A and B are selected at most one time is boxed{42}.

🧠:[<think>Okay, let's tackle this problem step by step. The question is about an ellipse, a line passing through a vertex and a focus, and the distance from the center to this line being one quarter of the minor axis length. We need to find the eccentricity of the ellipse. The answer choices are given as fractions, so the answer should be one of those.First, let me recall some basic properties of an ellipse. An ellipse is defined as the set of points where the sum of the distances from two fixed points (the foci) is constant. The standard equation of an ellipse centered at the origin is (x²/a²) + (y²/b²) = 1, where 'a' is the semi-major axis, 'b' is the semi-minor axis, and the distance from the center to each focus is 'c', with c² = a² - b². The eccentricity 'e' is given by e = c/a, which is a measure of how "stretched" the ellipse is.Now, the problem mentions a line passing through a vertex and a focus. Let's visualize this. If the ellipse is centered at the origin, the vertices are at (±a, 0). The foci are at (±c, 0) along the major axis. So, if we take one vertex, say (a, 0), and one focus, say (c, 0), the line connecting them is along the major axis. Wait, but if the line is along the major axis, then the distance from the center (which is on the major axis) to this line would be zero, since the line passes through the center. But the problem states that this distance is one quarter of the minor axis length. The minor axis length is 2b, so one quarter of that is (2b)/4 = b/2. Therefore, the distance from the center to the line is b/2.Hmm, this seems contradictory. If the line is passing through the vertex and the focus on the major axis, then the line is the major axis itself, so the distance from the center (which is on that line) should be zero. But the problem says it's b/2. Therefore, maybe the line isn't along the major axis? Wait, but a vertex and a focus are on the major axis. So any line passing through both a vertex and a focus must lie on the major axis. Therefore, the distance from the center to the line should be zero. But the problem states it's b/2. That doesn't make sense. So perhaps the line is not the major axis. Maybe the line passes through a different vertex and focus? Wait, but all vertices and foci are on the major axis. Unless the ellipse is rotated, but the problem doesn't mention that. So perhaps there's a misinterpretation here.Wait, maybe the line is passing through a vertex and a focus, but not necessarily the ones on the same side. Wait, but in the standard ellipse, vertices are at (a,0) and (-a,0), foci at (c,0) and (-c,0). So, if we take a vertex (a,0) and a focus (-c,0), then the line connects (a,0) to (-c,0). But this line is still along the major axis. So the distance from the center (0,0) to this line is still zero. That can't be. Therefore, maybe the line is not along the major axis? How is that possible? Unless the line is not horizontal. Wait, but if the ellipse is standard, the major axis is horizontal. If we take a vertex (a,0) and a focus (c,0), the line is horizontal. If we take a vertex (a,0) and the other focus (-c,0), the line is still horizontal? Wait, no. If we connect (a,0) to (-c,0), that line is still along the major axis, which is horizontal. The distance from the center (0,0) to that line is zero. So this seems inconsistent with the problem's statement.Wait, maybe the problem is referring to a different vertex and focus? The ellipse has vertices at (0, ±b) as well? No, the vertices are only on the major axis. The endpoints of the minor axis are called co-vertices, I think. So maybe the problem is using the term "vertex" to include both the major and minor axis endpoints? That could be a confusion. Let's check the problem statement again: "a line l passes through a vertex and a focus of an ellipse". So, "a vertex" – in standard terminology, the vertices are the endpoints of the major axis. The endpoints of the minor axis are called co-vertices. So, the line passes through a vertex (on major axis) and a focus (also on major axis). Therefore, the line is the major axis. Then, distance from center to line is zero. But problem says it's one quarter of the minor axis length, which is b/2. So this contradicts. Therefore, perhaps my understanding is wrong.Wait, maybe the line is not along the major axis. Wait, how can a line pass through a vertex and a focus not lie on the major axis? Unless the vertex is on the minor axis? But vertices are only on the major axis. The co-vertices are on the minor axis. So maybe the problem uses "vertex" to refer to both? If so, then if the line passes through a co-vertex (endpoint of minor axis) and a focus, then the line would not be on the major axis. That could be possible. Let's consider that possibility.Wait, the problem says "a vertex" – in standard terms, vertices are on the major axis. But sometimes, people refer to both major and minor axis endpoints as vertices. Maybe the problem is using that terminology. Let me check standard definitions. According to standard definitions, the major axis has length 2a, with vertices at (±a, 0), and the minor axis has length 2b, with co-vertices at (0, ±b). So, perhaps the problem is using "vertex" to refer to both? If that's the case, then the line could pass through a co-vertex (on minor axis) and a focus (on major axis). Then, such a line would not be on the major axis, and the distance from the center to the line would be non-zero. That makes sense. Let's proceed with this interpretation.So, assuming the line passes through a co-vertex (0, b) and a focus (c, 0). Let's write the equation of this line. The line connects (0, b) and (c, 0). The slope of this line is (0 - b)/(c - 0) = -b/c. Therefore, the equation of the line is y - b = (-b/c)(x - 0), which simplifies to y = (-b/c)x + b.Now, we need to find the distance from the center (0,0) to this line. The formula for the distance from a point (x₀, y₀) to the line Ax + By + C = 0 is |Ax₀ + By₀ + C| / sqrt(A² + B²). Let's write the line equation in standard form. Starting from y = (-b/c)x + b, we can rearrange:y + (b/c)x - b = 0So, (b/c)x + y - b = 0Therefore, A = b/c, B = 1, C = -b.The distance from (0,0) to this line is |(b/c)(0) + 1*(0) - b| / sqrt( (b/c)^2 + 1^2 ) = | -b | / sqrt( (b²/c²) + 1 ) = b / sqrt( (b² + c²)/c² ) ) = b / ( sqrt(b² + c²)/c ) = (b c)/sqrt(b² + c²)According to the problem, this distance is one quarter of the minor axis length. The minor axis length is 2b, so one quarter is (2b)/4 = b/2. Therefore, we have:(b c)/sqrt(b² + c²) = b/2We can divide both sides by b (assuming b ≠ 0, which it isn't in an ellipse):c / sqrt(b² + c²) = 1/2Multiply both sides by sqrt(b² + c²):c = (1/2) sqrt(b² + c²)Square both sides to eliminate the square root:c² = (1/4)(b² + c²)Multiply both sides by 4:4c² = b² + c²Subtract c² from both sides:3c² = b²So, b² = 3c²But in an ellipse, we have the relationship a² = b² + c². Let's substitute b²:a² = 3c² + c² = 4c²Therefore, a² = 4c² => a = 2cSince the eccentricity e = c/a, substitute a = 2c:e = c/(2c) = 1/2Therefore, the eccentricity is 1/2, which is option B.Wait, let me check the steps again to make sure I didn't make a mistake.1. Assumed the line connects a co-vertex (0, b) to a focus (c, 0). Is this a valid assumption? The problem states "a vertex", but in standard terms, a vertex is on the major axis. However, if the problem is using "vertex" to include co-vertices, then this is possible. Alternatively, maybe the line is passing through a major vertex and a focus, but the line is not the major axis. Wait, but the major vertex and focus are colinear with the center, so the line would be the major axis. But as discussed earlier, the distance from center to that line is zero, which contradicts the problem's statement. Therefore, to satisfy the problem's condition, the line must pass through a co-vertex and a focus, hence assuming the problem uses "vertex" to mean any axis endpoint.But that might be non-standard terminology. Let me check again.In standard terminology, the vertices are only the endpoints of the major axis, and the endpoints of the minor axis are called co-vertices. Therefore, if the problem says "a vertex", it should refer to one of the major axis endpoints. If that's the case, then connecting a vertex (a,0) to a focus (c,0) gives a line along the major axis, with distance zero from the center. But the problem says the distance is b/2. Therefore, this is impossible. Therefore, either the problem has a typo, or it's using non-standard terminology.But given that the answer is among the options, and our calculation leads to 1/2, which is an option (B), maybe that's the correct approach. Alternatively, perhaps the line is passing through a vertex and a focus but is not the major axis. How?Wait, unless the ellipse is not axis-aligned? If the major axis is not aligned with the coordinate axes, then the line connecting a vertex and a focus might not pass through the center. But the problem doesn't mention anything about the ellipse being rotated. So, we can assume it's a standard ellipse centered at the origin, with major axis along the x-axis.Alternatively, maybe the line passes through the vertex (a,0) and the focus (c,0), but since they are colinear with the center, the line is the x-axis. The distance from the center (0,0) to the x-axis is zero. Therefore, the problem must be referring to a different line. Wait, maybe the line is not the x-axis. Wait, but passing through (a,0) and (c,0) is the x-axis. Therefore, the problem must be referring to a different vertex and focus.Wait, unless the vertex is not on the major axis. But in standard terminology, vertices are on the major axis. Therefore, this is confusing. Maybe there's a mistake in the problem statement? Or perhaps my misunderstanding.Alternatively, perhaps the line is passing through the vertex (a,0) and the other focus (-c,0). Then, the line would be from (a,0) to (-c,0). The equation of this line is still the x-axis, so the distance is zero. Not helpful.Alternatively, maybe the line is passing through a vertex and a focus but in a different orientation. Wait, perhaps the ellipse is vertical? If the major axis is vertical, then vertices would be at (0, ±a), foci at (0, ±c). Then, connecting a vertex (0,a) to a focus (0,c) would be along the y-axis, distance from center is zero. Alternatively, connecting (0,a) to (c,0), which is a focus on the major axis? Wait, no. If the major axis is vertical, foci are on the vertical axis. So foci would be at (0, ±c). Then, connecting a vertex (0,a) to a focus (0,c) is along the y-axis. Distance from center is zero. If the major axis is vertical, but connecting a vertex (0,a) to a focus (c,0) (but foci are on the vertical axis), so that's not possible. Hmm, getting confused.Wait, maybe the problem is referring to a three-dimensional ellipse? No, it's a standard ellipse in the plane.Alternatively, maybe the line is not straight? No, lines are straight.Wait, perhaps I need to re-examine the problem statement again: "A line l passes through a vertex and a focus of an ellipse. If the distance from the center of the ellipse to l is one quarter of its minor axis length, then the eccentricity of the ellipse is (　　)."So, key points:1. Line passes through a vertex and a focus.2. Distance from center to line is 1/4 of minor axis length.Assuming standard terminology, vertex is on major axis, focus is also on major axis, so line is major axis, distance zero. Not possible. Therefore, the problem must be using non-standard terminology, where "vertex" includes co-vertices. Then, line passes through co-vertex (0,b) and focus (c,0). As we calculated, leading to e = 1/2. Since that's an option, and the steps check out, perhaps that is the answer.Alternatively, maybe there's another approach. Let's try parameterizing the ellipse.Let me consider the ellipse in standard position: center at (0,0), major axis along x-axis. Vertices at (a,0), foci at (c,0) and (-c,0). Let's assume the line passes through (a,0) and (c,0). Then, as before, the line is the x-axis, distance from center (0,0) is zero. Not matching the problem. Therefore, perhaps the line passes through (a,0) and (-c,0). Then, the line is still the x-axis. Distance is zero. Not helpful.Alternatively, suppose the line is not along the x-axis. How can a line pass through a vertex and a focus not be along the x-axis? If the ellipse is rotated, but the problem doesn't mention rotation. Therefore, maybe the problem is in 3D? Unlikely.Wait, perhaps the line is oblique. Wait, but if the vertex and focus are on the x-axis, the line connecting them is the x-axis. So unless the vertex is not on the major axis. But in standard definition, vertices are on major axis.Wait, unless the vertex is (a,0) and the focus is (0,c). But foci are on the major axis. If major axis is x-axis, foci are at (c,0). If major axis is y-axis, foci are at (0,c). So, if the ellipse has major axis along y-axis, then vertices are at (0, ±a), foci at (0, ±c), co-vertices at (±b,0). Then, a line passing through a co-vertex (b,0) and a focus (0,c). Let's try this.Equation of the line connecting (b,0) and (0,c). The slope is (c - 0)/(0 - b) = -c/b. Equation: y - 0 = (-c/b)(x - b), which simplifies to y = (-c/b)x + c.Distance from center (0,0) to this line. The line can be written as (c/b)x + y - c = 0.Distance is |(c/b)(0) + 1*0 - c| / sqrt( (c/b)^2 + 1 ) = | -c | / sqrt( c²/b² + 1 ) = c / sqrt( (c² + b²)/b² ) = c / ( sqrt(c² + b²)/b ) = (c b)/sqrt(b² + c²)According to problem, this distance is (1/4) of minor axis length. The minor axis length is 2b, so (1/4)(2b) = b/2.Thus, set (c b)/sqrt(b² + c²) = b/2Cancel b (b ≠ 0):c / sqrt(b² + c²) = 1/2Square both sides:c² / (b² + c²) = 1/4Multiply both sides by (b² + c²):c² = (1/4)(b² + c²)Multiply both sides by 4:4c² = b² + c²3c² = b²But in this case, since the major axis is along y-axis, the relationship is a² = b² + c². Wait, no. Wait, if major axis is along y-axis, then the semi-major axis is 'a' along y, semi-minor 'b' along x. Then, the relationship is a² = b² + c², where c is the distance from center to focus along y-axis.But in this case, we have b² = 3c². Then, substituting into a² = b² + c², we get a² = 3c² + c² = 4c², so a = 2c. Eccentricity e = c/a = c/(2c) = 1/2. Same result.Therefore, regardless of whether the major axis is along x or y, we get e = 1/2. Hence, the answer is B: 1/2.But just to confirm, since we initially considered the major axis along x-axis and co-vertex, then major axis along y-axis and co-vertex, both leading to e = 1/2. Therefore, the answer is consistent.Alternatively, if the problem uses "vertex" as the endpoint of the major axis, then this approach is invalid. But since following standard terminology leads to a contradiction, and assuming "vertex" includes co-vertices gives a consistent answer, which is among the options, I think this is the intended solution.Thus, the eccentricity is 1/2, which is option B.Final Answerboxed{B}</think>Given an ellipse, a line ( l ) passes through a vertex and a focus. The distance from the center of the ellipse to this line is one quarter of its minor axis length. We need to find the eccentricity of the ellipse.1. Understanding the Ellipse Properties: - The standard equation of an ellipse centered at the origin is (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), where (a) is the semi-major axis, (b) is the semi-minor axis, and (c) is the distance from the center to each focus, with (c^2 = a^2 - b^2). - The eccentricity (e) is given by (e = frac{c}{a}).2. Line Passing Through a Vertex and a Focus: - Assuming the line passes through a co-vertex (on the minor axis) and a focus (on the major axis). For example, the line connects ((0, b)) and ((c, 0)).3. Equation of the Line: - The slope of the line connecting ((0, b)) and ((c, 0)) is (-frac{b}{c}). - The equation of the line is (y = -frac{b}{c}x + b), which can be rewritten as (frac{b}{c}x + y - b = 0).4. Distance from the Center to the Line: - The distance from the center ((0, 0)) to the line (frac{b}{c}x + y - b = 0) is calculated as: [ frac{|0 + 0 - b|}{sqrt{left(frac{b}{c}right)^2 + 1}} = frac{b}{sqrt{frac{b^2}{c^2} + 1}} = frac{bc}{sqrt{b^2 + c^2}} ] - This distance is given as (frac{b}{2}) (one quarter of the minor axis length).5. Solving for (c): - Setting the distance equal to (frac{b}{2}): [ frac{bc}{sqrt{b^2 + c^2}} = frac{b}{2} ] - Simplifying, we get: [ frac{c}{sqrt{b^2 + c^2}} = frac{1}{2} ] - Squaring both sides: [ frac{c^2}{b^2 + c^2} = frac{1}{4} implies 4c^2 = b^2 + c^2 implies 3c^2 = b^2 ] - Using the relationship (a^2 = b^2 + c^2): [ a^2 = 3c^2 + c^2 = 4c^2 implies a = 2c ] - Eccentricity (e = frac{c}{a} = frac{c}{2c} = frac{1}{2}).Thus, the eccentricity of the ellipse is (boxed{B}).,<think>Okay, so I need to find the eccentricity of an ellipse given some conditions about a line passing through a vertex and a focus, and the distance from the center to that line being one quarter of the minor axis length. Let me start by recalling some properties of ellipses.An ellipse is defined by its major and minor axes. The standard equation of an ellipse centered at the origin with major axis along the x-axis is (x²/a²) + (y²/b²) = 1, where a is the semi-major axis, b is the semi-minor axis, and the foci are located at (±c, 0), where c = ae, with e being the eccentricity. The relationship between a, b, and c is c² = a² - b², so the eccentricity e is given by e = c/a.The problem states that a line l passes through a vertex and a focus of the ellipse. Let me visualize this. The vertices are at (±a, 0), and the foci are at (±c, 0). So if the line passes through a vertex and a focus, let's take the right vertex (a, 0) and the right focus (c, 0). Wait, but both of these are on the x-axis. If the line passes through (a, 0) and (c, 0), then the line is just the x-axis itself. But the distance from the center (which is at the origin) to the x-axis is zero, which can't be one quarter of the minor axis length unless the minor axis is zero, which isn't possible. So that must mean the line passes through a vertex and a different focus?Wait, no. Wait, the foci are on the major axis, so if the line passes through a vertex and a focus, then if the ellipse is major along the x-axis, the vertex is (a, 0) and the focus is (c, 0). But those are colinear along the x-axis. Similarly, the left vertex (-a, 0) and left focus (-c, 0) are colinear on the x-axis. So if the line is passing through a vertex and a focus, it must be the major axis itself, but as I saw, the distance from the center to the major axis is zero. Therefore, this can't be the case. So maybe the line is passing through a vertex and a focus that are not aligned on the major axis? Wait, but foci are always on the major axis. So maybe the line is passing through a co-vertex and a focus? Wait, the co-vertex is on the minor axis. Let me check the problem again.The problem says: "A line l passes through a vertex and a focus of an ellipse." So vertex and a focus. The vertices are on the major axis, the foci are also on the major axis, so the line connecting a vertex and a focus is the major axis itself. But then, as before, the distance from the center to the line would be zero. So this seems contradictory. But the problem states that the distance from the center to the line is one quarter of the minor axis length, which is non-zero. Therefore, my initial assumption must be wrong. Maybe the line passes through a vertex and a focus, but not both on the major axis? Wait, but foci are on the major axis. Wait, unless the ellipse is rotated? But the problem doesn't mention a rotated ellipse.Wait, maybe the vertex is a co-vertex? Sometimes, people refer to the endpoints of both major and minor axes as vertices, but usually, the term "vertex" specifically refers to the endpoints of the major axis, while "co-vertex" refers to the endpoints of the minor axis. Let me check that.In standard terminology, the major axis has length 2a, with vertices at (±a, 0), and the minor axis has length 2b, with co-vertices at (0, ±b). So maybe the problem is using "vertex" to include co-vertices? That could be a translation issue or a terminology difference. If that's the case, then a line passing through a co-vertex (0, b) and a focus (c, 0). Then the line connects (0, b) and (c, 0). Let me see.So, let me assume that the line passes through a co-vertex (0, b) and a focus (c, 0). Then, the equation of the line l can be found using these two points. Let's compute its equation.The line passing through (c, 0) and (0, b). The slope m is (b - 0)/(0 - c) = -b/c. So the equation is y = (-b/c)x + b. Wait, because when x = 0, y = b, and when x = c, y = 0.Alternatively, in standard form: (x/c) + (y/b) = 1. Because when x = c, y = 0, and when y = b, x = 0. So that's the intercept form.But maybe it's easier to write it in the general form. Let's do that. Starting with slope -b/c, so equation is y = (-b/c)x + b. Converting to standard form: (b/c)x + y = b. Multiply both sides by c to eliminate the denominator: bx + cy = bc. So the equation of the line is bx + cy = bc.Now, we need the distance from the center (0,0) to this line. The formula for the distance from a point (x0, y0) to the line Ax + By + C = 0 is |Ax0 + By0 + C| / sqrt(A² + B²). Here, the line is bx + cy - bc = 0 (since bx + cy = bc can be written as bx + cy - bc = 0). So A = b, B = c, C = -bc.Therefore, the distance from (0,0) to the line is |b*0 + c*0 - bc| / sqrt(b² + c²) = | -bc | / sqrt(b² + c²) = bc / sqrt(b² + c²).According to the problem, this distance is one quarter of the minor axis length. The minor axis length is 2b, so one quarter of that is (2b)/4 = b/2. Therefore, we have:bc / sqrt(b² + c²) = b/2Divide both sides by b (assuming b ≠ 0, which it isn't):c / sqrt(b² + c²) = 1/2Multiply both sides by sqrt(b² + c²):c = (1/2) sqrt(b² + c²)Square both sides to eliminate the square root:c² = (1/4)(b² + c²)Multiply both sides by 4:4c² = b² + c²Subtract c² from both sides:3c² = b²But we know that for an ellipse, c² = a² - b². So substituting b² = 3c² into this:c² = a² - 3c²Then:c² + 3c² = a²4c² = a²Thus, c² = a²/4Therefore, c = a/2But eccentricity e = c/a = (a/2)/a = 1/2Wait, so the eccentricity is 1/2, which is option B. Let me check if this is correct.Wait, but let's verify the steps again. Starting from the distance being b/2, we had:bc / sqrt(b² + c²) = b/2Canceling b gives c / sqrt(b² + c²) = 1/2Then cross-multiplying gives 2c = sqrt(b² + c²)Squaring both sides: 4c² = b² + c² => 3c² = b²Then from the ellipse equation: c² = a² - b² => c² = a² - 3c² => 4c² = a² => c = a/2 => e = c/a = 1/2.Yes, that seems correct. Therefore, the answer is B: 1/2.But let me just check if the initial assumption that the line passes through a co-vertex and a focus is valid. Because the problem says "a vertex and a focus". If "vertex" refers only to the endpoints of the major axis, then the line passing through (a,0) and (c,0) is the major axis, which as we saw earlier would have distance zero from the center. But the problem states the distance is one quarter of the minor axis, so that can't be. Therefore, the problem must be referring to a co-vertex as a vertex. Maybe in some contexts, all endpoints of axes are called vertices. Let me check standard definitions.According to standard definitions, in an ellipse, the two points where the major axis intersects the ellipse are called the vertices, and the endpoints of the minor axis are called the co-vertices. However, sometimes in some sources, all four points (the two on major and two on minor axes) are referred to as vertices, with the major ones being primary. But given that the problem mentions "a vertex", which is singular, it's ambiguous. However, if the line is passing through a major vertex and a focus, that line is the major axis, but the distance from the center is zero, which contradicts the problem's statement. Therefore, the problem must be referring to a line passing through a co-vertex (minor axis endpoint) and a focus, which would not be on the same axis, hence forming a line with a non-zero distance from the center. Therefore, the steps I followed are correct. Therefore, the answer is B: 1/2.Alternatively, to ensure there's no mistake, let's suppose the line is passing through a vertex (on the major axis) and a focus. But as we saw, that line would be the major axis, leading to distance zero. Therefore, the problem must be referring to a co-vertex. So, my reasoning holds. Thus, the answer is B.Final Answerboxed{B}</think>Given an ellipse with a line ( l ) passing through a vertex and a focus, we need to determine the eccentricity of the ellipse given that the distance from the center to the line ( l ) is one quarter of the minor axis length.1. Understanding the Ellipse Properties: - The standard equation of the ellipse is (frac{x^2}{a^2} + frac{y^2}{b^2} = 1). - The foci are located at ((pm c, 0)) where ( c = ae ) and ( e ) is the eccentricity. - The relationship between ( a ), ( b ), and ( c ) is ( c^2 = a^2 - b^2 ).2. Line Passing Through a Vertex and a Focus: - Assuming the line passes through a co-vertex ((0, b)) and a focus ((c, 0)), the equation of the line is derived. - The line passing through ((c, 0)) and ((0, b)) has the equation ( bx + cy = bc ).3. Distance from Center to the Line: - The distance from the center ((0, 0)) to the line ( bx + cy - bc = 0 ) is calculated using the formula for distance from a point to a line: [ frac{| -bc |}{sqrt{b^2 + c^2}} = frac{bc}{sqrt{b^2 + c^2}} ] - This distance is given as one quarter of the minor axis length, which is (frac{b}{2}).4. Solving for Eccentricity: - Setting up the equation: [ frac{bc}{sqrt{b^2 + c^2}} = frac{b}{2} ] - Simplifying, we get: [ frac{c}{sqrt{b^2 + c^2}} = frac{1}{2} ] - Squaring both sides and solving: [ 4c^2 = b^2 + c^2 implies 3c^2 = b^2 ] - Using the relationship ( c^2 = a^2 - b^2 ): [ c^2 = a^2 - 3c^2 implies 4c^2 = a^2 implies c = frac{a}{2} ] - Eccentricity ( e = frac{c}{a} = frac{1}{2} ).Thus, the eccentricity of the ellipse is (boxed{B}).]

🔑:AnalysisThis question examines the simple geometric properties of an ellipse and the distance from a point to a line, as well as the intercept form of a line equation. By knowing the relationship between a, b, and c, we can solve the problem.SolutionLet's assume the equation of the ellipse is: dfrac{x^2}{a^2} + dfrac{y^2}{b^2} = 1 (a > b > 0),The line l passes through a vertex and a focus of the ellipse,Assume the equation of the line is: dfrac{x}{c} + dfrac{y}{b} = 1,Since the distance from the center of the ellipse to l is one quarter of its minor axis length, we have: dfrac{1}{sqrt{dfrac{1}{c^2} + dfrac{1}{b^2}}} = dfrac{b}{4}, therefore dfrac{b^2}{c^2} = 3, which means dfrac{a^2 - c^2}{c^2} = 3, therefore e = dfrac{c}{a} = dfrac{1}{2}. Therefore, the correct answer is boxed{text{B}}.

🧠:[<think>Okay, so I need to find the nth term, a_n, and the sum of the first n terms, S_n, of a geometric sequence where the first term a_1 is 3 and the sum of the first two terms S_2 is 9. Hmm, let's start by recalling what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted as r. So, the terms go like a_1, a_1*r, a_1*r^2, and so on. Therefore, the nth term is a_n = a_1 * r^(n-1).Given that a_1 is 3, so the first term is 3. The second term would then be 3*r. The sum of the first two terms S_2 is a_1 + a_2 = 3 + 3*r. And they told us that S_2 is 9. So, 3 + 3r = 9. Let me write that down:3 + 3r = 9.To find r, I can solve this equation. Subtract 3 from both sides:3r = 9 - 3,3r = 6.Then divide both sides by 3:r = 6 / 3,r = 2.Okay, so the common ratio r is 2. That means each term is double the previous one. So the sequence is 3, 6, 12, 24, and so on. Let me confirm that. The first term is 3, the second term is 3*2 = 6, and the sum of the first two terms is 3 + 6 = 9, which matches the given information. Perfect, so r is indeed 2.Now, to find a_n, the nth term, we use the formula for the nth term of a geometric sequence:a_n = a_1 * r^(n-1).We know a_1 is 3 and r is 2, so substituting those in:a_n = 3 * 2^(n-1).So that's the expression for the nth term. Let me check this formula with the first few terms. For n=1, a_1 should be 3. Plugging in n=1:a_1 = 3 * 2^(1-1) = 3 * 2^0 = 3*1 = 3. Correct.For n=2, a_2 should be 6:a_2 = 3 * 2^(2-1) = 3 * 2^1 = 3*2 = 6. Correct.n=3, a_3 = 3*2^(3-1) = 3*4 = 12. Which is also correct as per the sequence. So that seems good.Now, moving on to S_n, the sum of the first n terms. The formula for the sum of the first n terms of a geometric series is:S_n = a_1 * (1 - r^n) / (1 - r), when r ≠ 1.Since our common ratio is 2, which is not equal to 1, we can use this formula. Let's plug in the values:a_1 = 3, r = 2. So,S_n = 3 * (1 - 2^n) / (1 - 2).Simplify the denominator: (1 - 2) is -1, so:S_n = 3 * (1 - 2^n) / (-1) = 3 * (-1)*(1 - 2^n) = 3*(2^n - 1).Alternatively, distributing the negative sign:S_n = 3*(2^n - 1).Let me verify this sum formula with the given S_2. If n=2, S_2 should be 9 according to the problem. Let's compute it using the formula:S_2 = 3*(2^2 - 1) = 3*(4 - 1) = 3*3 = 9. Perfect, that matches.Let me check another term. For n=3, the sum S_3 should be 3 + 6 + 12 = 21. Using the formula:S_3 = 3*(2^3 - 1) = 3*(8 - 1) = 3*7 = 21. Correct again.So, the formula for S_n seems to check out. Therefore, the expressions are:a_n = 3*2^(n-1),S_n = 3*(2^n - 1).Let me just recap to make sure I didn't make any mistakes. The problem gave a geometric sequence with a first term of 3 and the sum of the first two terms as 9. I found the common ratio r by setting up the equation 3 + 3r = 9, solved for r to get 2. Then used the standard formulas for the nth term and the sum, substituting the known values of a_1 and r. Verified both formulas with n=1, 2, 3, and they worked out. So, I think that's it.Final AnswerThe nth term is boxed{3 cdot 2^{n-1}} and the sum of the first n terms is boxed{3(2^n - 1)}.</think>Given a geometric sequence ({a_n}) with the first term (a_1 = 3) and the sum of the first two terms (S_2 = 9), we need to find the nth term (a_n) and the sum of the first (n) terms (S_n).First, we determine the common ratio (r). The sum of the first two terms is given by:[S_2 = a_1 + a_2 = 3 + 3r = 9]Solving for (r):[3 + 3r = 9 implies 3r = 6 implies r = 2]The nth term of a geometric sequence is given by:[a_n = a_1 cdot r^{n-1}]Substituting (a_1 = 3) and (r = 2):[a_n = 3 cdot 2^{n-1}]The sum of the first (n) terms of a geometric sequence is given by:[S_n = a_1 cdot frac{1 - r^n}{1 - r}]Substituting (a_1 = 3) and (r = 2):[S_n = 3 cdot frac{1 - 2^n}{1 - 2} = 3 cdot frac{1 - 2^n}{-1} = 3 cdot (2^n - 1)]Thus, the nth term (a_n) and the sum (S_n) are:[a_n = boxed{3 cdot 2^{n-1}}][S_n = boxed{3(2^n - 1)}],<think>Okay, let's see. I need to find the nth term of a geometric sequence, a_n, and the sum of the first n terms, S_n. The given information is that the first term a_1 is 3, and the sum of the first two terms S_2 is 9. Alright, let's start by recalling what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio, r. So the terms go like a_1, a_1*r, a_1*r^2, and so on. Therefore, the nth term a_n should be a_1 * r^(n-1). Since a_1 is given as 3, then a_n = 3 * r^(n-1). But I don't know the value of r yet. That's something I need to find using the information about S_2.The sum of the first n terms of a geometric sequence is given by the formula S_n = a_1*(1 - r^n)/(1 - r) when r ≠ 1. Since we have S_2 = 9, let's apply that formula for n=2.So, S_2 = a_1 + a_2 = 3 + 3*r = 9. Wait, that's straightforward. Let me write that equation:3 + 3r = 9Subtract 3 from both sides:3r = 6Divide by 3:r = 2Wait, so the common ratio is 2? Let me check that. If r=2, then the first two terms are 3 and 6. Their sum is 3 + 6 = 9, which matches S_2=9. Okay, that checks out. So the common ratio is indeed 2.Therefore, the nth term is a_n = 3 * 2^(n-1). Let me test that. For n=1, a_1 = 3*2^(0) = 3*1=3, correct. For n=2, a_2=3*2^(1)=6, which is what we had before. So that seems right.Now, the sum S_n. Since we have r=2, which is not equal to 1, we can use the formula S_n = a_1*(1 - r^n)/(1 - r). Plugging in the values, that becomes S_n = 3*(1 - 2^n)/(1 - 2). Let's simplify the denominator: 1 - 2 = -1. So the formula becomes S_n = 3*(1 - 2^n)/(-1) = 3*(2^n - 1)/1 = 3*(2^n - 1). Therefore, S_n = 3*(2^n - 1).Let me verify this with n=2. S_2 should be 9. Plugging n=2 into the formula: 3*(2^2 -1) = 3*(4 -1)=3*3=9. Correct. For n=1, S_1 should be 3. The formula gives 3*(2^1 -1)=3*(2 -1)=3*1=3. Also correct. Let's check n=3. The terms would be 3, 6, 12. Sum is 3+6+12=21. Using the formula: 3*(2^3 -1)=3*(8 -1)=3*7=21. Perfect. So the formula works.So, putting it all together. The nth term is 3*2^(n-1), and the sum of the first n terms is 3*(2^n -1).But wait, let me just make sure there's no alternative formula or another way to represent it. For example, sometimes people write the sum formula differently, but in this case, since r >1, the formula S_n = a_1*(r^n -1)/(r -1). Which is equivalent to what I have here. So, 3*(2^n -1)/(2 -1) =3*(2^n -1). Yep, that's correct.Alternatively, if I had left it as 3*(1 -2^n)/(-1), that's the same as 3*(2^n -1). So either way, same result. So I think that's solid.Therefore, the answers are a_n = 3*2^(n-1) and S_n=3*(2^n -1).Final AnswerThe nth term is boxed{3 cdot 2^{n-1}} and the sum of the first n terms is boxed{3(2^n - 1)}.</think>Given a geometric sequence ({a_n}) with the first term (a_1 = 3) and the sum of the first two terms (S_2 = 9), we need to find the nth term (a_n) and the sum of the first (n) terms (S_n).First, we determine the common ratio (r). The sum of the first two terms is given by:[S_2 = a_1 + a_2 = 3 + 3r = 9]Solving for (r):[3 + 3r = 9 implies 3r = 6 implies r = 2]The nth term of a geometric sequence is given by (a_n = a_1 cdot r^{n-1}). Substituting (a_1 = 3) and (r = 2):[a_n = 3 cdot 2^{n-1}]The sum of the first (n) terms of a geometric sequence is given by the formula (S_n = a_1 cdot frac{1 - r^n}{1 - r}) for (r neq 1). Substituting (a_1 = 3) and (r = 2):[S_n = 3 cdot frac{1 - 2^n}{1 - 2} = 3 cdot frac{1 - 2^n}{-1} = 3 cdot (2^n - 1)]Thus, the nth term (a_n) and the sum (S_n) are:[a_n = boxed{3 cdot 2^{n-1}}][S_n = boxed{3(2^n - 1)}]]

🔑:Since the sum of the first n terms of the geometric sequence {a_n} is S_n, and given a_1=3, S_2=9, we have S_2=3+3q=9. Solving this equation, we find q=2, thus, a_n=3cdot 2^{n-1}, and S_n= frac{3(1-2^n)}{1-2}=3cdot(2^n-1). Therefore, the answers are: a_n=boxed{3cdot2^{n-1}}; S_n=boxed{3cdot(2^n-1)}. By using the formula for the sum of the first n terms of a geometric sequence, we find the common ratio q=2, from which we can derive the results. This problem tests the application of the general term formula and the sum formula of a geometric sequence, which is fundamental. When solving the problem, it is important to carefully read the question and properly apply the properties of geometric sequences.

🧠:[<think>Okay, so I need to find the volume of T, which is the set of points within a distance of 1 or less from a brick S that has dimensions a x b x c. Hmm, let me visualize this. The brick is a rectangular prism, right? So, if I take all the points within 1 unit of this brick, T would be like expanding the brick by 1 unit in every direction. But wait, expanding in every direction would mean adding a sort of "buffer" around the brick. However, when you expand a 3D object uniformly, the resulting shape is similar to the original but with rounded edges and corners. But wait, the problem says "a distance of 1 or less from S". So, T is the Minkowski sum of S and a closed unit ball. But since S is a rectangular prism, the Minkowski sum would result in a sort of rounded rectangle, but I need to calculate its volume.Alternatively, maybe I can think of it as the original brick S expanded by 1 unit in all directions, but then the overlapping regions where the expansion occurs at the edges and corners need to be accounted for properly. So, if I just expand each dimension by 2 (1 unit on each side), the volume would be (a + 2)(b + 2)(c + 2). But this would overcount because when you expand the brick, the edges and corners where multiple faces meet would have overlapping regions that are spheres (quarter-cylinders or eighth-spheres). Wait, actually, expanding a polyhedron by a radius r in all directions results in a volume equal to the volume of the original polyhedron plus the surface area times r plus the length of all edges times (π r²)/4 plus the number of vertices times (4π r³)/3. Is that right? Let me recall the formula for the volume of an offset or expanded polyhedron.Yes, the volume of the expanded shape is given by:Volume = Volume_original + SurfaceArea_original * r + Length_edges_original * (π r²)/4 + Number_vertices_original * (4π r³)/3 / 8Wait, I might be mixing up some factors here. Let me recall that when you offset a polyhedron by radius r, the volume added can be broken down into three parts:1. The original volume.2. The volume added by extending each face outward, which is the surface area times r.3. The volume added by rounding each edge, which is the length of each edge times a quarter-cylinder (since each edge is part of a cylinder when expanded, but only a quarter-cylinder because the original edge is a right angle).4. The volume added by rounding each vertex, which is like an eighth of a sphere at each vertex (since each vertex is a corner where three faces meet, so expanding it would create an eighth-sphere).Therefore, the total volume should be:Volume = (a + 2)(b + 2)(c + 2) - (the parts that are overcounted if we just add 2 to each dimension). Wait, no. Let me think again. If we naively expand each dimension by 2 (1 unit on each side), that would give (a+2)(b+2)(c+2). However, this naive expansion doesn't account for the fact that the actual shape isn't a larger rectangular prism but has rounded edges and corners. However, wait, the problem is asking for all points within distance 1 from S. So, actually, the shape T is the original brick S expanded by a radius of 1 in all directions. So, this shape is called the "offset" or "Minkowski sum" of S with a closed ball of radius 1. The formula for the volume of such a Minkowski sum is given by the Steiner formula, which in 3D is:Volume = Volume(S) + SurfaceArea(S) * r + Length_edges(S) * π r² / 4 + Number_vertices(S) * (4/3)π r³ / 8Wait, let's check this.In 3D, the Steiner formula for the volume of the Minkowski sum of a convex body K with a ball of radius r is:Vol(K + rB) = Vol(K) + SurfaceArea(K) * r + Length_edges(K) * π r² / 4 + Number_vertices(K) * (4/3)π r³ / 8Wait, but I need to confirm the coefficients here. Let me recall that in 3D, the Steiner formula is:Vol(K + rB) = Vol(K) + SurfaceArea(K) * r + (π/4) * Length_edges(K) * r² + (4π/3) * Number_vertices(K) * (r³)/8Wait, actually, the general Steiner formula is expressed in terms of intrinsic volumes. The formula is:Vol(K + rB) = sum_{i=0}^3 ω_i μ_{3-i}(K) r^{3-i}Where ω_i is the volume of the i-dimensional unit ball, and μ_j(K) are the intrinsic volumes. For convex polytopes, intrinsic volumes can be expressed in terms of the number of vertices, edges, faces, etc. For a 3D convex polytope, the intrinsic volumes are:μ_0(K) = Number of vertices / 8,μ_1(K) = Length of edges / (4 * π),μ_2(K) = Surface area / (2 * π),μ_3(K) = Volume.But I might need to check this. Alternatively, maybe I can approach it more directly.If we consider expanding the brick S by a unit distance in all directions, the resulting shape T will consist of:1. The original brick S.2. Rectangular prisms extending from each face of S, each with a thickness of 1. The volume of each of these prisms is the area of the face times 1. Since there are 6 faces (each pair of opposite faces), the total volume added by the faces is 2(ab + ac + bc).3. Quarter-cylinders along each edge of S. Each edge is of length equal to the corresponding dimension (a, b, or c), and when expanded by 1 unit, the quarter-cylinder (since each edge is where two faces meet at a right angle, so the expansion around the edge is a quarter-cylinder) has a volume of (π * 1²) * length * 1/4. Since there are 12 edges (4 of each length corresponding to a, b, c), the total volume from edges is 4(a + b + c) * (π/4) = π(a + b + c).Wait, hold on. Each edge is part of two faces. When you expand the two faces, they each contribute a half-cylinder along the edge, but since the expansion is in 3D, the overlapping around the edge would form a quarter-cylinder? Wait, maybe it's a quarter-cylinder. Let me think. If you have two adjacent faces, each extended by 1 unit, the region around the edge is like a quarter-cylinder. So, for each edge, the expansion adds a quarter-cylinder of radius 1 and length equal to the edge length. Since there are 12 edges (4 of each type: a, b, c), the total volume from edges is 12 edges * (quarter-cylinder volume). Wait, but edges of the same length: 4 edges of length a, 4 of length b, 4 of length c. Wait, no: a rectangular prism has 12 edges: 4 of each length. Wait, no: 12 edges total, 4 of each length a, b, c. For example, on a brick, there are 12 edges: 4 lengths, 4 widths, 4 heights. Wait, actually, no. Each rectangular prism has 12 edges: 4 edges along the length (each of length a), 4 along the width (each of length b), and 4 along the height (each of length c). Wait, actually, no. A rectangular prism has 12 edges: 12 edges in total, divided into 3 groups of 4. Each group corresponds to length, width, height. So, for a brick with dimensions a, b, c, there are 4 edges of length a, 4 edges of length b, and 4 edges of length c. Wait, no. Let me correct this. For a rectangular prism, each dimension has 4 edges. Wait, no. For example, take a standard brick: there are 12 edges. For each pair of opposite faces, there are 4 edges. Wait, perhaps it's better to count: 8 vertices, each connected to 3 edges, but each edge is shared by 2 vertices. So total edges: (8 * 3)/2 = 12. Now, how many edges of each length? For a brick with length a, width b, height c, there are 4 edges of length a (the ones along the length), 4 edges of length b (the ones along the width), and 4 edges of length c (the ones along the height). So, yes, 4 edges each of a, b, c. Therefore, the total length of all edges is 4(a + b + c). Each edge contributes a quarter-cylinder. So the volume from each edge is (1/4) * π * r² * length, where r=1. Therefore, each edge contributes (π/4) * length. Therefore, total volume from edges is (π/4) * 4(a + b + c) = π(a + b + c).Wait, hold on. Since each edge is length a, b, or c, and there are 4 edges of each. So total edges length is 4(a + b + c). For each edge, the volume added is a quarter-cylinder. The volume of a full cylinder is πr²h, so a quarter-cylinder is (πr²h)/4. Here, r=1, h is the length of the edge. So for each edge, the volume is (π * 1² * length)/4. Therefore, total volume from edges is sum over all edges of (π * length)/4. Since there are 12 edges, 4 of each length a, b, c, so sum is (π/4)(4a + 4b + 4c) = π(a + b + c). So that's the volume contributed by the edges.Then, the third component is the volume added by the vertices. Each vertex is a corner where three edges meet. When expanded by 1 unit, each vertex contributes an eighth-sphere, since the three adjacent faces each contribute a quarter-cylinder, but the corner itself is like an eighth of a sphere. A full sphere would be formed by all eight octants, but each vertex is in one octant, so expanding it would create an eighth-sphere. Since there are 8 vertices, the total volume from vertices is 8 * (1/8) * (4/3)π r³ = (4/3)π r³. Since r=1, this is (4/3)π.Putting it all together, the total volume of T is:Volume(T) = Volume(S) + SurfaceArea(S)*r + π*(a + b + c)*r² + (4/3)π*r³But in our case, r=1, so:Volume(T) = abc + 2(ab + bc + ac) + π(a + b + c) + (4/3)πWait, but hold on, SurfaceArea(S) is 2(ab + bc + ac). So the expansion gives:Original volume: abcAdded volume from faces: 2(ab + bc + ac)*1Added volume from edges: π(a + b + c)*1²Added volume from vertices: (4/3)π*1³Therefore, total Volume(T) = abc + 2(ab + bc + ac) + π(a + b + c) + (4/3)πBut let me check if this formula is correct. Let me test it with a simple case where a, b, c are all 0. Then the brick is a point, and T should be a sphere of radius 1. But substituting a=0, b=0, c=0, we get Volume(T) = 0 + 0 + π(0) + 4π/3 = 4π/3, which is the volume of a unit ball. Wait, but actually, if S is a point, then T is a ball of radius 1, which has volume (4/3)π(1)^3 = 4π/3. So that checks out.Another test case: Let’s take a 1x1x1 cube. Then the expanded volume should be:Original volume: 1Surface area: 6, so added volume from faces: 6*1=6Edges: 12 edges, each of length 1. So edges contribute π(1+1+1) = π*3 ≈9.4248Vertices: 8 vertices, contributing 4π/3 ≈4.1888Total Volume(T) = 1 + 6 + 3π + 4π/3 = 7 + (9π + 4π)/3 = 7 + 13π/3 ≈7 + 13.617=20.617But what's the actual volume of the Minkowski sum of a unit cube with a unit ball? Alternatively, we can compute it directly. If we have a cube of 1x1x1, expanding it by 1 in all directions. The Minkowski sum would be a cube with rounded edges and corners. The volume should be:Original cube: 1Plus extensions on each face: each face has area 1, extended by 1, so 6*1=6Plus quarter-cylinders on each edge: 12 edges, each with length 1, so total volume from edges is 12*(1/4)*π*1^2*1= 3πPlus eighth-spheres at each vertex: 8*(1/8)*(4/3)π*1^3= (4/3)πSo total volume is 1 + 6 + 3π + (4/3)π = 7 + (13/3)π ≈7 + 13.617≈20.617, which matches the previous calculation. So the formula seems correct.Therefore, in general, for an a x b x c brick, the volume of T is:abc + 2(ab + bc + ac) + π(a + b + c) + (4/3)πBut let me check if this matches the formula I derived earlier. Yes:Volume(T) = Volume(S) + SurfaceArea(S)*r + π*(sum of edge lengths)/4 * r^2 + (number of vertices)*(1/8)*(4/3 π r^3)But since r=1, and sum of edge lengths is 4(a + b + c), then:SurfaceArea(S)*r = 2(ab + bc + ac)*1 = 2(ab + bc + ac)Sum of edge lengths is 4(a + b + c), so π*(sum of edge lengths)/4 *1^2 = π(a + b + c)Number of vertices is 8, so (8)*(1/8)*(4/3 π) = 4/3 πSo indeed, the formula is:Volume(T) = abc + 2(ab + bc + ac) + π(a + b + c) + 4π/3Therefore, that's the answer.But let me think again. Wait, when you expand the brick, the regions that are added are:1. Extensions from each face: these are like rectangular prisms extending out from each face, which have volume equal to the area of the face times 1. Since there are two of each face, the total is 2(ab + bc + ac).2. The quarter-cylinders along each edge. Each edge is adjacent to two faces, and when you expand each face, the overlapping region around the edge is a quarter-cylinder. Since there are 12 edges, each of length a, b, or c, the total volume from edges is 12*(1/4)*π*1^2*edge_length. Wait, but 12 edges, but 4 edges of each length a, b, c. So total edge length is 4a + 4b + 4c = 4(a + b + c). So each edge contributes (1/4)*π*1^2*length, so total volume is (π/4)*(4(a + b + c)) = π(a + b + c). Which matches.3. The eighth-spheres at each vertex. Each vertex is where three edges meet, and expanding each face leads to an eighth-sphere at each vertex. There are 8 vertices, each contributing (1/8)*(4/3 π r^3) = (4/3 π)/8 per vertex. So total for 8 vertices: 8*(4/3 π)/8 = 4/3 π.So combining all these:Original volume: abcAdded by faces: 2(ab + bc + ac)Added by edges: π(a + b + c)Added by vertices: 4π/3Thus, total volume of T is:abc + 2(ab + bc + ac) + π(a + b + c) + (4/3)πTherefore, the answer should be this expression.But let me check another test case where a, b, c are large. If a, b, c are very large, then the dominant term would be abc + 2(ab + bc + ac) + π(a + b + c) + 4π/3. But if the brick is very large, the volume of T should approximate a rectangular prism expanded by 1 in each dimension, which is (a + 2)(b + 2)(c + 2). Let's expand that:(a + 2)(b + 2)(c + 2) = abc + 2ab + 2ac + 2bc + 4a + 4b + 4c + 8Comparing this to our formula:abc + 2(ab + bc + ac) + π(a + b + c) + 4π/3The difference is in the linear terms and constants. In the expanded prism, the linear terms are 4(a + b + c) and the constant is 8, whereas in our formula, the linear terms are π(a + b + c) and the constant is 4π/3. So for large a, b, c, the two formulas are different. But this is expected, because when the brick is large, the rounded edges and corners contribute less significantly compared to the overall volume. However, the formula (a + 2)(b + 2)(c + 2) overestimates the volume because it doesn't account for the rounded edges and corners. The actual volume is less than that, because the expansion isn't a full rectangular prism but has rounded parts. So when a, b, c are large, the main terms would be abc + 2(ab + bc + ac) + π(a + b + c). Comparing to the expanded prism, which is abc + 2(ab + bc + ac) + 4(a + b + c) + 8. So the difference comes from the coefficients of the linear terms (π vs. 4) and the constants (4π/3 vs. 8). Since π is approximately 3.14, which is less than 4, so the actual volume is less than the expanded prism, as expected.Therefore, the formula we derived is correct.Hence, the volume of T is abc + 2(ab + bc + ac) + π(a + b + c) + (4/3)π.Final AnswerThe volume of ( T ) is (boxed{abc + 2(ab + bc + ac) + pi(a + b + c) + frac{4}{3}pi}).</think>To find the volume of the set ( T ), which consists of all points within a distance of 1 or less from a brick ( S ) with dimensions ( a times b times c ), we need to consider the Minkowski sum of ( S ) and a closed unit ball. This results in expanding the brick by 1 unit in all directions, accounting for the rounded edges and corners.The volume of ( T ) can be calculated using the Steiner formula, which breaks down the volume into the original volume of the brick, the volume added by expanding each face, the volume added by rounding each edge, and the volume added by rounding each vertex.1. Original Volume: The volume of the brick ( S ) is ( abc ).2. Volume from Faces: Each face of the brick is expanded by 1 unit, contributing a volume equal to the surface area of the brick times 1. The surface area is ( 2(ab + bc + ac) ), so this adds ( 2(ab + bc + ac) ).3. Volume from Edges: Each edge of the brick contributes a quarter-cylinder of radius 1. The total length of all edges is ( 4(a + b + c) ), and each edge contributes ( frac{pi}{4} times text{length} ). Thus, the volume from edges is ( pi(a + b + c) ).4. Volume from Vertices: Each vertex contributes an eighth of a sphere of radius 1. There are 8 vertices, so the total volume from vertices is ( 8 times frac{1}{8} times frac{4}{3}pi = frac{4}{3}pi ).Combining these contributions, the volume of ( T ) is:[boxed{abc + 2(ab + bc + ac) + pi(a + b + c) + frac{4}{3}pi}],<think>Alright, so I need to find the volume of T, which is the set of points within a distance of 1 or less from the brick S. The brick S has dimensions a x b x c. Hmm, okay. Let's think about how to model this.First, visualizing the brick. It's a rectangular prism with length a, width b, and height c. The set T is all points that are at most 1 unit away from S. So, T is like a "buffer" or "offset" around the brick. If I imagine expanding the brick by 1 unit in every direction, but then also rounding the edges and corners where the expansion would overlap. Wait, but actually, since we're dealing with a three-dimensional object, expanding it in all directions by 1 unit would create a sort of rounded shape. However, I need to be careful here because the set of points within a distance of 1 from S includes not just expanding the brick, but also adding hemispherical ends to each face, edge, and corner.Let me recall that in 3D, when you take all points within a distance r from a polyhedron, the volume of that region is equal to the volume of the original polyhedron expanded by r, plus the volume contributed by the hemispheres on each face, quarter-cylinders along each edge, and eighth-spheres at each vertex. Wait, actually, let's break it down:1. Original Brick Volume: The original volume is straightforward, which is a * b * c. But since we're expanding the brick by 1 unit in all directions, the expanded brick would have dimensions (a + 2) x (b + 2) x (c + 2). Wait, is that correct? Because if you expand 1 unit in every direction, each dimension increases by 2 (1 on each side). So the expanded brick would be (a + 2)(b + 2)(c + 2). However, that's just the expanded rectangular prism part, but we also have rounded parts. Wait, no. Actually, when you take all points within 1 unit of the brick, it's not just expanding the brick. The shape is called an "offset" or "Minkowski sum" with a sphere of radius 1. The Minkowski sum of a rectangular prism and a sphere results in a shape that has the expanded prism, hemispherical ends on each face, quarter-cylinders along each edge, and eighth-spheres at each corner. So, the total volume is the sum of the expanded prism, plus the hemispheres on each face, plus the quarter-cylinders on each edge, plus the eighth-spheres at each vertex.But wait, actually, if you take the Minkowski sum with a sphere, the resulting volume can be computed by the Steiner formula. For convex bodies, the volume of the Minkowski sum with a ball of radius r is given by:Volume = Volume(S) + SurfaceArea(S) * r + (Perimeter(S) * π * r²)/4 + (4/3)πr³ * NumberOfVertices/8 ?Wait, maybe I need to recall the correct formula. The Steiner formula for 3D is:Vol(S + rB) = Vol(S) + SurfaceArea(S) * r + Length(Edges) * π r² / 4 + (4/3 π r³) * NumberOfVertices / 8Wait, let me verify. For each face, when you offset the brick, you are adding a "prism" of height r over each face. But actually, since the offset is a Minkowski sum with a ball, the formula involves the surface area, the integral of the mean curvature, and the Euler characteristic. But maybe I need a simpler approach here.Alternatively, think of the Minkowski sum as the union of all balls of radius 1 centered at points in S. So, the volume of T is the volume of S expanded by 1 in all directions, but with overlapping regions accounted for. However, since S is convex, the Minkowski sum is just the expanded convex set, which can be calculated as:Volume(T) = Volume(S) + SurfaceArea(S) * 1 + Length(Edges) * (π * 1²)/4 + NumberOfVertices * (4/3 π * 1³)/8Wait, this seems similar to the 2D case where when you take the Minkowski sum of a polygon with a disk, the area is Area(P) + Perimeter(P) * r + π r². In 3D, the formula is analogous but involves the surface area, the total length of edges, and the number of vertices. Each term corresponds to the volume contributions from the original solid, the "extrusion" along the surface, the "extrusion" along the edges, and the "extrusion" at the vertices.So, for a rectangular prism:1. Original Volume: abc2. Surface Area Contribution: The surface area of the brick is 2(ab + bc + ac). When you offset by 1 unit, each face contributes a volume equal to the face area times 1 (like extending outward), but since each edge and corner is shared by multiple faces, those overlaps are accounted for by the next terms.Wait, but in 3D, the Minkowski sum volume is given by:Vol(S + B) = Vol(S) + SurfaceArea(S) * r + Length(Edges) * (π r²)/4 + NumberOfVertices * (4/3 π r³)/8Where r is the radius. Here, r = 1. So plugging in, we get:Vol(T) = abc + 2(ab + bc + ac)*1 + (12 edges)*(length of each edge)*(π*1²)/4 + (8 vertices)*(4/3 π*1³)/8Wait, let's compute each term:1. Original Volume: abc2. Surface Area: 2(ab + bc + ac). So SurfaceArea * r = 2(ab + bc + ac)*1 = 2(ab + bc + ac)3. Length of Edges: A rectangular prism has 12 edges: 4 of each length a, b, c. So total edge length is 4(a + b + c). Then, Length(Edges) * π r² /4. Since r=1, that becomes 4(a + b + c) * π *1² /4 = (a + b + c)π4. Number of Vertices: 8. Each vertex contributes (4/3 π r³)/8. So total vertices contribution is 8*(4/3 π)/8 = 4/3 πSo summing all terms:Vol(T) = abc + 2(ab + bc + ac) + (a + b + c)π + (4/3)πWait, but hold on. Let me confirm this. The formula from the Steiner formula for a convex polyhedron in 3D is:Vol(S + rB) = Vol(S) + SurfaceArea(S) * r + Length(Edges) * (π r²)/4 + NumberOfVertices * (π r³)/3Wait, different coefficients here. Wait, perhaps my previous decomposition was incorrect.Wait, according to the Steiner formula for 3D:Vol(S + rB) = Vol(S) + A * r + M * r² + C * r³Where A is the surface area, M is the integral of the mean curvature over the surface, and C is the Euler characteristic times 4π/3. Wait, maybe.Alternatively, for convex polyhedra, the Steiner formula is:Vol(S + rB) = Vol(S) + SurfaceArea(S) * r + Length(Edges) * (π r²)/4 + NumberOfVertices * (π r³)/3Wait, let's check this with a cube. Let's take a cube with side length s, so edges length 12s, surface area 6s², volume s³, 8 vertices. Then,Vol(T) = s³ + 6s² r + 12s * π r² /4 + 8 * π r³ /3 = s³ + 6s² r + 3s π r² + (8/3)π r³But if we take s=0, then we get a cube of zero size, which would just be a point, and the volume of T would be a sphere of radius r, which is (4/3)π r³. But according to the formula above, it would be 0 + 0 + 0 + (8/3)π r³, which is (8/3)π r³, which is not equal to (4/3)π r³. Hmm, so there's a problem here.Wait, that suggests that the coefficient for the vertices is incorrect. If the original cube is a point (s=0), then the Minkowski sum is a sphere of radius r, so volume (4/3)π r³. But according to the formula above with 8 vertices, each contributing π r³ /3, we get 8*(π r³ /3) = 8/3 π r³, which is double the actual volume. Therefore, that formula can't be correct.So perhaps my initial decomposition is wrong. Let's think differently.When you take the Minkowski sum of a convex polyhedron with a ball of radius r, the volume can be expressed as:Vol(S + rB) = Vol(S) + SurfaceArea(S) * r + Length(Edges) * (π r²)/4 + NumberOfVertices * (π r³)/6Wait, let's test this with s=0, a single vertex. Then, S is a point, Volume(S)=0, SurfaceArea=0, Length(Edges)=0, NumberOfVertices=1. Then the formula gives 0 + 0 + 0 + (π r³)/6. But the Minkowski sum should be a sphere of volume (4/3)π r³. So this is still inconsistent. Therefore, my approach is flawed.Alternatively, maybe the correct formula is:Vol(S + rB) = Vol(S) + SurfaceArea(S) * r + Length(Edges) * π r² /4 + (Number of Vertices) * π r³ /6But even that, for a point (vertex), would give π r³ /6, but the actual volume is (4/3)π r³. So, not matching. Therefore, my decomposition is wrong.Perhaps a better approach is to think of the Minkowski sum as the union of all balls of radius r centered at points of S. Since S is convex, the Minkowski sum is the same as the offset. So, the volume of T is equal to the volume of the original brick expanded by 1 in all directions, plus hemispherical caps on each face, quarter-cylinders along each edge, and eighth-spheres at each vertex.Wait, let's try to compute it by parts.Imagine expanding the brick by 1 unit in all directions. The expanded brick would have dimensions (a + 2) x (b + 2) x (c + 2). However, this expanded brick includes regions that are outside the actual Minkowski sum. Wait, no. Actually, the Minkowski sum with a ball includes the expanded brick plus the rounded parts. Wait, no. If you take the Minkowski sum of the brick with a ball of radius 1, the shape is the set of all points within distance 1 from the brick. This is equivalent to the union of all balls of radius 1 centered at points of the brick. This shape will have:1. The original brick.2. For each face of the brick, a region that is a rectangular prism extending 1 unit outward from the face, but with a hemispherical end. However, since the brick is convex, these regions overlap with the expanded brick.Wait, maybe it's simpler to decompose the Minkowski sum into several parts:- The original brick.- Extensions along each face (like padding each face with a 1-unit thick layer).- Extensions along each edge (like quarter-cylinders of radius 1 along each edge).- Extensions at each vertex (like eighth-spheres of radius 1 at each vertex).But when you pad each face, you are adding a 1-unit thick slab to each face. However, where two slabs meet at an edge, their intersection is a quarter-cylinder (since the slabs are each offset by 1 unit). Similarly, where three slabs meet at a vertex, their intersection is an eighth-sphere.Therefore, the total volume can be calculated using inclusion-exclusion:Volume(T) = Volume(expanded brick) + Volume(hemispherical ends) + Volume(quarter-cylinders) + Volume(eighth-spheres)But actually, when you expand the brick by 1 in all directions, you get a larger brick (a+2)(b+2)(c+2). But that's actually overcounting because the corners and edges of this expanded brick are sharp, but in reality, the Minkowski sum rounds those edges and corners. Therefore, the expanded brick is not the correct volume. Wait, perhaps the correct way is:The Minkowski sum of a convex polyhedron and a ball can be thought of as the union of:1. The original polyhedron.2. A 1-unit thick layer around all faces.3. Quarter-cylinders along each edge.4. Eighth-spheres at each vertex.But if we think in terms of the Minkowski sum, it's equivalent to:Volume(T) = Volume(S) + SurfaceArea(S) * r + Length(Edges) * π r² /4 + NumberOfVertices * π r³ /6But as we saw earlier, this formula gives an incorrect result for a single point. However, for a single point, the Minkowski sum is a full sphere, so the volume should be (4/3)π r³. But according to the formula, NumberOfVertices * π r³ /6 = 1 * π r³ /6, which is π r³ /6, which is less than the actual volume. So this formula is not correct.Alternatively, maybe I need to think of each vertex contributing an eighth of a sphere, each edge contributing a quarter of a cylinder, and each face contributing a half-cylinder?Wait, maybe the correct way is:- Each face of the brick, when offset by 1 unit, adds a volume equal to the area of the face times 1 (a slab), but this overlaps with the adjacent face slabs. So, to avoid double-counting the edges where two slabs meet, we need to subtract the overlapping regions. However, the edges themselves, when expanded, form quarter-cylinders (since two adjacent slabs each contribute a half-cylinder, but overlapping). Wait, this is getting confusing. Maybe inclusion-exclusion is the way to go.Alternatively, here's a different approach:The set T is all points within distance 1 from the brick S. So, T can be visualized as the union of all spheres of radius 1 centered at points in S. Since S is a rectangular prism, T will have:1. The original brick S.2. Six rectangular prisms extending from each face of S, each of length equal to the dimensions of the face and height 1 (but these prisms are actually "extruded" outward from each face).3. Twelve quarter-cylinders along each edge of S, each of radius 1 and length equal to the length of the edge.4. Eight eighth-spheres at each corner of S, each of radius 1.Wait, that seems plausible. Let's verify:- Each face of the brick, when expanded outward by 1 unit, forms a rectangular prism. However, the edges where two faces meet would have overlapping regions. To account for the rounded edges, instead of having overlapping rectangular prisms, we replace the sharp edges with quarter-cylinders. Similarly, the corners, where three faces meet, are replaced by eighth-spheres.So, the total volume is:Volume(T) = Volume of original brick + Volume of six extruded prisms + Volume of twelve quarter-cylinders + Volume of eight eighth-spheres.But wait, actually, if we extrude each face outward by 1 unit, the volume added by each face is (area of face) * 1. However, the extrusion from each face would overlap with the extrusions from adjacent faces along their edges. Therefore, we need to subtract the overlapping edges, which are the quarter-cylinders. But then again, subtracting those might remove too much, so we need to add back the corners, which are the eighth-spheres.This is classic inclusion-exclusion principle. So, let's compute:1. Volume of the original brick: abc2. Volume from extruding each face: The brick has 6 faces. Each pair of opposite faces has area ab, bc, or ac. Extruding each face by 1 unit adds a volume of (ab + bc + ac) * 1 * 2 (since each pair of opposite faces is extruded). Wait, no. Each face is extruded outward by 1 unit. For example, the front face (area ab) is extruded forward 1 unit, and the back face (area ab) is extruded backward 1 unit. So total volume from face extrusions is 2ab + 2bc + 2ac.But if we just add these, the edges where two extruded faces meet would have overlapping volumes. Specifically, each edge is where two faces meet, and their extrusions overlap in a quarter-cylinder. Since the original extrusion assumes rectangular prisms, but the actual Minkowski sum rounds these edges, we need to subtract the overlapping rectangular parts and add the quarter-cylinders.Wait, perhaps it's simpler to model the offset as follows:When you take all points within 1 unit of the brick, the shape T can be decomposed into:- The original brick.- Six rectangular prisms (one for each face) of dimensions (a x b x 1), (a x b x 1), (a x c x 1), (a x c x 1), (b x c x 1), (b x c x 1). Wait, no, actually, each face extrusion would have one dimension increased by 1. Wait, maybe not. Let me think.Alternatively, the volume around the brick can be considered as:- The main expanded brick: (a + 2)(b + 2)(c + 2). But this includes the original brick and all extruded parts. However, the expanded brick has sharp edges and corners, but in reality, T has rounded edges and corners. Therefore, the difference between the expanded brick and T is that the expanded brick has rectangular prisms at the edges and corners, while T has cylindrical and spherical parts. Therefore, the volume of T can be calculated by subtracting the overlapping rectangular edge and corner regions from the expanded brick and adding the corresponding cylindrical and spherical volumes.So:Volume(T) = Volume of expanded brick - Volume of rectangular edges and corners + Volume of quarter-cylinders and eighth-spheres.Let me elaborate.The expanded brick has dimensions (a + 2)(b + 2)(c + 2). This includes the original brick, plus slabs on each face, plus rectangular prisms along each edge (where two slabs intersect), plus cubical regions at each corner (where three slabs intersect). However, in the Minkowski sum T, the edges should be quarter-cylinders instead of rectangular prisms, and the corners should be eighth-spheres instead of cubes. Therefore, to correct the volume, we need to:1. Start with the expanded brick: (a + 2)(b + 2)(c + 2).2. Subtract the volume of the rectangular prisms along each edge. Each edge of the original brick has a rectangular prism of length equal to the edge's length and cross-section 1x1. There are 12 edges: 4 of each length a, b, c. So total volume to subtract is 4(a + b + c) * (1 * 1) = 4(a + b + c).3. Subtract the volume of the cubical corners. There are 8 corners, each a cube of 1x1x1, so total volume to subtract is 8 * 1 = 8.But now, instead of those rectangular prisms and cubes, we need to add back the volume of the quarter-cylinders along each edge and eighth-spheres at each corner.4. Add the volume of quarter-cylinders along each edge. A quarter-cylinder has volume (1/4) * π r² * length. Here, r = 1, and length is the edge length. There are 12 edges, each contributing (1/4)π *1² * length. So total volume from quarter-cylinders is (1/4)π * (4(a + b + c)) ) = π(a + b + c). Wait, 12 edges: 4 edges of length a, 4 of length b, 4 of length c? Wait, no. Wait, a rectangular prism has 12 edges: 4 of each length a, b, c. So total edge length is 4(a + b + c). Each edge contributes (1/4)π *1² * edge length. So total quarter-cylinder volume is (1/4)π * edge length for each edge. Summing over all edges: (1/4)π * [4(a + b + c)] = π(a + b + c).5. Add the volume of eighth-spheres at each corner. An eighth-sphere has volume (1/8)*(4/3)π r³ = (1/6)π r³. With r=1, each eighth-sphere is (1/6)π. There are 8 corners, so total volume added is 8*(1/6)π = (4/3)π.Therefore, putting it all together:Volume(T) = (a + 2)(b + 2)(c + 2) - 4(a + b + c) - 8 + π(a + b + c) + (4/3)πLet's expand the first term:(a + 2)(b + 2)(c + 2) = abc + 2ab + 2ac + 2bc + 4a + 4b + 4c + 8Then subtract 4(a + b + c) and 8:= abc + 2ab + 2ac + 2bc + 4a + 4b + 4c + 8 - 4a - 4b - 4c -8Simplifies to:abc + 2ab + 2ac + 2bcThen add π(a + b + c) and (4/3)π:Volume(T) = abc + 2ab + 2ac + 2bc + π(a + b + c) + (4/3)πSo, combining constants:= abc + 2(ab + ac + bc) + π(a + b + c + 4/3)Wait, but (4/3)π is a constant term. Wait, no. Wait, π(a + b + c) + (4/3)π = π(a + b + c + 4/3). But 4/3 is a scalar, not a term with a, b, c. So actually, it's:Volume(T) = abc + 2(ab + bc + ac) + π(a + b + c) + (4/3)πHence, the final answer is:abc + 2(ab + bc + ac) + π(a + b + c) + (4/3)πBut let's verify this formula with a simple case where a, b, c are 0. Suppose a = b = c = 0. Then the brick is a single point. The volume of T should be a sphere of radius 1, which is (4/3)π. Plugging into the formula:0 + 2(0 + 0 + 0) + π(0 + 0 + 0) + (4/3)π = (4/3)π. Correct.Another test case: Let’s take a = 1, b = 1, c = 1 (a unit cube). Then Volume(T) should be:Original cube volume: 1Expanded cube: (1 + 2)^3 = 27But subtract edges and corners:Edges: 4(1 + 1 + 1) = 12. So subtract 12.Corners: 8. Subtract 8.But then add quarter-cylinders: π(1 + 1 + 1) = 3πAdd eighth-spheres: (4/3)πBut according to the formula:abc + 2(ab + bc + ac) + π(a + b + c) + (4/3)π= 1 + 2(1 + 1 + 1) + π(3) + (4/3)π= 1 + 6 + 3π + (4/3)π= 7 + (13/3)π ≈ 7 + 13.617 = 20.617Alternatively, using the expanded brick method:Original expanded brick: 27Subtract edges: 12*1=12 and corners:8, so 27 -12 -8=7Add quarter-cylinders: 3π and eighth-spheres: (4/3)π, so 7 + 3π + (4/3)π=7 + (13/3)π≈20.617But let's compute the actual volume of T for a unit cube. The Minkowski sum of a unit cube and a unit ball. It should be the volume of the cube expanded by 1 in all directions, but with rounded edges and corners. The exact volume is:Expanded cube: (3)^3=27But subtract the 12 edges (each a rectangular prism 1x1x1, but length 1): total subtracted edges volume: 12*1=12Subtract the 8 corners (each 1x1x1 cube): 8*1=8But then add back the 12 quarter-cylinders (each of volume (1/4)*π*1^2*1=π/4): 12*(π/4)=3πAdd back the 8 eighth-spheres (each of volume (1/8)*(4/3)π*1^3=π/6): 8*(π/6)=4π/3So total volume:27 -12 -8 +3π +4π/3=7 + (9π/3 +4π/3)=7 +13π/3≈7 +13.617≈20.617Which matches the formula. So the formula seems correct.Therefore, the answer is abc + 2(ab + bc + ac) + π(a + b + c) + (4/3)πBut let me check if that can be written more concisely. Let's factor out π:abc + 2(ab + bc + ac) + π(a + b + c + 4/3)But in the original problem, the answer is requested in boxed notation. So likely, the answer is:boxed{abc + 2(ab + ac + bc) + frac{4}{3}pi + pi(a + b + c)}Alternatively, they might factor π as:boxed{abc + 2(ab + ac + bc) + pi(a + b + c) + frac{4}{3}pi}Either form is acceptable, but to match standard answers, perhaps it's written as:Volume = (a + 2)(b + 2)(c + 2) + adjustments, but as we saw, the expanded form is abc + 2(ab + bc + ac) + π(a + b + c) + (4/3)π.Yes, that's the correct formula.Final AnswerThe volume of ( T ) is boxed{abc + 2(ab + bc + ac) + pi(a + b + c) + frac{4}{3}pi}.</think>To find the volume of the set ( T ), which consists of all points within a distance of 1 unit from the brick ( S ) with dimensions ( a times b times c ), we need to consider the contributions from different parts of the brick. The volume of ( T ) can be decomposed into several components:1. Original Volume: The volume of the original brick ( S ), which is ( abc ).2. Surface Area Contribution: Expanding each face of the brick by 1 unit in all directions, contributing ( 2(ab + bc + ac) ).3. Edge Contributions: Each edge of the brick contributes a quarter-cylinder of radius 1. The total length of all edges is ( 4(a + b + c) ), and the volume contributed by these quarter-cylinders is ( pi(a + b + c) ).4. Vertex Contributions: Each vertex of the brick contributes an eighth-sphere of radius 1. There are 8 vertices, and the total volume contributed by these eighth-spheres is ( frac{4}{3}pi ).Combining all these contributions, the volume of ( T ) is calculated as follows:[text{Volume}(T) = abc + 2(ab + bc + ac) + pi(a + b + c) + frac{4}{3}pi]Thus, the final answer is:[boxed{abc + 2(ab + bc + ac) + pi(a + b + c) + frac{4}{3}pi}]]

🔑:: abc + 2(ab + bc + ca) + π(a + b + c + 4/3). Easy. T comprises: S, volume abc; a slab on each face, two volume ab, two volume bc, two volume ca; a quarter cylinder on each edge, four with length a, total volume πa, four length b, total volume πb, and four length c, total volume πc; and an eighth sphere radius 1 at each vertex, total volume 4/3 π. 45th Putnam 1984 © John Scholes [email protected] 3 Nov 1999 Last corrected/updated 25 Nov 03

🧠:[<think>Okay, let's see. The problem is asking for the monotonic decreasing interval of the function ( y = log_{frac{1}{5}} |x - 2| ). The options are given as (A) through (D). Hmm, I need to figure out where this function is decreasing. First, let me recall some logarithm properties. The function ( log_b a ) is decreasing if the base ( b ) is between 0 and 1. Since the base here is ( frac{1}{5} ), which is less than 1, the logarithm function itself is a decreasing function. But wait, that's just the basic logarithm. Here, the argument inside the log is ( |x - 2| ), so the function is actually a composition of the absolute value function and the logarithm. Let me break it down. Let me consider the function ( y = log_{frac{1}{5}} |x - 2| ). To find where this is decreasing, I need to analyze the behavior of the composite function. Since the logarithm with base less than 1 is decreasing, the overall function will be decreasing where the inner function ( |x - 2| ) is increasing. Conversely, the outer log function being decreasing means that when the inner function is increasing, the composition will be decreasing, and when the inner function is decreasing, the composition will be increasing. So, the key here is to figure out the intervals where ( |x - 2| ) is increasing or decreasing. Let's recall that ( |x - 2| ) is a V-shaped graph shifted to the right by 2 units. The vertex is at (2, 0). To the left of x = 2, the function ( |x - 2| ) is decreasing, and to the right of x = 2, it's increasing. Therefore, the inner function ( |x - 2| ) is increasing on ( (2, +infty) ) and decreasing on ( (-infty, 2) ). Since the outer function ( log_{frac{1}{5}} ) is decreasing, the composition ( log_{frac{1}{5}} |x - 2| ) will have its monotonicity reversed. So, when ( |x - 2| ) is increasing (on ( (2, +infty) )), the log function will be decreasing. When ( |x - 2| ) is decreasing (on ( (-infty, 2) )), the log function will be increasing. Therefore, the function ( y = log_{frac{1}{5}} |x - 2| ) is decreasing on ( (2, +infty) ). That corresponds to option (C). Wait a second, but let me make sure. Let me also check the domain of the function. The argument of the logarithm must be positive, so ( |x - 2| > 0 ), which implies ( x neq 2 ). So, the domain is ( (-infty, 2) cup (2, +infty) ). Therefore, the function is defined for all real numbers except x = 2. Now, as per the previous analysis, on ( (2, +infty) ), the inner function is increasing, so the composition is decreasing. On ( (-infty, 2) ), the inner function is decreasing, so the composition is increasing. Hence, the function is decreasing only on ( (2, +infty) ). Looking at the options, (C) is ( (2, +infty) ), which should be the correct answer. But wait, let me double-check by considering specific points. Let's pick x = 3, which is in ( (2, +infty) ). Then ( |3 - 2| = 1, so y = log_{1/5} 1 = 0 ). If x increases to 4, ( |4 - 2| = 2, so y = log_{1/5} 2 ). Since ( log_{1/5} 2 ) is negative because 2 is greater than 1 and the base is less than 1. So as x increases from 3 to 4, y decreases from 0 to a negative number. So that's decreasing. What about x = 1, which is in ( (-infty, 2) ). ( |1 - 2| = 1, so y = 0. If x increases to 1.5, ( |1.5 - 2| = 0.5, so y = log_{1/5} 0.5 ). Since 0.5 is less than 1, ( log_{1/5} 0.5 ) is positive. Wait, because when the base is between 0 and 1, the logarithm of a number less than 1 is positive. So as x increases from 1 to 1.5, the inner function decreases from 1 to 0.5, so the log value increases from 0 to a positive number. So y is increasing here. Thus, on ( (-infty, 2) ), the function is increasing. Therefore, the function is decreasing only on ( (2, +infty) ), so option (C) is correct. But let me check option (D): ( (0,2) cup (2, +infty) ). Wait, why is (0,2) included here? The function isn't defined at x=2, but the interval (0,2) is part of ( (-infty, 2) ). We saw that on ( (-infty, 2) ), the function is increasing, so (0,2) would also be increasing. So option (D) is incorrect. Similarly, option (A) is ( (-infty, 2) ), but we saw the function is increasing there. Option (B) is a union of two intervals, but we need only ( (2, +infty) ). So yeah, (C) is correct. Therefore, the answer should be option (C). Final Answerboxed{C}</think>To determine the monotonic decreasing interval of the function ( y = log_{frac{1}{5}} |x - 2| ), we need to analyze the behavior of the composite function. 1. Base of the logarithm: The function ( log_{frac{1}{5}} ) is a decreasing function because its base ( frac{1}{5} ) is between 0 and 1.2. Inner function: The argument of the logarithm is ( |x - 2| ), which is a V-shaped function with a vertex at ( x = 2 ). - For ( x > 2 ), ( |x - 2| ) is increasing. - For ( x < 2 ), ( |x - 2| ) is decreasing.3. Composite function behavior: - Since ( log_{frac{1}{5}} ) is decreasing, the composite function ( log_{frac{1}{5}} |x - 2| ) will decrease where ( |x - 2| ) is increasing and increase where ( |x - 2| ) is decreasing.4. Domain consideration: The function ( log_{frac{1}{5}} |x - 2| ) is defined for all ( x neq 2 ), i.e., on ( (-infty, 2) cup (2, +infty) ).By analyzing the intervals:- For ( x in (2, +infty) ), ( |x - 2| ) is increasing, so ( log_{frac{1}{5}} |x - 2| ) is decreasing.- For ( x in (-infty, 2) ), ( |x - 2| ) is decreasing, so ( log_{frac{1}{5}} |x - 2| ) is increasing.Thus, the function ( y = log_{frac{1}{5}} |x - 2| ) is monotonically decreasing on the interval ( (2, +infty) ).[boxed{C}],<think>Okay, let's see. The problem is asking for the monotonic decreasing interval of the function y equals log base 1/5 of the absolute value of (x minus 2). The options are given as A to D. Hmm, I need to figure out where this function is decreasing. Let me start by recalling some properties of logarithmic functions and absolute values.First, the general shape of a logarithmic function. Since the base here is 1/5, which is between 0 and 1, the logarithm function is decreasing. Normally, log base a, where a < 1, is a decreasing function. But here, the argument of the logarithm is an absolute value, |x - 2|. So that complicates things a bit. The absolute value will affect the domain and the behavior of the function on different intervals.First, let me determine the domain of the function. The argument of the logarithm must be positive, so |x - 2| > 0. That means x - 2 ≠ 0, so x ≠ 2. Therefore, the domain is all real numbers except x = 2. So the function is defined on (-∞, 2) union (2, ∞).Now, since the logarithm has a base less than 1, the function log_{1/5}(u) is decreasing in u. That is, as u increases, the log decreases. So if we can figure out where |x - 2| is increasing or decreasing, we can determine where the original function is decreasing or increasing.Let me break it down into the two intervals where the absolute value expression |x - 2| is defined without the absolute value. Specifically:1. For x - 2 ≥ 0, which is x ≥ 2, |x - 2| = x - 2. So the function becomes y = log_{1/5}(x - 2).2. For x - 2 < 0, which is x < 2, |x - 2| = -(x - 2) = -x + 2. So the function becomes y = log_{1/5}(-x + 2).So now I can consider these two cases separately.First, for x > 2: The function is y = log_{1/5}(x - 2). Since the base is between 0 and 1, the logarithm is a decreasing function. So as x - 2 increases, y decreases. But x - 2 is increasing when x is increasing. Therefore, in this interval (2, ∞), as x increases, x - 2 increases, so log_{1/5}(x - 2) decreases. So on (2, ∞), the function is decreasing.Now, for x < 2: The function is y = log_{1/5}(-x + 2). Let's analyze this. Let me set u = -x + 2. Then, as x increases towards 2 from the left (i.e., from -∞ to 2), u decreases from +∞ to 0. Since the logarithm base 1/5 is decreasing, as u decreases, log_{1/5}(u) increases. Therefore, in this interval (-∞, 2), as x increases (moving towards 2), u decreases, so y increases. Therefore, the function is increasing on (-∞, 2).Therefore, combining both intervals: the function is increasing on (-∞, 2) and decreasing on (2, ∞). The question is asking for the monotonic decreasing interval, so that would be (2, ∞), which is option C.Wait, but let me double-check. Let me think again. The key is that the log base 1/5 is a decreasing function. So when the argument increases, the log decreases. So for the interval where the argument |x - 2| is increasing, the function y will be decreasing. Conversely, where |x - 2| is decreasing, the function y will be increasing.So I need to analyze the behavior of |x - 2|. Let's graph |x - 2|. It's a V-shaped graph with the vertex at (2, 0). To the left of x=2, the function is -x + 2, which is decreasing. To the right of x=2, it's x - 2, which is increasing. Therefore, |x - 2| is decreasing on (-∞, 2) and increasing on (2, ∞). Therefore, since log_{1/5} of something is a decreasing function, then:- When |x - 2| is increasing (i.e., x > 2), log_{1/5}(|x - 2|) is decreasing.- When |x - 2| is decreasing (i.e., x < 2), log_{1/5}(|x - 2|) is increasing.Therefore, the function is decreasing on (2, ∞) and increasing on (-∞, 2). So the monotonic decreasing interval is (2, ∞), which is option C. Therefore, the answer should be C.But let me check the options again. The options are:(A) (-∞, 2)(B) (-∞,-2) ∪ (2, +∞)(C) (2, +∞)(D) (0,2) ∪ (2, +∞)So according to my reasoning, the correct answer is C. Option C is (2, +∞). But let me make sure there isn't a trick here. For example, perhaps the function is decreasing in both intervals? Wait, that can't be. Since the log is decreasing with respect to its argument, but the argument is |x - 2|. So, as x increases towards 2 from the left, |x - 2| decreases, so log_{1/5}(|x - 2|) increases. So on (-∞, 2), y is increasing. As x increases beyond 2, |x - 2| increases, so log_{1/5}(|x - 2|) decreases. So y is decreasing there. Therefore, yes, the function is decreasing on (2, ∞), which is option C. So the answer is C.Wait, but let me check the options once more. Option D is (0,2) union (2, +∞). Maybe there's a confusion with the domain? The domain is all real numbers except 2, but the function is only decreasing on (2, ∞). So D would include (0,2) where the function is increasing. So D is not correct. Similarly, option B is (-∞, -2) union (2, +∞). But why -2? The critical point is at x=2, not x=-2. So B is incorrect. Option A is (-∞, 2), but the function is increasing there. So the answer is indeed C.Therefore, after careful analysis, I conclude the correct answer is C.Final Answerboxed{C}</think>To determine the monotonic decreasing interval of the function ( y = log_{frac{1}{5}} |x - 2| ), we start by analyzing the properties of the logarithmic function and the absolute value expression.1. Domain of the function: The argument of the logarithm must be positive, so ( |x - 2| > 0 ). This implies ( x neq 2 ). Therefore, the domain is ( (-infty, 2) cup (2, +infty) ).2. Behavior of the logarithmic function: Since the base ( frac{1}{5} ) is between 0 and 1, the function ( log_{frac{1}{5}} u ) is decreasing in ( u ). This means as ( u ) increases, ( log_{frac{1}{5}} u ) decreases.3. Analyzing the absolute value expression: - For ( x > 2 ), ( |x - 2| = x - 2 ). Here, ( x - 2 ) is increasing as ( x ) increases. Therefore, ( log_{frac{1}{5}} (x - 2) ) is decreasing. - For ( x < 2 ), ( |x - 2| = -(x - 2) = -x + 2 ). Here, ( -x + 2 ) is decreasing as ( x ) increases. Therefore, ( log_{frac{1}{5}} (-x + 2) ) is increasing.4. Conclusion: The function ( y = log_{frac{1}{5}} |x - 2| ) is decreasing on the interval where ( |x - 2| ) is increasing, which is ( (2, +infty) ). On the interval ( (-infty, 2) ), the function is increasing.Thus, the monotonic decreasing interval of the function is ( (2, +infty) ), which corresponds to option C.[boxed{C}]]

🔑:begin{array}{l}text { - } 、 1 .(mathrm{C}) text {. } text { Let } f(x)=|x-2| text {, its }end{array}graph is shown in the right figure. When x>2, y=log _{frac{1}{3}}|x-2| is a decreasing function.