🧠:[<think>Okay, so I have this geometry problem here: "Two lines that are perpendicular to the same line can ( ) A. be parallel B. intersect C. be skew D. all of the above are possible." Hmm, let's see. I need to figure out which of these options is possible. Let me start by recalling some basic geometry concepts.First, perpendicular lines are lines that intersect at a right angle, which is 90 degrees. Now, the question is about two lines that are both perpendicular to the same line. So, imagine there's a line, let's say line L, and then two other lines, say line M and line N, both of which are perpendicular to line L. The question is asking what the possible relationship between M and N can be. The options are that they can be parallel, intersect, be skew, or all of the above.Let me start by visualizing this. If all three lines are in the same plane, like on a flat surface, then if two lines are both perpendicular to the same line, they must be parallel. Because in a plane, the two lines would have the same slope as each other if they're both perpendicular to line L, which would mean they never meet—so they'd be parallel. For example, if line L is horizontal, then lines M and N would both be vertical, and if they are both vertical, they can't intersect unless they're the same line, but since they are two distinct lines, they must be parallel. So in a plane, the answer would be A. But wait, the problem doesn't specify that the lines are in the same plane. Hmm, that's important.In three-dimensional space, lines can be skew, which means they don't intersect and aren't parallel. So, maybe in 3D, two lines that are both perpendicular to the same line could be skew? Let me think. If line L is in some direction, and lines M and N are both perpendicular to L, but they are not in the same plane as each other, then they could be skew. For instance, imagine line L is along the z-axis. Then, in the x-y plane, any horizontal line (along x or y) is perpendicular to the z-axis. But if you take two lines in different planes, like one along the x-axis in the x-z plane and another along the y-axis in the y-z plane, both would be perpendicular to the z-axis, but they don't intersect and aren't parallel. Wait, but actually, if they are in different planes, would they be skew?Wait, the x-axis and y-axis in 3D are perpendicular to the z-axis, right? The x-axis is in the x-z plane, and the y-axis is in the y-z plane. Wait, no, actually, the x-axis and y-axis are in their own planes. But the x-axis and y-axis intersect at the origin. Wait, but in three-dimensional space, if you have two lines that are both perpendicular to the z-axis, they must either intersect or be parallel. Because if they are both perpendicular to the z-axis, their direction vectors have a z-component of zero. So, for example, if you have two lines in 3D space that are both horizontal (perpendicular to the vertical z-axis), they can either be parallel (like two different lines along the x-axis) or they can intersect (like the x-axis and y-axis at the origin). But can they be skew?Hmm, if two lines are both perpendicular to the z-axis, their direction vectors are in the x-y plane. If they are skew, they would have to not intersect and not be parallel. But in 3D, if two lines have direction vectors in the same plane (the x-y plane), can they still be skew? Let's consider two lines: Line M is along the x-axis at z=0, and Line N is along the line y=1, z=1, with direction vector along the x-axis. Wait, that line N would be in the plane z=1, y=1, direction along x-axis. So, Line M is (x, 0, 0) and Line N is (x, 1, 1). These lines are parallel because they both have direction vector (1, 0, 0). So they are parallel. If I make Line N have a different direction, say along the y-axis. Wait, but if Line N is along the y-axis, then at z=0, it's the same as the original y-axis, which intersects Line M at the origin. But if Line N is shifted, say along the line x=1, z=1, direction along the y-axis. Then Line N is (1, y, 1). Line M is (x, 0, 0). These two lines don't intersect because Line M is at z=0 and Line N is at z=1. Their direction vectors are along the x-axis and y-axis, which are perpendicular, so they aren't parallel. Therefore, they must be skew. Wait, but are they both perpendicular to the same line?Wait, Line M is along the x-axis, so it's perpendicular to the z-axis. Line N is (1, y, 1), direction vector (0,1,0). Is Line N perpendicular to the z-axis? The z-axis has direction vector (0,0,1). The direction vector of Line N is (0,1,0), which is perpendicular to (0,0,1) because their dot product is 0*0 + 1*0 + 0*1 = 0. So yes, Line N is perpendicular to the z-axis. Line M is also perpendicular to the z-axis. So here we have two lines, both perpendicular to the z-axis, which are skew. So in 3D space, they can be skew. Therefore, option C is possible.Wait, but earlier I thought that if two lines are both perpendicular to the same line in 3D, they could be skew. So that adds another possibility. Then, in 3D, two lines perpendicular to the same line can be parallel, intersecting, or skew. Therefore, all of the above are possible. But hold on, in my previous example, when I had two lines both perpendicular to the z-axis, one along the x-axis and another shifted line along the y-axis but at a different z-coordinate, they were skew. Then, if they are in the same plane, like both in the x-y plane, then they would either be parallel or intersecting. So depending on their positions, they can be parallel, intersecting, or skew. Therefore, all options are possible, which would be answer D.But let me verify again. Let's take specific examples.Case 1: Both lines are in the same plane and perpendicular to the third line. For example, in the x-y plane, line L is the x-axis. Then lines M and N, both perpendicular to L, would be vertical lines (parallel to the y-axis). If M and N are different vertical lines, they are parallel. If they are the same line, they coincide, but since the problem says "two lines," assuming they are distinct, they are parallel. So that's option A.Case 2: In 3D space, take line L as the z-axis. Then, line M could be the x-axis (in the x-y plane) and line N could be the y-axis (also in the x-y plane). These two lines intersect at the origin and are both perpendicular to the z-axis. So that's option B.Case 3: Again in 3D space, line L is the z-axis. Line M is the x-axis at z=0, and line N is a line parallel to the y-axis but shifted along the z-axis, say at (0, y, 1) for all y. Wait, no, that line N would have direction vector (0,1,0), and since it's at (0, y, 1), it's along the y-axis but at z=1. Then, line M is (x, 0, 0) with direction (1,0,0). These two lines are both perpendicular to the z-axis. Are they skew? Let's check. They don't intersect because line M is at z=0 and line N is at z=1. Their direction vectors are (1,0,0) and (0,1,0), which are not parallel. Therefore, they are skew. So that's option C.Therefore, all three options A, B, and C are possible. Hence, the correct answer should be D: all of the above are possible.But wait, the problem didn't specify whether we are working in 2D or 3D space. Usually, when skew lines are mentioned, it's in 3D because skew lines don't exist in 2D. Since the option C is about skew lines, the problem must be considering 3D space. So, in 3D, two lines perpendicular to the same line can be parallel, intersecting, or skew. Therefore, D is correct.But let me double-check. Another example: If two lines are both perpendicular to the same line in 3D, they can be in different planes. For instance, take line L as the z-axis. Then, line M can be in the x-z plane, going along the x-axis, and line N can be in the y-z plane, going along the y-axis. These two lines both are perpendicular to the z-axis, and they intersect at the origin. But if I shift line N along the z-axis, say line N is the y-axis shifted up by 1 unit in the z-direction, then line N is at (0, y, 1). Then, line M is the x-axis at z=0. These two lines are skew. If line M and N are both along the x-axis but at different z positions, they are parallel. So yes, all possibilities exist.Therefore, the answer is D.Final Answerboxed{D}</think>To determine the possible relationships between two lines that are both perpendicular to the same line, we need to consider both two-dimensional and three-dimensional space.1. In a plane (2D space): - If two lines are perpendicular to the same line, they must be parallel. For example, if line ( L ) is horizontal, any two vertical lines perpendicular to ( L ) will be parallel.2. In three-dimensional (3D) space: - Parallel Lines: If two lines are perpendicular to the same line and lie in the same plane, they are parallel. For example, two vertical lines in the x-y plane perpendicular to the z-axis. - Intersecting Lines: If two lines are perpendicular to the same line and intersect at a point, they can intersect. For example, the x-axis and y-axis in the x-y plane both perpendicular to the z-axis intersect at the origin. - Skew Lines: If two lines are perpendicular to the same line but lie in different planes, they can be skew. For example, a line along the x-axis at ( z = 0 ) and a line along the y-axis at ( z = 1 ) are both perpendicular to the z-axis but do not intersect and are not parallel.Since all three possibilities (parallel, intersecting, skew) can occur in three-dimensional space, the correct answer is:[boxed{D}],<think>Okay, so I have this geometry problem here: "Two lines that are perpendicular to the same line can ( )" with options A. be parallel, B. intersect, C. be skew, D. all of the above are possible. Hmm, let me think through this step by step.First, I need to recall what it means for two lines to be perpendicular to the same line. If two lines are both perpendicular to a third line, what possible relationships can they have with each other? The options are parallel, intersecting, or skew. And then there's the option that all of these are possible. Let me visualize this.Let me start in a two-dimensional plane because that's simpler. If I have a horizontal line, say the x-axis, then lines perpendicular to it would be vertical. So, if I draw two different vertical lines, both would be perpendicular to the x-axis. In this case, those two vertical lines are parallel to each other. So in 2D, two lines perpendicular to the same line must be parallel. So in a plane, option A is possible, but B and C wouldn't be, right? Because in 2D, lines either intersect or are parallel. Skew lines don't exist in 2D since they have to be non-parallel and non-intersecting but in different planes.But wait, the problem doesn't specify that the lines are in a plane. So maybe they are in three-dimensional space. In 3D, lines can be skew. So let me think in 3D.Suppose we have a line L. Two lines that are both perpendicular to L. Let me imagine line L as the z-axis. So lines perpendicular to the z-axis would lie in planes that are horizontal (assuming z-axis is vertical). Wait, but in 3D, a line perpendicular to the z-axis can be in any direction in the xy-plane. But actually, to be perpendicular to the z-axis, the line must be in a horizontal direction. Wait, no. The direction vector of the z-axis is (0,0,1). So a line that's perpendicular to the z-axis must have a direction vector that is orthogonal to (0,0,1). So their direction vectors must have a dot product of zero. So direction vectors like (a, b, 0) where a and b are real numbers.Therefore, in 3D, two lines perpendicular to the z-axis can be in different planes. For example, one could be along the x-axis and another along the y-axis. These two lines are both perpendicular to the z-axis. But in 3D, the x-axis and y-axis are perpendicular to each other and intersect at the origin. So here, they intersect. So that's case B.Alternatively, if I take two lines parallel to the x-axis but at different heights along the z-axis. Wait, but if they are parallel to the x-axis, their direction vector is (1,0,0), which is perpendicular to the z-axis. So if they are at different z positions, like one at z=0 and another at z=1, both along the x-direction. Then these two lines are parallel to each other. So that's case A.But can they be skew? Skew lines are lines that are not parallel and do not intersect. So, for two lines to be skew, they need to be non-parallel and non-intersecting. Let me see. Suppose line M is along the x-axis at z=0, direction vector (1,0,0). Line N is along the line y=1, z=1, with direction vector (0,1,0). Wait, but direction vector (0,1,0) is also perpendicular to the z-axis. So line N is along the y-axis but shifted up in z. So line M is along the x-axis at z=0, and line N is along the y-axis at z=1. These two lines are both perpendicular to the z-axis, but are they skew?Yes, because they are not parallel (their direction vectors are (1,0,0) and (0,1,0), which are not scalar multiples of each other) and they don't intersect. Since line M is at z=0 and line N is at z=1, and their x and y coordinates don't overlap in a way that they would meet. So yes, skew lines can exist in this case. Therefore, option C is also possible.So in three-dimensional space, two lines perpendicular to the same line can be parallel, intersect, or skew. Therefore, the correct answer would be D. All of the above are possible.But wait, let me confirm. If two lines are both perpendicular to the same line, can they intersect? In 3D, if two lines are both perpendicular to line L, can they intersect? Yes. For example, take the z-axis again. The x-axis and y-axis are both perpendicular to the z-axis and intersect at the origin. So that's case B.Alternatively, if they are parallel but not intersecting, like two different lines along the x-axis but shifted along the y or z, but wait, if they are along the x-axis, shifting along y or z would make them skew or parallel? If they are both along the x-axis direction but at different z levels, then they are parallel. If they are along different directions, like one along x and another along y, but shifted, they can be skew.Wait, but if they are both in the same plane, say the xy-plane, and both perpendicular to the z-axis, then they would have to be parallel or intersecting. But in 3D, if they are not in the same plane, they can be skew. So depending on their positions, they can be parallel, intersecting, or skew.Therefore, all options A, B, C are possible. So D is the correct answer.Final Answerboxed{D}</think>To determine the possible relationships between two lines that are perpendicular to the same line, we need to consider both two-dimensional and three-dimensional spaces.1. Two-dimensional space (plane): - If two lines are perpendicular to the same line in a plane, they must be parallel. For example, two vertical lines perpendicular to a horizontal line (x-axis) are parallel.2. Three-dimensional space: - Parallel: Two lines can be parallel if they are both perpendicular to the same line and lie in parallel planes. For example, two lines along the x-axis at different z levels. - Intersecting: Two lines can intersect if they are both perpendicular to the same line and lie in the same plane. For example, the x-axis and y-axis, both perpendicular to the z-axis, intersect at the origin. - Skew: Two lines can be skew if they are not parallel and do not intersect, lying in different planes. For example, one line along the x-axis at z=0 and another line along the y-axis at z=1.Thus, in three-dimensional space, two lines perpendicular to the same line can be parallel, intersecting, or skew. Therefore, all options A, B, and C are possible.[boxed{D}]]

🔑:We are given two lines that are both perpendicular to a third line. We can consider the following cases:1. Parallel: If the two lines are perpendicular to a third line at different points and they are in the same plane, then by the definition of parallel lines, the two lines will never meet and thus are parallel.2. Intersecting: If the two lines are perpendicular to a third line at the same point, then they are in the same plane and they will intersect exactly at the point where they are perpendicular to the third line.3. Skew: If the two lines are perpendicular to a third line but they are in different planes, then they are neither parallel nor do they intersect each other. Such lines are called skew lines.Thus, the correct answer is: boxed{D} (all of the above are possible)

🧠:[<think>Okay, so I have this problem here: An odd function f(x) defined on all real numbers, and when you shift it left by 1 unit, making f(x + 1), that becomes an even function. They also tell me that f(-1) = 2, and I need to find f(2017). Hmm, interesting. Let me try to unpack this step by step.First, let's recall what it means for a function to be odd and even. An odd function satisfies f(-x) = -f(x) for all x in its domain. So, if I plug in -x, I get the negative of the function at x. An even function, on the other hand, satisfies f(-x) = f(x) for all x. Now, the problem says that f(x + 1) is even. That means if I take the function f and shift it to the left by 1 unit, the resulting function is even. So, the shifted function f(x + 1) should satisfy f(-x + 1) = f(x + 1) for all x. Wait, let me make sure that's right. If the function f(x + 1) is even, then replacing x with -x in the argument should give the same value. So, f(-x + 1) = f(x + 1). Yeah, that seems correct.So, given that f is odd, which gives us another equation: f(-x) = -f(x). And the shifted function is even, so f(-x + 1) = f(x + 1). Let me write these two properties down:1. f(-x) = -f(x) for all x (since f is odd).2. f(-x + 1) = f(x + 1) for all x (since f(x + 1) is even).Also, we know that f(-1) = 2. Since f is odd, this implies f(1) = -f(-1) = -2. So, f(1) is -2. That's a concrete value. The question is about f(2017). Maybe we can find a pattern or a periodicity in the function?Let me see if I can manipulate the two equations above to find a relationship. Let's start with the second equation: f(-x + 1) = f(x + 1). Let me substitute a variable here to make it clearer. Let's set y = x + 1. Then, when x = y - 1, the equation becomes f(-(y - 1) + 1) = f((y - 1) + 1). Simplify both sides:Left side: f(-y + 1 + 1) = f(-y + 2)Right side: f(y - 1 + 1) = f(y)So, f(-y + 2) = f(y). So, replacing y with x, we can write f(-x + 2) = f(x) for all x. Hmm, that's an interesting relation. So, f(2 - x) = f(x). Let's note that down.Now, since f is odd, we also have f(-x) = -f(x). Let's see if we can combine these two properties. Let's substitute x with 2 - x in the oddness condition. Wait, maybe not. Let's think.We have two equations:1. f(-x) = -f(x) (oddness)2. f(2 - x) = f(x) (from the even shift)Let me see if I can plug one into the other. For example, from equation 2, f(2 - x) = f(x). But from equation 1, f(2 - x) can be rewritten if we let z = 2 - x. Wait, perhaps not. Let me try substituting x with 2 - x in equation 1. Hmm, equation 1 says f(-x) = -f(x). Let me replace x with 2 - x in equation 1. Then:f(-(2 - x)) = -f(2 - x)Simplify left side: f(-2 + x) = -f(2 - x)But from equation 2, f(2 - x) = f(x). So, substituting that in:f(-2 + x) = -f(x)So, f(x - 2) = -f(x). Hmm, so f(x - 2) = -f(x). Let's write that down:3. f(x - 2) = -f(x) for all x.Is that correct? Let me check with x = 1. If x = 1, then f(1 - 2) = f(-1) = 2, and -f(1) = -(-2) = 2. So, that works. Let me check another value. Suppose x = 2. Then f(2 - 2) = f(0) = 0 (since f is odd, f(0) must be 0), and -f(2) should also equal 0. Wait, if x = 2, then f(2 - 2) = f(0) = 0, and -f(2) = -f(2). So, 0 = -f(2), which implies f(2) = 0. Hmm, that's an interesting result. Let me verify this.If x = 2 in equation 3: f(2 - 2) = f(0) = 0 = -f(2). Therefore, f(2) = 0. That's correct. Similarly, if I take x = 3, then f(3 - 2) = f(1) = -2 = -f(3), so -2 = -f(3) implies f(3) = 2. Then, x = 4: f(4 - 2) = f(2) = 0 = -f(4), so f(4) = 0. Wait, this seems like a pattern. Let me list some values:f(0) = 0 (from oddness)f(1) = -2f(-1) = 2f(2) = 0f(3) = 2f(4) = 0f(5) = -2? Wait, let's check x=3: f(3 - 2) = f(1) = -2 = -f(3) ⇒ f(3) = 2. Then x=4: f(4 - 2) = f(2) = 0 = -f(4) ⇒ f(4) = 0. x=5: f(5 - 2)=f(3)=2 = -f(5) ⇒ f(5) = -2. Then x=6: f(6 - 2)=f(4)=0 = -f(6) ⇒ f(6)=0. Hmm, so the pattern seems like:n | f(n)0 | 01 | -22 | 03 | 24 | 05 | -26 | 07 | 2... etc.So, every even integer is 0, and the odd integers alternate between -2 and 2. Starting from f(1) = -2, then f(3) = 2, f(5) = -2, etc. So, it's periodic with period 4? Let's see. From n=1 to n=5, which is 4 units apart, the value repeats. But wait, from n=1 to n=5, it goes from -2 to -2, but with 0s in between. Wait, maybe the non-zero values have period 4? Let's check f(1) = -2, f(5) = -2, f(9) = -2, etc. Similarly, f(3) = 2, f(7) = 2, f(11)=2, etc. So, the non-zero terms at odd integers have a period of 4. So, 1, 5, 9,... all have f(n) = -2, and 3,7,11,... have f(n) = 2.Therefore, for any integer k, f(4k + 1) = -2, f(4k + 3) = 2, and even integers are 0. So, 2017 divided by 4. Let's compute 2017 divided by 4. 4*504 = 2016, so 2017 = 4*504 + 1. Therefore, 2017 is of the form 4k + 1. Therefore, according to the pattern, f(2017) = -2. But wait, let's verify this with the equations we have to make sure there's no mistake.But let's step back. The key relation we derived is f(x - 2) = -f(x). So, this recurrence relation suggests that shifting x by 2 units to the right changes the sign of the function. If we apply this relation again, f((x - 2) - 2) = -f(x - 2) ⇒ f(x - 4) = -f(x - 2) = -(-f(x)) = f(x). Therefore, f(x - 4) = f(x). So, the function is periodic with period 4. So, f(x + 4) = f(x). Therefore, the function has period 4. So, every 4 units, the function repeats its values.Given that, to find f(2017), we can find the remainder when 2017 is divided by 4. 2017 divided by 4 is 504*4 + 1, as I mentioned before. So, 2017 ≡ 1 mod 4. Therefore, f(2017) = f(1) = -2. But f(1) is given as -2 from the initial condition. Therefore, the answer is -2. However, wait, in the problem statement, they said f(-1) = 2. Since f is odd, f(1) = -f(-1) = -2. So, that's consistent.But let me verify this with the function's properties to be thorough. Let's check if f(x + 4) = f(x). Given that f(x - 4) = f(x). So, replacing x with x + 4, we get f((x + 4) - 4) = f(x + 4) ⇒ f(x) = f(x + 4). Therefore, period 4. So, yes, the function is periodic with period 4. Therefore, f(2017) = f(1 + 4*504) = f(1) = -2.Therefore, the answer should be boxed{-2}. Let me just check once again.Wait, another way to see: using the relation f(x - 2) = -f(x). So, f(x) = -f(x - 2). Then, f(x + 2) = -f(x). So, shifting by 2 gives the negative. Then, shifting by 4 would be f(x + 4) = -f(x + 2) = -(-f(x)) = f(x). So, period 4.Therefore, the function's period is 4. So, as 2017 = 4*504 + 1, f(2017) = f(1) = -2. Therefore, boxed{-2} is the answer.But wait, let me check if there is another possible way to get this result. Suppose we use the original two properties. Since f(x + 1) is even, f(x + 1) = f(-x + 1). So, replacing x with x - 1, we get f(x) = f(-x + 2). So, f(x) = f(2 - x). That's the same as equation 2. And since f is odd, f(2 - x) = f(x) = -f(-x). So, f(2 - x) = -f(-x). So, replacing x with -x, we get f(2 + x) = -f(x). Therefore, f(x + 2) = -f(x). So, this gives the same recurrence: shifting by 2 gives negative. Therefore, f(x + 4) = -f(x + 2) = --f(x) = f(x). So, period 4. Therefore, the same conclusion.Another way: Let's take specific values. We know f(-1) = 2. Then, f(1) = -2. From the relation f(x) = f(2 - x), so f(1) = f(2 - 1) = f(1). Not helpful. Wait, f(2) = f(2 - 0) = f(0) = 0. So, f(2) = 0. Then, f(3) = f(2 - (-1)) = f(-1 + 2) = f(1) = -2. Wait, hold on. Wait, f(3) = f(2 - (-1))? Wait, according to f(x) = f(2 - x). So, f(3) = f(2 - 3) = f(-1) = 2. Wait, that contradicts earlier. Wait, maybe I messed up.Wait, using f(x) = f(2 - x). So, for x = 3, f(3) = f(2 - 3) = f(-1) = 2. But from the recurrence relation, f(3) = -f(1) = -(-2) = 2. So, same result. So, f(3) = 2. Then, f(4) = f(2 - 2) = f(0) = 0. Then, f(5) = f(2 - 5) = f(-3) = -f(3) = -2. Then, f(5) = -2. Similarly, f(6) = f(2 - 6) = f(-4) = -f(4) = 0. So, same pattern as before.Therefore, every odd integer is alternating between -2 and 2, starting with f(1) = -2, f(3) = 2, f(5) = -2, etc. So, the odd integers: 1, 5, 9,... which are 1 mod 4, have f(n) = -2, and 3,7,11,... which are 3 mod 4, have f(n) = 2. Therefore, 2017 divided by 4 gives remainder 1, so f(2017) = -2. Therefore, the answer is boxed{-2}.I think that's solid. All the relations point to the function being periodic with period 4, alternating between 0 and the non-zero values at odd integers, which themselves alternate between -2 and 2. Since 2017 is 1 mod 4, it's in the same position as 1, 5, 9, etc., so f(2017) = f(1) = -2. Yep, that seems right. I don't see any holes in this reasoning.Final Answerboxed{-2}</think>Given that the function ( f(x) ) is an odd function defined on (mathbb{R}), and ( f(x+1) ) is an even function. We know ( f(-1) = 2 ) and need to find ( f(2017) ).1. Odd Function Property: ( f(-x) = -f(x) ) for all ( x ).2. Even Function Property for ( f(x+1) ): ( f(-x + 1) = f(x + 1) ) for all ( x ).By substituting ( y = x + 1 ) into the even function property, we get:[ f(-y + 2) = f(y) ]Replacing ( y ) with ( x ), this becomes:[ f(2 - x) = f(x) ]Combining the odd function property with the above result:[ f(2 - x) = f(x) = -f(-x) ]Substituting ( x ) with ( -x ), we get:[ f(2 + x) = -f(x) ]This implies:[ f(x + 2) = -f(x) ]Applying this recurrence relation twice, we find:[ f(x + 4) = -f(x + 2) = -(-f(x)) = f(x) ]Thus, ( f(x) ) is periodic with period 4.Given ( f(-1) = 2 ) and using the odd function property:[ f(1) = -f(-1) = -2 ]To find ( f(2017) ), we note that 2017 divided by 4 leaves a remainder of 1:[ 2017 = 4 times 504 + 1 ]Since ( f ) is periodic with period 4:[ f(2017) = f(1) = -2 ]Thus, the final answer is:[boxed{-2}],<think>Okay, so I have this problem here: Given that the function ( f(x) ) is an odd function defined on ( mathbb{R} ), and ( f(x+1) ) is an even function. If ( f(-1) = 2 ), then ( f(2017) ) equals _______. Hmm, let me think. Let's break it down step by step.First, let's recall what it means for a function to be odd and even. An odd function satisfies ( f(-x) = -f(x) ) for all ( x ), right? And an even function satisfies ( f(-x) = f(x) ). But here, it's not ( f(x) ) that's even, but ( f(x+1) ). So, the function shifted left by 1 unit is even. Let me write that down.Since ( f(x+1) ) is even, that means ( f(-x + 1) = f(x + 1) ). Wait, is that right? Let me double-check. If we have a function ( g(x) = f(x+1) ), and ( g(x) ) is even, then ( g(-x) = g(x) ). So, substituting back, ( g(-x) = f(-x + 1) ) and ( g(x) = f(x + 1) ). Therefore, ( f(-x + 1) = f(x + 1) ). Yes, that's correct. So, for all ( x ), ( f(1 - x) = f(1 + x) ). Got it.So, this is a symmetry condition. The function ( f ) is symmetric around ( x = 1 ). Because if you replace ( x ) with ( x + 1 ), the function value at ( 1 - (x + 1) = -x ) is equal to the function value at ( 1 + (x + 1) = x + 2 ). Wait, maybe not. Let me think again. The condition ( f(1 - x) = f(1 + x) ) for all ( x ) does imply that the graph of ( f ) is symmetric about the line ( x = 1 ). So, if you reflect any point across the vertical line ( x = 1 ), you get another point with the same function value. For example, ( x = 1 + a ) and ( x = 1 - a ) have the same value. So, the function is symmetric about ( x = 1 ).But also, ( f(x) ) is odd. So, ( f(-x) = -f(x) ). Let's see if we can combine these two properties.Given that ( f ) is odd and has symmetry about ( x = 1 ), maybe we can derive some functional equations.Let me write down both properties:1. ( f(-x) = -f(x) ) for all ( x ).2. ( f(1 - x) = f(1 + x) ) for all ( x ).Our goal is to find ( f(2017) ), given that ( f(-1) = 2 ).First, let's see what ( f(-1) = 2 ) tells us. Since ( f ) is odd, ( f(-1) = -f(1) ). Therefore, ( 2 = -f(1) ), so ( f(1) = -2 ).Okay, so ( f(1) = -2 ). Now, let's try to use the symmetry about ( x = 1 ). Let's take ( x = 0 ) in the symmetry condition. Then, ( f(1 - 0) = f(1 + 0) ), which gives ( f(1) = f(1) ). Hmm, that's trivial.How about taking ( x = 1 ). Then, ( f(1 - 1) = f(1 + 1) ), so ( f(0) = f(2) ). Since ( f ) is odd, ( f(0) = 0 ), so ( f(2) = 0 ). So, ( f(2) = 0 ).Wait, ( f(0) = 0 ) because for odd functions, ( f(-0) = -f(0) ), which implies ( f(0) = -f(0) ), hence ( f(0) = 0 ). So, that's a good point.Now, let's see if we can find a recurrence relation or some periodicity here. Let's consider the two properties.From the symmetry about ( x = 1 ), ( f(1 + x) = f(1 - x) ). Let's let ( t = 1 + x ), so ( x = t - 1 ). Then, substituting back, ( f(t) = f(1 - (t - 1)) = f(2 - t) ). So, we have ( f(t) = f(2 - t) ) for all ( t ).But also, since ( f ) is odd, ( f(-t) = -f(t) ).So, combining these two, we have ( f(2 - t) = f(t) = -f(-t) ). Therefore, ( f(2 - t) = -f(-t) ). Let's write that as:( f(2 - t) = -f(-t) ). Hmm. Let me see if I can find a recurrence relation here. Let's set ( u = -t ), so ( t = -u ). Then, substituting into the equation:( f(2 - (-u)) = -f(-(-u)) implies f(2 + u) = -f(u) ).Therefore, ( f(u + 2) = -f(u) ).So, this gives a functional equation: ( f(u + 2) = -f(u) ).That's interesting. So, the function ( f ) has a periodicity of sorts here. If we shift by 2, we get the negative of the function. Then, shifting by 4 would give:( f(u + 4) = -f(u + 2) = -(-f(u)) = f(u) ).Therefore, the function ( f ) is periodic with period 4. So, every 4 units, the function repeats.So, ( f(u + 4) = f(u) ), so period 4. Therefore, ( f(2017) = f(2017 mod 4) ). Let's calculate 2017 divided by 4.4 * 504 = 2016, so 2017 = 4*504 + 1. Therefore, 2017 mod 4 is 1. Therefore, ( f(2017) = f(1) ).But we already found that ( f(1) = -2 ). Therefore, ( f(2017) = -2 ).Wait, but let me check if this is correct. Let's verify the steps again.We started with the given that ( f(x+1) ) is even, so ( f(1 - x) = f(1 + x) ). Then, using substitution, we found that ( f(t) = f(2 - t) ). Then, using the oddness of ( f ), we found that ( f(2 - t) = -f(-t) ), leading to ( f(t + 2) = -f(t) ). Then, applying this twice gives ( f(t + 4) = f(t) ). Therefore, period 4.So, since ( f(2017) = f(1) ), and we found ( f(1) = -2 ), the answer should be -2.But let's test this with smaller numbers to see if the periodicity holds. For example, let's compute ( f(3) ). Since 3 = 1 + 2, then according to the relation ( f(u + 2) = -f(u) ), so ( f(3) = -f(1) = -(-2) = 2 ).Now, check using the symmetry. Since ( f(1 + x) = f(1 - x) ), let x = 2: ( f(1 + 2) = f(1 - 2) implies f(3) = f(-1) ). But ( f(-1) = 2 ), and we found ( f(3) = 2 ), which matches. Also, since ( f ) is odd, ( f(-3) = -f(3) = -2 ). Let's check using the periodicity: ( f(-3) = f(-3 + 4) = f(1) = -2 ), which matches. So, that's consistent.Similarly, ( f(4) = f(0) = 0 ), as period 4. Let's check using ( f(2 + 2) = -f(2) = -0 = 0 ), which matches.Another test: ( f(5) = f(1) = -2 ), and via periodicity, ( f(5) = f(1 + 4) = f(1) = -2 ). Also, using the relation ( f(5) = f(1 + 4) = -f(3) = -2 ), which matches.Okay, seems consistent. Therefore, the conclusion is that the function has period 4, so ( f(2017) = f(1) = -2 ).Wait, but let's make sure there are no contradictions here. Let me check if the function being odd and having period 4 is possible. Suppose we define f(0) = 0, f(1) = -2, f(2) = 0, f(3) = 2, f(4) = 0, etc., following the periodicity. Let's check if this is odd.Check f(-1) = -f(1) = -(-2) = 2, which matches the given condition. f(-2) = -f(2) = -0 = 0, which should be equal to f(2) since f(2) = 0? Wait, no. For oddness, f(-2) = -f(2). But f(2) = 0, so f(-2) = 0. That's okay. Then, using the symmetry about x =1: f(1 - x) = f(1 + x). Let's take x = 3: f(1 - 3) = f(-2) = 0; f(1 + 3) = f(4) = 0. So, that's okay. Take x = 1: f(0) = f(2) = 0. Take x = 2: f(-1) = f(3). f(-1) = 2, f(3) = 2, which works. Take x = 0.5: f(0.5) = f(1.5). But according to the function's periodicity, f(0.5) is... Wait, but we only defined integer points here. Hmm, but the function is defined on all real numbers. So, perhaps the function is defined such that it's periodic with period 4, and satisfies the given conditions. However, the problem is only asking for f(2017), which is an integer, so maybe we don't need to worry about non-integer points.But the problem states that f is defined on all real numbers. So, the function is determined by these conditions. Let me verify if such a function actually exists. Suppose f is periodic with period 4, odd, and satisfies f(1) = -2, f(-1) = 2, etc. Let's attempt to construct such a function.First, since f is periodic with period 4, we can define f on the interval [-2, 2], and then extend it periodically. But f is also odd, so f(-x) = -f(x). So, if we define f on [0, 2], then f on [-2, 0] is determined by oddness. Additionally, f is symmetric about x =1. So, f(1 + x) = f(1 - x). Therefore, on the interval [0, 2], f(1 + x) = f(1 - x). So, this is a reflection around x=1. So, for x in [0,1], we can define f(1 + x) = f(1 - x). Therefore, f is symmetric around x=1 on [0,2]. But since f is also odd, let's see how that interacts.Wait, maybe it's better to start building the function step by step.Given f is odd: f(-x) = -f(x). And f is periodic with period 4: f(x + 4) = f(x). Also, it's symmetric about x=1: f(1 + x) = f(1 - x). Let's see if these conditions are compatible.First, let's define the function on [0,2]. Since f is symmetric about x=1, we can define f(1 + t) = f(1 - t) for t in [0,1]. Therefore, from x=1 to x=2, the function mirrors the part from x=0 to x=1.But f is also odd: f(-x) = -f(x). So, for x in [0,1], f(x) = -f(-x). Let's connect this with the symmetry about x=1.Wait, this might get complicated. Let's take specific values.We know f(0) = 0, since f is odd.We know f(1) = -2.From the symmetry about x=1: f(1 + t) = f(1 - t). So, for t=1, f(2) = f(0) = 0. So, f(2)=0. Then, for t=2, f(3) = f(-1) = 2 (since f(-1) = 2). But since f is periodic with period 4, f(3) = f(-1). But f(-1) = 2, and since f is odd, f(-1) = -f(1) = 2. Which is consistent with f(1) = -2. So, f(3) = 2. Then, f(4) = f(0) = 0, and so on.Similarly, f(-2) = -f(2) = 0, which is also f(2) because f is periodic? Wait, f(-2) = -f(2) = 0, and f(2) = 0, so that's consistent. So, f(-2) = 0 = f(2). So, f(-2) = f(2) because of periodicity (since f(-2) = f(-2 + 4) = f(2)), but also f(-2) = -f(2). Therefore, f(2) must be 0. Which matches our previous result. So, all these conditions are consistent.Therefore, such a function exists, and its values at integers are determined as follows:- f(0) = 0- f(1) = -2- f(2) = 0- f(3) = 2- f(4) = 0- f(5) = -2- etc., repeating every 4 units.Therefore, since 2017 divided by 4 gives a remainder of 1 (since 4*504=2016, 2017=2016+1), then f(2017) = f(1) = -2.Thus, the answer should be boxed{-2}.Final Answerboxed{-2}</think>Given that the function ( f(x) ) is an odd function defined on ( mathbb{R} ), and ( f(x+1) ) is an even function. We need to find ( f(2017) ) given that ( f(-1) = 2 ).1. Odd Function Property: ( f(-x) = -f(x) ) for all ( x ).2. Even Function Property for ( f(x+1) ): ( f(1 - x) = f(1 + x) ) for all ( x ).Given ( f(-1) = 2 ), using the odd function property, we get:[ f(1) = -f(-1) = -2 ]Using the symmetry property ( f(1 - x) = f(1 + x) ), we substitute ( x = t - 1 ):[ f(2 - t) = f(t) ]Combining this with the odd function property, we get:[ f(2 - t) = -f(-t) ]Letting ( t = u ), we derive:[ f(u + 2) = -f(u) ]This implies the function ( f ) has a periodicity of 4:[ f(u + 4) = f(u) ]To find ( f(2017) ), we compute the remainder of 2017 divided by 4:[ 2017 div 4 = 504 text{ remainder } 1 ]Thus, ( 2017 equiv 1 mod 4 ), so:[ f(2017) = f(1) ]Since we found ( f(1) = -2 ), the final answer is:[ boxed{-2} ]]

🔑:According to the problem, since f(x) is an odd function defined on mathbb{R}, we have f(-x)=-f(x). Furthermore, since f(x+1) is an even function, we have f(-x+1)=f(x+1), which implies f(-x)=f(2+x). Combining these, we get: f(x+2)=-f(x). Then, f(x+4)=f[(x+2)+2]=-f(x+2)=f(x), which means the function has a period of 4. Therefore, f(2017)=f(1+4times504)=f(1)=-f(-1)=-2. So, f(2017)=-2. Hence, the answer is: boxed{-2}. Based on the problem, by analyzing the odd and even properties of the function and the fact that f(x+1) is an even function, we can deduce f(x+2)=-f(x), and further deduce f(x+4)=f(x). By analyzing the periodicity of the function, we can conclude that f(x) is a periodic function with a period of 4, and further analysis leads to f(2017)=f(1+4times504)=f(1)=-f(-1), which gives us the answer. This problem examines the application of the properties of odd and even functions and periodicity, with the key being the analysis of the function's period.

🧠:[<think>Okay, so I have this problem here. It says that the roots of the quadratic equation ( a x^2 - x - frac{1}{2} = 0 ) are ( x = frac{1 pm sqrt{3}}{2} ). I need to find the value of ( a ). Hmm, let me think. First, I remember that for a quadratic equation of the form ( ax^2 + bx + c = 0 ), the sum of the roots is ( -b/a ) and the product of the roots is ( c/a ). Maybe I can use Vieta's formulas here. Let me write down what I know.The given equation is ( a x^2 - x - frac{1}{2} = 0 ). Comparing this to the standard form ( ax^2 + bx + c = 0 ), here the coefficients are:- ( a ) (the coefficient of ( x^2 ))- ( b = -1 ) (the coefficient of ( x ))- ( c = -frac{1}{2} ) (the constant term)But wait, actually in the standard form, it's ( ax^2 + bx + c = 0 ), so in our case, the equation is ( a x^2 + (-1)x + (-frac{1}{2}) = 0 ). So yes, ( a ) is the coefficient of ( x^2 ), ( b = -1 ), and ( c = -frac{1}{2} ).The roots are given as ( x = frac{1 pm sqrt{3}}{2} ). Let me denote the roots as ( r_1 = frac{1 + sqrt{3}}{2} ) and ( r_2 = frac{1 - sqrt{3}}{2} ).According to Vieta's formulas:1. Sum of roots: ( r_1 + r_2 = -b/a )2. Product of roots: ( r_1 times r_2 = c/a )So maybe I can compute the sum and product of the given roots and set them equal to ( -b/a ) and ( c/a ), then solve for ( a ).Let me compute the sum first. ( r_1 + r_2 = frac{1 + sqrt{3}}{2} + frac{1 - sqrt{3}}{2} )Adding the numerators:( (1 + sqrt{3}) + (1 - sqrt{3}) = 1 + 1 + sqrt{3} - sqrt{3} = 2 )So the sum is ( 2/2 = 1 ). According to Vieta's formula, the sum should be ( -b/a ). Here, ( b = -1 ), so ( -b = 1 ). Therefore, sum = ( 1/a ).Wait, hold on. Vieta's formula says sum = ( -b/a ). Since ( b = -1 ), then ( -b = 1 ), so sum = ( 1/a ). But we calculated the sum as 1. Therefore, ( 1 = 1/a ). Solving for ( a ), we get ( a = 1 ).But let me check with the product as well to confirm. Maybe I made a mistake.Product of roots: ( r_1 times r_2 = left( frac{1 + sqrt{3}}{2} right) times left( frac{1 - sqrt{3}}{2} right) )This is a difference of squares: ( (1)^2 - (sqrt{3})^2 = 1 - 3 = -2 ). Then divide by 4: ( -2/4 = -1/2 ).So the product is ( -1/2 ). According to Vieta's formula, the product is ( c/a ). Here, ( c = -1/2 ), so ( c/a = (-1/2)/a ). Setting that equal to the product we found:( -1/2 = (-1/2)/a )Multiply both sides by ( a ): ( (-1/2)a = -1/2 )Then divide both sides by ( -1/2 ): ( a = 1 )So both the sum and the product give me ( a = 1 ). That seems consistent. Let me just double-check by plugging ( a = 1 ) into the original equation and check if the roots are indeed ( frac{1 pm sqrt{3}}{2} ).Original equation with ( a = 1 ):( x^2 - x - 1/2 = 0 )Let's solve this quadratic equation using the quadratic formula. For an equation ( ax^2 + bx + c = 0 ), roots are:( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} )Here, ( a = 1 ), ( b = -1 ), ( c = -1/2 ):Discriminant ( D = (-1)^2 - 4 times 1 times (-1/2) = 1 + 2 = 3 )Therefore, the roots are:( x = frac{-(-1) pm sqrt{3}}{2 times 1} = frac{1 pm sqrt{3}}{2} )Which matches the given roots. So yes, ( a = 1 ) is correct.But wait, the problem says "with respect to ( x )", but I think that's just specifying the variable. So the answer is ( a = 1 ).I think that's solid. Both Vieta's formulas and solving the equation with ( a = 1 ) give the correct roots. So the value of ( a ) is 1.Final AnswerThe value of ( a ) is boxed{1}.</think>Given the quadratic equation ( a x^2 - x - frac{1}{2} = 0 ) with roots ( x = frac{1 pm sqrt{3}}{2} ), we need to find the value of ( a ).First, we use Vieta's formulas, which state that for a quadratic equation ( ax^2 + bx + c = 0 ), the sum of the roots is ( -frac{b}{a} ) and the product of the roots is ( frac{c}{a} ).The given roots are ( r_1 = frac{1 + sqrt{3}}{2} ) and ( r_2 = frac{1 - sqrt{3}}{2} ).1. Sum of the roots: [ r_1 + r_2 = frac{1 + sqrt{3}}{2} + frac{1 - sqrt{3}}{2} = frac{(1 + sqrt{3}) + (1 - sqrt{3})}{2} = frac{2}{2} = 1 ] According to Vieta's formula, the sum of the roots is ( -frac{b}{a} ). Here, ( b = -1 ), so: [ 1 = -frac{-1}{a} implies 1 = frac{1}{a} implies a = 1 ]2. Product of the roots: [ r_1 times r_2 = left( frac{1 + sqrt{3}}{2} right) left( frac{1 - sqrt{3}}{2} right) = frac{(1)^2 - (sqrt{3})^2}{4} = frac{1 - 3}{4} = frac{-2}{4} = -frac{1}{2} ] According to Vieta's formula, the product of the roots is ( frac{c}{a} ). Here, ( c = -frac{1}{2} ), so: [ -frac{1}{2} = frac{-frac{1}{2}}{a} implies -frac{1}{2} = -frac{1}{2a} implies a = 1 ]Both the sum and product of the roots confirm that ( a = 1 ). Verifying by solving the quadratic equation with ( a = 1 ) using the quadratic formula also yields the given roots.Thus, the value of ( a ) is boxed{1}.,<think>Okay, so I need to find the value of 'a' in the quadratic equation ( a x^2 - x - frac{1}{2} = 0 ), given that the roots are ( x = frac{1 pm sqrt{3}}{2} ). Hmm, let's see. First, I remember that for a quadratic equation of the form ( ax^2 + bx + c = 0 ), the sum of the roots is ( -b/a ) and the product of the roots is ( c/a ). Maybe I can use these relationships to find 'a'. Let me write down the given roots. The roots are ( frac{1 + sqrt{3}}{2} ) and ( frac{1 - sqrt{3}}{2} ). Let me denote them as ( r_1 ) and ( r_2 ). So, ( r_1 = frac{1 + sqrt{3}}{2} ) and ( r_2 = frac{1 - sqrt{3}}{2} ). First, let's compute the sum of the roots. Sum ( S = r_1 + r_2 ). Let's add them:( S = frac{1 + sqrt{3}}{2} + frac{1 - sqrt{3}}{2} ). Since the denominators are the same, I can add the numerators:Numerator: ( (1 + sqrt{3}) + (1 - sqrt{3}) = 1 + 1 + sqrt{3} - sqrt{3} = 2 ). So, ( S = frac{2}{2} = 1 ). Okay, the sum of the roots is 1. According to the formula, the sum of the roots is also ( -b/a ). In the given equation, the quadratic is ( a x^2 - x - frac{1}{2} = 0 ). Comparing to the standard form ( ax^2 + bx + c = 0 ), here, the coefficient of x is -1, so b = -1. So, sum of roots should be ( -b/a = -(-1)/a = 1/a ). But we found that the sum of the roots is 1. Therefore:( 1/a = 1 ). So solving for 'a', we get ( a = 1 ). Wait, is that right? Let me verify with the product of the roots as well to make sure. Product of the roots ( P = r_1 times r_2 ). Let's compute that:( P = left( frac{1 + sqrt{3}}{2} right) times left( frac{1 - sqrt{3}}{2} right) ). This looks like a difference of squares. So, ( (a + b)(a - b) = a^2 - b^2 ). Applying that here:( P = left( frac{1}{2} right)^2 - left( frac{sqrt{3}}{2} right)^2 = frac{1}{4} - frac{3}{4} = -frac{2}{4} = -frac{1}{2} ). So the product of the roots is ( -1/2 ). According to the standard formula, the product of the roots is ( c/a ). In the given equation, the constant term is ( -1/2 ), so c = -1/2. Therefore, product of the roots should be ( c/a = (-1/2)/a = -1/(2a) ). We found that the product is ( -1/2 ). Therefore:( -1/(2a) = -1/2 ). Solving for 'a', multiply both sides by 2a:( -1 = -a )Multiply both sides by -1:( 1 = a )So again, we get a = 1. Both the sum and product of the roots lead to a = 1. That seems consistent. But just to make sure, let me plug a = 1 back into the original equation and check if the given roots satisfy it. Original equation with a = 1: ( x^2 - x - 1/2 = 0 ).Let's test ( x = frac{1 + sqrt{3}}{2} ):Left-hand side (LHS): ( left( frac{1 + sqrt{3}}{2} right)^2 - left( frac{1 + sqrt{3}}{2} right) - frac{1}{2} ).First compute the square term:( left( frac{1 + sqrt{3}}{2} right)^2 = frac{1 + 2sqrt{3} + 3}{4} = frac{4 + 2sqrt{3}}{4} = frac{2 + sqrt{3}}{2} ).Then subtract ( frac{1 + sqrt{3}}{2} ):( frac{2 + sqrt{3}}{2} - frac{1 + sqrt{3}}{2} = frac{2 + sqrt{3} - 1 - sqrt{3}}{2} = frac{1}{2} ).Now subtract 1/2:( frac{1}{2} - frac{1}{2} = 0 ). Perfect, that works.Now check the other root ( x = frac{1 - sqrt{3}}{2} ):LHS: ( left( frac{1 - sqrt{3}}{2} right)^2 - left( frac{1 - sqrt{3}}{2} right) - frac{1}{2} ).Compute the square:( left( frac{1 - sqrt{3}}{2} right)^2 = frac{1 - 2sqrt{3} + 3}{4} = frac{4 - 2sqrt{3}}{4} = frac{2 - sqrt{3}}{2} ).Subtract ( frac{1 - sqrt{3}}{2} ):( frac{2 - sqrt{3}}{2} - frac{1 - sqrt{3}}{2} = frac{2 - sqrt{3} - 1 + sqrt{3}}{2} = frac{1}{2} ).Subtract 1/2:( frac{1}{2} - frac{1}{2} = 0 ). That also works. So both roots satisfy the equation when a = 1. Therefore, the value of a is indeed 1. Alternatively, another approach could be using the quadratic formula. Let's see. The quadratic equation is ( a x^2 - x - 1/2 = 0 ). Using the quadratic formula, the roots are:( x = frac{1 pm sqrt{1 + 2a}}{2a} ). Wait, let's verify that. The quadratic formula is ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ). In our equation, a coefficient is 'a', b coefficient is -1, and c coefficient is -1/2. So plugging into the formula:( x = frac{-(-1) pm sqrt{(-1)^2 - 4 times a times (-1/2)}}{2 times a} )Simplify:( x = frac{1 pm sqrt{1 + 2a}}{2a} )But the problem states that the roots are ( frac{1 pm sqrt{3}}{2} ). Therefore, we can set the expressions equal:( frac{1 pm sqrt{1 + 2a}}{2a} = frac{1 pm sqrt{3}}{2} )Therefore, comparing both sides:First, the denominators must be equal. So, 2a = 2 => a = 1. Let's check that. If a = 1, then the denominator is 2*1 = 2, which matches the denominator on the right-hand side. Then the numerators must be equal. Let's check the discriminant part:( sqrt{1 + 2a} = sqrt{3} ). If a = 1, then:( sqrt{1 + 2*1} = sqrt{3} ), which is true. Therefore, both numerator and denominator match when a = 1. So that confirms again that a = 1. Therefore, through multiple methods: sum and product of roots, substituting back into the equation, and using the quadratic formula, we find that a = 1. I think that covers all possible ways to verify. All methods consistently point to a = 1. I don't see any contradictions or errors in the reasoning. So the answer must be 1.Final AnswerThe value of ( a ) is boxed{1}.</think>Given the quadratic equation ( a x^2 - x - frac{1}{2} = 0 ) with roots ( x = frac{1 pm sqrt{3}}{2} ), we need to find the value of ( a ).First, we use the relationships for the sum and product of the roots of a quadratic equation. For the roots ( r_1 = frac{1 + sqrt{3}}{2} ) and ( r_2 = frac{1 - sqrt{3}}{2} ):1. Sum of the roots: [ r_1 + r_2 = frac{1 + sqrt{3}}{2} + frac{1 - sqrt{3}}{2} = frac{2}{2} = 1 ] The sum of the roots for the quadratic equation ( ax^2 + bx + c = 0 ) is given by ( -frac{b}{a} ). Here, ( b = -1 ), so: [ -frac{-1}{a} = frac{1}{a} = 1 implies a = 1 ]2. Product of the roots: [ r_1 times r_2 = left( frac{1 + sqrt{3}}{2} right) left( frac{1 - sqrt{3}}{2} right) = frac{1 - 3}{4} = -frac{1}{2} ] The product of the roots for the quadratic equation is given by ( frac{c}{a} ). Here, ( c = -frac{1}{2} ), so: [ frac{-frac{1}{2}}{a} = -frac{1}{2a} = -frac{1}{2} implies a = 1 ]3. Verification using the quadratic formula: The roots of the equation ( a x^2 - x - frac{1}{2} = 0 ) are given by: [ x = frac{1 pm sqrt{1 + 2a}}{2a} ] Comparing with the given roots ( frac{1 pm sqrt{3}}{2} ), we equate the discriminants and denominators: [ sqrt{1 + 2a} = sqrt{3} implies 1 + 2a = 3 implies 2a = 2 implies a = 1 ] [ 2a = 2 implies a = 1 ]All methods confirm that the value of ( a ) is boxed{1}.]

🔑:To find the value of a given the roots of the quadratic equation ax^2 - x - frac{1}{2} = 0, we use the relationship between the roots and coefficients of a quadratic equation. The sum of the roots, alpha + beta, is equal to -frac{b}{a}, where a, b, and c are the coefficients of ax^2 + bx + c = 0. In our case, b = -1, and thus, the sum of the roots is -frac{-1}{a} = frac{1}{a}.Given the roots x = frac{1 pm sqrt{3}}{2}, we calculate the sum of the roots as follows:[frac{1 + sqrt{3}}{2} + frac{1 - sqrt{3}}{2} = frac{1 + sqrt{3} + 1 - sqrt{3}}{2} = frac{2}{2} = 1]Therefore, we have:[frac{1}{a} = 1]Solving this equation for a gives:[a = 1]Thus, the value of a is boxed{1}.

🧠:[<think>Okay, so I need to figure out the cube root of -125. Hmm, let's start by recalling what a cube root is. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. So, if I'm looking for the cube root of -125, I need to find a number x such that x³ = -125.First, I know that cube roots can be negative because if you multiply a negative number by itself three times, the result is negative. For example, (-2)³ is -8. So, in this case, since -125 is negative, the cube root should also be negative. That makes sense.Now, let's think about the number 125. I remember that 5³ is 125 because 5 * 5 is 25, and 25 * 5 is 125. So, 5 is the cube root of 125. But we're dealing with -125 here. If 5³ is 125, then (-5)³ should be -125. Let me check that. Multiplying -5 by itself: (-5) * (-5) is 25, and then 25 * (-5) is -125. Yep, that works out. So, (-5)³ equals -125, which means the cube root of -125 is -5. Wait, let me make sure there aren't any other possible solutions. Cube roots of real numbers are unique, right? Unlike square roots, which can have both positive and negative roots, cube roots only have one real solution. For example, the cube root of 8 is 2, and the cube root of -8 is -2. So, similarly, for -125, the only real cube root should be -5. Is there a possibility of a complex number solution? Hmm, the question doesn't specify whether it's looking for real or complex roots. But usually, when someone asks for the cube root without any context, they're referring to the principal cube root, which in the case of real numbers is the real one. If they wanted complex roots, they would probably mention it. So, sticking with real numbers here. Let me verify once again. Let's compute (-5)³ step by step. First, multiply -5 by -5: that's 25. Then multiply 25 by -5: 25 * 5 is 125, and since one of the numbers is negative, it's -125. Exactly. So, (-5)³ = -125. Therefore, the cube root of -125 is indeed -5. I think that's solid. No other numbers come to mind. For example, if I try -4: (-4)³ is -64, which is not -125. If I try -6: (-6)³ is -216, which is way too low. So, -5 is right in between. So, yes, definitely -5. Just to recap: The cube root of a negative number is negative, and since 5³ is 125, (-5)³ is -125. Therefore, the cube root of -125 is -5. I can't think of any mistakes in that reasoning. Seems straightforward. So, the answer should be -5.Final AnswerThe cube root of -125 is boxed{-5}.</think>To find the cube root of (-125), we need to determine a number (x) such that (x^3 = -125). 1. Recognize that the cube root of a negative number is negative.2. Calculate the cube root of 125, which is 5, because (5^3 = 125).3. Therefore, the cube root of (-125) must be (-5), since ((-5)^3 = -125).Verification:[(-5) times (-5) = 25 quad text{and then} quad 25 times (-5) = -125]Thus, the cube root of (-125) is (boxed{-5}).,<think>Okay, so I need to find the cube root of -125. Hmm, cube roots... Let me remember. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. Since we're dealing with a negative number here, -125, I should recall that the cube root of a negative number is also negative. That makes sense because if you multiply three negative numbers together, the result is negative. First, let me think about the cube root of 125. What number multiplied by itself three times equals 125? Well, 5 times 5 is 25, and then 25 times 5 is 125. So the cube root of 125 is 5. But since the original number is -125, the cube root should be -5. Let me verify that. If I take -5 and multiply it by itself three times: (-5) * (-5) is 25, and then 25 * (-5) is -125. Yep, that checks out. Wait a second, could there be any other cube roots? I remember that in some cases, especially with even roots like square roots, there can be both positive and negative answers. But cube roots are different because they're odd roots. For cube roots, there's only one real root. So even though 125 has a cube root of 5, when dealing with -125, the cube root is simply -5. There aren't any other real numbers that would satisfy that equation. Let me just make sure there's no trick here. The problem is straightforward, right? It's asking for the cube root, not a square root or any other root. So cube root specifically. And since -5 cubed is indeed -125, that's got to be the answer. Another thing to consider: sometimes people get confused between exponents and roots. For example, squaring a negative number gives a positive result, but cubing keeps the sign. So, similar to how squaring a negative number becomes positive, cubing it retains the negative. Therefore, the cube root of a negative number is negative. I might also think about how -125 is a perfect cube. Since 125 is 5 cubed, -125 is (-5) cubed. That's exactly what's needed here. If the number wasn't a perfect cube, the cube root might be an irrational number, but in this case, it's nice and neat. Just to recap: cube root of -125 is the number that when multiplied by itself three times equals -125. That number is -5. Verified by calculation: (-5)^3 = -5 * -5 * -5 = 25 * -5 = -125. No other numbers would work here. If I tried a different number, like -4, then (-4)^3 would be -64, which is not -125. If I tried -6, (-6)^3 is -216, which is way too low. So, -5 is the only integer that fits. So, after going through all these steps and checking, I'm confident the answer is -5.Final AnswerThe cube root of (-125) is boxed{-5}.</think>To find the cube root of (-125), we need a number that, when multiplied by itself three times, equals (-125). 1. The cube root of (125) is (5) because (5 times 5 times 5 = 125).2. Since (-125) is negative, the cube root must also be negative. Therefore, the cube root of (-125) is (-5).3. Verifying, we calculate ((-5) times (-5) times (-5)): - First, ((-5) times (-5) = 25) - Then, (25 times (-5) = -125)Thus, the cube root of (-125) is (boxed{-5}).]

🔑:To find the cube root of -125, we follow these steps:1. Identify a number which, when cubed, equals -125.2. We know that -5 times -5 times -5 = -125.Therefore, the cube root of -125 is -5.Hence, the final answer is boxed{-5}.

🧠:[<think>Okay, let me try to figure out how to solve this triangle problem. So, we have triangle ABC where angle B is 60 degrees, side a (opposite angle A) is 4 units, and the area of the triangle is 20√3. We need to find the value of side c, which is opposite angle C. The options are given as A:15, B:16, C:20, D:4√21.First, let me recall the formula for the area of a triangle when two sides and the included angle are known. The formula is: Area = (1/2) * side1 * side2 * sin(included angle)In this case, if we can find two sides that include the known angle B (which is 60 degrees), we can plug into this formula. But here, we know angle B is 60 degrees, and side a is opposite angle A, which is 4. Wait, side a is opposite angle A, so side a is BC. Side b is opposite angle B, which is 60 degrees, so side b is AC. And side c is opposite angle C, which is AB. Hmm, maybe I need to label the triangle to visualize better.Let me sketch triangle ABC with angle B at vertex B. So, angle B is 60 degrees. Side opposite angle A is a = 4, so that's BC. So BC = 4. The area is 20√3. We need to find side c, which is AB.So, if we can find another side or angle, maybe we can use the area formula. Let me see. The area formula with angle B: if we can find sides adjacent to angle B, which are sides AB and BC. Wait, angle B is at vertex B, so the sides adjacent to angle B are AB and BC. Therefore, sides AB and BC form angle B. So, sides AB and BC are sides that include angle B. The formula for area would then be (1/2)*AB*BC*sin(60°). Since AB is side c, BC is side a = 4.So, substituting into the area formula:20√3 = (1/2) * c * 4 * sin(60°)Simplify sin(60°) which is √3/2. So:20√3 = (1/2)*c*4*(√3/2)Let me compute the right-hand side step by step. First, multiply (1/2)*4 = 2. Then multiply by (√3/2): 2*(√3/2) = √3. So, the equation becomes:20√3 = c * √3Divide both sides by √3:20 = cSo, c is 20. Hmm, that's option C. So, is that the answer? Wait, but let me check again because that seems straightforward, but maybe I made a mistake.Wait, if the area is given by (1/2)*AB*BC*sin(60°), where AB is c and BC is a=4. Then:20√3 = (1/2)*c*4*(√3/2)So, simplifying:Left side: 20√3Right side: (1/2)*4*c*(√3/2) = (4/2)*c*(√3/2) = 2*c*(√3/2) = c*√3So, equation: c*√3 = 20√3 => c = 20. So, yes, that's correct. So, answer is C:20.Wait, but the options include D:4√21. Is there a possibility I used the wrong formula? Let me check again.Alternatively, maybe the problem is that angle B is 60 degrees, but sides a and c are not the ones including angle B. Wait, in triangle ABC, angle B is 60 degrees. The sides adjacent to angle B are sides AB and BC. Since side a is BC (opposite angle A), and side c is AB (opposite angle C). So yes, sides AB and BC are adjacent to angle B, so the formula is correct.Alternatively, maybe the area is given by another formula. Let me confirm. The area can also be calculated using Heron's formula, but that requires knowing all three sides. Or, using (1/2)*base*height. But here, if we take base BC = 4, then the height from A to BC would be h, such that (1/2)*4*h = 20√3, so h = 10√3. But then using trigonometry, maybe?Alternatively, using the Law of Sines. The Law of Sines states that a/sin A = b/sin B = c/sin C.We know angle B is 60°, and side a is 4. So, 4/sin A = b/sin 60° = c/sin C.But we don't know angles A and C. But maybe we can use the area formula in another way. The area is also given by (1/2)*ab*sin C. Wait, no. Wait, the area can be expressed in terms of different sides and angles. For example, (1/2)*ab*sin C, (1/2)*bc*sin A, or (1/2)*ac*sin B.Wait, in this case, (1/2)*a*c*sin B would be another formula. Wait, but a is BC =4, c is AB. Wait, angle B is between sides AB and BC. So, if we take sides AB and BC (which are c and a=4) and the included angle B=60°, then the area is (1/2)*c*a*sin B. Which is exactly the formula I used earlier. So, that's the same as before. Therefore, the area is (1/2)*c*4*sin 60° = 20√3. Which gives c=20. So that's correct.Therefore, answer is C:20. So maybe the answer is C. However, let me check if there is another possible interpretation.Wait, maybe the problem is that angle B is 60°, but sides a and c are not the ones adjacent to angle B. Wait, but in triangle ABC, angle B is at vertex B, so sides adjacent to angle B are AB and BC. Which are sides c and a respectively. So, yes, sides c and a are adjacent to angle B. Therefore, the formula (1/2)*a*c*sin B is correct.Alternatively, perhaps the student might get confused between sides and angles. Let me check with Law of Sines.From Law of Sines: a/sin A = b/sin B = c/sin C.Given that a=4, B=60°, and area is 20√3.But if we can find another side or angle, maybe using Law of Sines. But how?Alternatively, the area can also be written as (1/2)*a*b*sin C. So, if I can find another expression for the area.But perhaps we can use two different formulas for the area and set them equal.So, we have:20√3 = (1/2)*a*c*sin B = (1/2)*4*c*sin 60° = (1/2)*4*c*(√3/2) = c*√3Therefore, c=20.Alternatively, if we use another formula for the area, maybe involving sides a and b and angle C, but since we don't know side b or angles A and C, that might complicate things.Wait, maybe using the Law of Cosines? For angle B=60°, Law of Cosines says:b² = a² + c² - 2ac cos BSo, if we can find side b, then we can use that equation.But we don't know side b. However, maybe we can find side b through the area formula.But the area is 20√3. If we can express the area in terms of sides a, b, c, and angle B, then we can relate them.Wait, but if we use the Law of Sines:a/sin A = b/sin B = c/sin C = 2R, where R is the radius of the circumscribed circle.But again, this might not be straightforward.Alternatively, let's try to use the Law of Sines and the area formula.Given that a=4, angle B=60°, area=20√3.Let’s denote:From Law of Sines: 4/sin A = b/sin 60° = c/sin CBut area can also be written as (1/2)*a*b*sin C. Wait, but that might not be helpful. Alternatively, (1/2)*b*c*sin A. Hmm.Alternatively, let me denote the angles. Let’s say angle A is α, angle C is γ. Then, α + 60° + γ = 180°, so α + γ = 120°.From the Law of Sines:4/sin α = b/sin 60° = c/sin γSo, let's denote that ratio as 2R (the diameter of the circumscribed circle). Then,b = 2R sin 60°, c = 2R sin γ, and 4 = 2R sin α.But perhaps this is getting too abstract. Let me see.Alternatively, express the area in terms of sides and angles. Since the area is 20√3, which we already used to find c=20. But maybe cross-check using Law of Cosines.If c=20, then using Law of Cosines to find side b:b² = a² + c² - 2ac cos BSo, plugging a=4, c=20, angle B=60°:b² = 4² + 20² - 2*4*20*cos 60°Calculate:4² = 16, 20²=400, 2*4*20=160, cos 60°=0.5, so:b² = 16 + 400 - 160*0.5 = 416 - 80 = 336Therefore, b=√336 = √(16*21) = 4√21So, side b is 4√21, which is option D. Wait, but the question asks for c, which we found as 20. But the answer D is 4√21, which is side b. So, maybe the answer is 20, which is option C. So, seems correct.Alternatively, maybe there's a mistake in my calculation? Let me check again.Area formula:Area = (1/2)*a*c*sin BSo:20√3 = 0.5*4*c*(√3/2)Multiply 0.5*4 = 2Multiply 2*(√3/2) = √3So, 20√3 = √3*c => c=20. Correct.Law of Cosines for side b:b² = 4² + 20² - 2*4*20*cos60°Calculate:16 + 400 = 4162*4*20 = 160cos60°=0.5, so 160*0.5=80Therefore, b²=416 - 80=336sqrt(336)=sqrt(16*21)=4√21, which is answer D. But the question is asking for c, which is 20, so answer C. Therefore, seems correct.Alternatively, maybe the problem is designed to trick people into solving for b instead of c? If someone confuses sides, they might compute b instead. But in this case, the answer options include both 20 and 4√21, so maybe need to be careful.But the problem statement says "Find the value of c". So, c is 20, which is option C. Thus, the answer should be C:20.But let me just check once again. Suppose instead, if I tried to use Heron's formula. But for Heron's formula, we need all three sides. But we only have sides a=4, c=20, and side b=4√21. Then, the semi-perimeter would be (4 + 20 + 4√21)/2. Then, area squared would be s(s-a)(s-b)(s-c). That seems complicated, but let's compute it.s = (4 + 20 + 4√21)/2 = (24 + 4√21)/2 = 12 + 2√21s - a = 12 + 2√21 - 4 = 8 + 2√21s - b = 12 + 2√21 - 4√21 = 12 - 2√21s - c = 12 + 2√21 - 20 = -8 + 2√21Then, the area squared would be:(12 + 2√21)(8 + 2√21)(12 - 2√21)(-8 + 2√21)Hmm, this seems complex. Maybe instead, compute the area using Heron's formula and see if it's 20√3.Alternatively, since we already found sides a=4, c=20, b=4√21, and angle B=60°, the area is 20√3 as given, which matches. Therefore, Heron's formula should confirm that.Alternatively, compute area using another formula: (1/2)*base*height. If we take base as BC=4, then the height h from A to BC must satisfy (1/2)*4*h=20√3 => h=10√3.Then, using Pythagoras in triangle ABD where D is the foot from A to BC. Then, in triangle ABD, h=10√3, BD is some length x, and AB is c=20. Then:h^2 + x^2 = c^2 => (10√3)^2 + x^2 = 400 => 300 + x^2 = 400 => x^2=100 =>x=10.Similarly, the entire base BC is 4, so BD + DC =4. If BD=10, but BD=10 is longer than BC=4. That's impossible. Wait, that can't be.Wait, hold on, this is a contradiction. If we take BC as the base with length 4, then the height h from A to BC is 10√3, but then BD (the segment from B to the foot of the height D) would be x, and DC=4 - x.But according to Pythagoras, in triangle ABD: h^2 + x^2 = AB^2 = c^2 = 400So, h^2 = (10√3)^2 = 300So, 300 + x^2 = 400 => x^2=100 =>x=10. But x is BD, which can't be 10 since BC is only 4. Therefore, this is impossible. Therefore, there must be a mistake here.Wait, this is a problem. If the height is 10√3, then BD would be 10, but BC is only 4. Therefore, this is impossible. Therefore, my previous conclusion that c=20 must be wrong. Therefore, there's an error in my initial reasoning.Wait, so where did I go wrong?Wait, let's go back. I assumed that sides AB and BC are the sides adjacent to angle B, so using the formula (1/2)*AB*BC*sin(angle B). But maybe AB and BC are not both adjacent? Wait, angle B is at vertex B, so the two sides adjacent to angle B are AB and BC. So, that's correct. So, sides AB and BC are adjacent to angle B, so the formula (1/2)*AB*BC*sin(angle B) is correct.But if AB is 20, BC is 4, then the height from A to BC is 10√3, but BD would be 10, which is longer than BC=4. That is impossible. Therefore, this suggests that the triangle cannot exist with these dimensions. Therefore, my answer must be wrong.Wait, but how is that possible? If using the area formula gave c=20, but then Heron's formula or height calculation gives an inconsistency. Therefore, where is the mistake?Wait, perhaps the formula (1/2)*AB*BC*sin(angle B) is correct, but in reality, if we have AB=20, BC=4, angle B=60°, then the height from A to BC would be AB*sin(angle between AB and BC). Wait, but angle at B is 60°, so the height h from A to BC is AB*sin(angle B). Wait, but AB is 20, angle B is 60°, so h = 20*sin(60°)=20*(√3/2)=10√3. Then, the area is (1/2)*BC*h=(1/2)*4*10√3=20√3, which matches. However, geometrically, this seems impossible because the foot of the height would lie outside the triangle.Wait, so in this case, if we have BC=4, and the height from A is 10√3, which is a vertical distance, but the horizontal distance from B to the foot of the height is AB*cos(60°)=20*(1/2)=10. So, the foot of the height is 10 units away from B along the direction perpendicular to BC. But since BC is only 4 units long, the foot of the height would lie outside the segment BC. Therefore, the triangle is obtuse, and the area is still valid, but the height is outside the triangle. Therefore, this is possible.So, in such a case, the area formula still holds, even if the foot of the height is outside the triangle. So, the area is indeed 20√3. However, in that case, angle at B is 60°, but the triangle is obtuse at some other angle. So, in reality, the triangle can exist, but the foot of the altitude is outside. Therefore, maybe the answer c=20 is correct, even though the height is outside. So, perhaps my confusion about BD being 10 is not a problem because the foot is outside.Therefore, maybe my initial answer c=20 is correct, and the problem is designed this way. Therefore, the answer is C:20.But to be thorough, let's check using the Law of Sines.If c=20, angle B=60°, and a=4. Then, from Law of Sines:a/sin A = c/sin C => 4/sin A = 20/sin CAlso, since angles A + C = 120°, we can write sin C = sin(120° - A) = sin(120° - A). Let's use that.So, 4/sin A = 20/sin(120° - A)Cross-multiplying:4*sin(120° - A) = 20*sin ADivide both sides by 4:sin(120° - A) = 5*sin AHmm, expand sin(120° - A) using sine subtraction formula:sin(120° - A) = sin 120° cos A - cos 120° sin AWe know sin 120° = √3/2, cos 120° = -1/2.Therefore:√3/2 cos A - (-1/2) sin A = 5 sin ASimplify:(√3/2) cos A + (1/2) sin A = 5 sin AMultiply both sides by 2 to eliminate denominators:√3 cos A + sin A = 10 sin ASubtract sin A from both sides:√3 cos A = 9 sin ADivide both sides by cos A:√3 = 9 tan ATherefore:tan A = √3 / 9Compute angle A:A = arctan(√3 / 9) ≈ arctan(0.19245) ≈ 10.9 degrees.Then angle C = 120° - 10.9° ≈ 109.1°, which is obtuse.Then, using Law of Sines, compute side b:a/sin A = b/sin B4/sin(10.9°) = b/sin(60°)Calculate sin(10.9°): approx 0.188, sin(60°)=√3/2≈0.866Therefore:4 / 0.188 ≈ b / 0.866 => 21.276 ≈ b / 0.866 => b≈21.276 * 0.866 ≈18.42But earlier we found b=4√21≈4*4.583≈18.33, which is close, considering the approximated angle A. So, this supports that our answer c=20 is correct.Therefore, despite the foot of the altitude being outside the triangle, the answer is c=20, option C. Therefore, the correct answer is C:20.Final Answerboxed{C}</think>Given triangle (ABC) with (angle B = 60^circ), side (a = 4) (opposite (angle A)), and area (S = 20sqrt{3}), we need to find the value of side (c) (opposite (angle C)).1. Area Formula: The area of a triangle can be calculated using the formula: [ S = frac{1}{2} times text{side1} times text{side2} times sin(text{included angle}) ] Here, sides (AB) and (BC) include (angle B). Let (AB = c) and (BC = a = 4). The area is given by: [ 20sqrt{3} = frac{1}{2} times c times 4 times sin(60^circ) ] Simplifying (sin(60^circ) = frac{sqrt{3}}{2}): [ 20sqrt{3} = frac{1}{2} times c times 4 times frac{sqrt{3}}{2} ] [ 20sqrt{3} = frac{4c}{2} times frac{sqrt{3}}{2} ] [ 20sqrt{3} = c times sqrt{3} ] Solving for (c): [ c = 20 ]2. Verification using Law of Cosines: Using the Law of Cosines to find side (b) (opposite (angle B)): [ b^2 = a^2 + c^2 - 2ac cos(B) ] Substituting (a = 4), (c = 20), and (cos(60^circ) = 0.5): [ b^2 = 4^2 + 20^2 - 2 times 4 times 20 times 0.5 ] [ b^2 = 16 + 400 - 80 ] [ b^2 = 336 ] [ b = sqrt{336} = 4sqrt{21} ] This confirms the consistency of the solution.3. Conclusion: The value of (c) is found to be 20, which matches option C. The verification steps confirm the correctness of the solution.[boxed{C}],<think>Okay, let me try to solve this problem step by step. So, we have triangle ABC with angle B equal to 60 degrees, side a opposite angle A is 4 units long, and the area of the triangle is 20√3. We need to find the value of side c, which is opposite angle C. The answer choices are given, so maybe I can use some trigonometric formulas here.First, I remember that the area of a triangle can be calculated using the formula: ( S = frac{1}{2}absin C )Wait, but in this case, we know angle B is 60 degrees. Maybe a better formula to use here is the one involving two sides and the included angle. The formula is:( S = frac{1}{2} times a times c times sin B )Wait, no. Wait, actually, if angle B is 60 degrees, then the sides adjacent to angle B would be sides a and c? Wait, hold on. Let me clarify.In triangle ABC, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. So, angle B is 60 degrees, and the sides adjacent to angle B would be sides a and c? Because sides are opposite their respective angles. So, angle B is between sides a and c. Therefore, the formula for the area using two sides and the included angle would indeed be:( S = frac{1}{2} times a times c times sin B )Given that S is 20√3, angle B is 60 degrees, and a is 4, we can plug these values into the formula to solve for c.Let me write that down:( 20sqrt{3} = frac{1}{2} times 4 times c times sin 60^circ )First, compute sin 60 degrees. I know that sin 60° is √3/2.So substituting that in:( 20sqrt{3} = frac{1}{2} times 4 times c times frac{sqrt{3}}{2} )Let me simplify the right side step by step. First, 1/2 multiplied by 4 is 2.So, 2 times c times √3/2.Wait, so 2 multiplied by √3/2 is √3. Then, √3 multiplied by c is c√3.Therefore, the equation simplifies to:( 20sqrt{3} = csqrt{3} )To solve for c, we can divide both sides by √3:( c = frac{20sqrt{3}}{sqrt{3}} )Which simplifies to:( c = 20 )Wait, but 20 is one of the answer choices, option C. So, is that the answer? Hmm. But let me double-check my reasoning because sometimes the area formula might involve different sides.Wait, maybe I got confused with which sides are adjacent to angle B. Since angle B is at vertex B, the sides adjacent to angle B would be sides a and c. Wait, but side a is opposite angle A, so sides adjacent to angle B would actually be sides AB and BC. Wait, maybe I need to label the triangle properly.Let me visualize triangle ABC. Let's say angle A is opposite side a (which is 4), angle B is 60 degrees opposite side b, and angle C is opposite side c. So, in this triangle, sides a, b, c are opposite angles A, B, C respectively.Therefore, angle B is between sides a and c. Because in a triangle, each angle is between the two sides that are not opposite it. So, angle B is between sides AB and BC. Wait, but side AB is opposite angle C, which is side c, and side BC is opposite angle A, which is side a. Wait, no, perhaps I need to recall the standard triangle labeling.In standard triangle notation, vertex A is opposite side a, vertex B is opposite side b, and vertex C is opposite side c. Therefore, sides adjacent to angle B (which is at vertex B) would be sides AB and BC. Side AB is opposite angle C, so its length is c, and side BC is opposite angle A, so its length is a. Therefore, angle B is between sides AB (length c) and BC (length a). Therefore, the formula for the area using sides a and c and the included angle B is correct:( S = frac{1}{2} times a times c times sin B )So substituting the known values:( 20sqrt{3} = frac{1}{2} times 4 times c times sin 60^circ )As sin 60° is √3/2, substituting that:( 20sqrt{3} = frac{1}{2} times 4 times c times frac{sqrt{3}}{2} )Calculating the right-hand side:First, 1/2 * 4 = 2Then, 2 * (√3 / 2) = √3Thus, the equation becomes:20√3 = c * √3Divide both sides by √3:20 = cSo, c = 20, which is option C. That seems straightforward, but let me verify with another method to ensure I didn't make a mistake.Alternatively, maybe using Heron's formula. But for Heron's formula, I need all three sides or semi-perimeter, which I don't have. Alternatively, the Law of Sines or Law of Cosines.Law of Sines states that:( frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C} )But we know angle B and side a. Maybe we can find another angle or side.Alternatively, Law of Cosines:For angle B, which is 60 degrees, the Law of Cosines states:( b^2 = a^2 + c^2 - 2ac cos B )But we don't know b. However, perhaps if we can find another relation using the area.Wait, the area is also equal to 1/2 * base * height. If we take side a as the base (length 4), then the height h from vertex B to side a would be:( S = frac{1}{2} times a times h )So,( 20sqrt{3} = frac{1}{2} times 4 times h )Solving for h:( 20sqrt{3} = 2h implies h = 10sqrt{3} )But how does that help? Maybe using trigonometry with the height.Alternatively, since angle B is 60 degrees, and sides adjacent to it are a and c, then maybe using the area formula as before was correct.Alternatively, if we consider the coordinates. Let me place point B at the origin, side BC along the x-axis. Then coordinates of B are (0,0). Let me denote side BC as length a = 4, so point C is at (4, 0). Angle at B is 60 degrees, so angle between BA and BC is 60 degrees. Then point A would be somewhere in the plane. The area is 20√3. Then, coordinates of point A can be found.Wait, if we consider point B at (0,0), point C at (4,0). Then, the coordinates of point A can be determined using the angle at B (60 degrees) and the area.The area is 1/2 * base * height. The base is BC = 4, so the height from A to BC is h = (2 * area)/base = (2 * 20√3)/4 = 10√3. So the height is 10√3. Since angle at B is 60 degrees, the height can also be related to side AB.Wait, the height from A to BC is the same as the length of the altitude from A to BC. Alternatively, if we construct triangle ABC with BC = 4, angle at B is 60 degrees, then point A must be somewhere such that the height is 10√3.Alternatively, using coordinates: if B is at (0,0), C is at (4,0), then point A is somewhere in the plane. The angle at B is 60 degrees, so the slope of BA is tan(60°) = √3. So the line BA has a slope of √3. Therefore, coordinates of A can be represented as (x, √3 x) for some x. Then, the distance from A to BC (which is the x-axis) is the y-coordinate, which is √3 x. But we know the height is 10√3, so √3 x = 10√3 => x = 10. Therefore, coordinates of A would be (10, 10√3). Wait, but then BA would be the distance from (0,0) to (10, 10√3), which is sqrt(10^2 + (10√3)^2) = sqrt(100 + 300) = sqrt(400) = 20. So BA is 20, which is side c (since side c is opposite angle C, which is at point C). Wait, but side BA is opposite angle C. Wait, in standard notation, side c is opposite angle C. So if angle C is at point C, then side AB is opposite angle C, so side AB is indeed side c. Therefore, length BA is 20, so c = 20. Therefore, that confirms the answer is 20, which is option C.Alternatively, this coordinate geometry approach also leads us to c = 20. So that's consistent with the previous method.Wait, let me check once more. If angle B is 60 degrees, and we placed B at (0,0), C at (4,0), then point A is at (10, 10√3). Then, the length BA is sqrt((10 - 0)^2 + (10√3 - 0)^2) = sqrt(100 + 300) = sqrt(400) = 20. So yes, BA is 20, which is side c. Therefore, c = 20.Alternatively, using Law of Sines. Let me try that approach.Law of Sines says:( frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C} )We know angle B is 60 degrees, side a is 4, and area is 20√3. Let's see if we can find another angle or side.But we need more information. Maybe using the area formula again. The area can also be expressed as:( S = frac{1}{2}ab sin C = frac{1}{2}bc sin A = frac{1}{2}ac sin B )We already used the last one. Let's try another one. Suppose we use:( 20sqrt{3} = frac{1}{2} times b times c times sin A )But we don't know b or angle A. Alternatively, maybe use Heron's formula? But we need all three sides for that. Hmm.Alternatively, use the Law of Cosines. If we can find side b, then we can relate sides a, c, and angle B.Wait, we have from the Law of Cosines:( b^2 = a^2 + c^2 - 2ac cos B )But we don't know b. But maybe we can express the area in another way to find another relation.Alternatively, area is 20√3, and we can also compute the area using sides a, b, and angle C. But that might complicate things.Wait, perhaps using the formula:( S = frac{1}{2}ab sin C )But again, we don't know angle C or side b.Alternatively, let's consider that we have two expressions for the area. We already have one:( 20sqrt{3} = frac{1}{2} times 4 times c times frac{sqrt{3}}{2} )Which simplifies to c = 20. So that gives c directly. Therefore, maybe this is the simplest way.Alternatively, perhaps the problem is designed so that you don't need to go through coordinate geometry or Law of Sines/Cosines. The direct formula gives the answer immediately.Therefore, given that angle B is 60 degrees, sides a and c are the ones enclosing angle B, and the area is 1/2 * a * c * sin B. So, solving for c gives 20. The answer is C.But just to confirm once again, maybe with Law of Cosines. If c = 20, a = 4, angle B = 60 degrees, then side b can be computed.Compute side b:( b^2 = a^2 + c^2 - 2ac cos B )Substituting:( b^2 = 4^2 + 20^2 - 2 times 4 times 20 times cos 60° )Calculate each term:4^2 = 1620^2 = 4002*4*20 = 160cos 60° = 0.5Thus:b^2 = 16 + 400 - 160 * 0.5160 * 0.5 = 80So:b^2 = 416 - 80 = 336Therefore, b = sqrt(336) = sqrt(16*21) = 4√21Wait, but answer choice D is 4√21. Hmm. So that's the length of side b. But the problem is asking for c, which we found to be 20. So that's okay. But just to check, if we compute the area using sides a, b, and angle C, would it also be 20√3?Alternatively, let's compute the area using sides a, c, and angle B, which is 60 degrees. That's what we did initially, and it gives 20√3. So that's consistent.Alternatively, using side b and other angles. But since we don't know angles A and C, perhaps not necessary. So everything checks out. So c is indeed 20, which is option C.Therefore, the correct answer is C: 20.Final Answerboxed{C}</think>Given triangle (ABC) with angle (B = 60^circ), side (a = 4) (opposite angle (A)), and area (S = 20sqrt{3}), we need to find the value of side (c) (opposite angle (C)).We use the formula for the area of a triangle involving two sides and the included angle:[S = frac{1}{2} times a times c times sin B]Substituting the given values:[20sqrt{3} = frac{1}{2} times 4 times c times sin 60^circ]Since (sin 60^circ = frac{sqrt{3}}{2}), we substitute this value:[20sqrt{3} = frac{1}{2} times 4 times c times frac{sqrt{3}}{2}]Simplifying the right-hand side:[20sqrt{3} = 2 times c times frac{sqrt{3}}{2}][20sqrt{3} = c sqrt{3}]Dividing both sides by (sqrt{3}):[c = 20]Thus, the value of (c) is (boxed{C}).]

🔑:AnalysisThis problem mainly tests the application of solving triangles. Being familiar with the formula for the area of a triangle is key to solving this problem. It is a basic question.SolutionGiven the problem, we have:therefore S= frac{1}{2}acsin B= frac{1}{2}times4times ctimessin 60^circ=20sqrt{3}, solving for c gives c=20.Therefore, the correct choice is boxed{C}.

🧠:[<think>Okay, let's see. We have four cousins: Alan, Bob, Carl, and Dan. Their ages are 3, 8, 12, and 14, but we don't know who is which age. We need to figure out the sum of Alan and Bob's ages based on the clues given.First, let's list out the clues:1. Alan is younger than Carl.2. The sum of the ages of Alan and Dan is divisible by 5.3. The sum of the ages of Carl and Dan is also divisible by 5.Our goal is to find Alan + Bob. Let's take it step by step.Let me assign variables to each person's age to make it clearer. Let A = Alan, B = Bob, C = Carl, D = Dan. Their ages are 3, 8, 12, 14, each a different age.First clue: A < C. So whatever age Alan is, Carl has to be older. So Carl can't be 3, because Alan has to be younger. So Carl could be 8, 12, or 14. Let's note that.Second clue: A + D is divisible by 5. The possible ages are 3, 8, 12, 14. Let's list all possible sums of A and D:Possible pairs (A, D):If A is 3: D could be 8, 12, 14. Then sums would be 11, 15, 17. Which of these are divisible by 5? 15. So if A is 3, D must be 12 (3 + 12 = 15). Wait, 3 + 12 is 15, which is divisible by 5. So that's one possibility.If A is 8: D could be 3, 12, 14. Sums: 11, 20, 22. Divisible by 5: 20. So D would have to be 12 (8 + 12 = 20). Wait, 8 + 12 is 20, which is divisible by 5. Alternatively, 8 + 12 is 20. So D is 12.If A is 12: Then D can be 3, 8, 14. Sums: 15, 20, 26. Divisible by 5: 15 and 20. So D could be 3 or 8. But if A is 12, then from the first clue, Carl must be older than Alan. So Carl would have to be 14. But if Carl is 14, then let's check the third clue.Third clue: C + D divisible by 5. If Carl is 14, then D must be such that 14 + D is divisible by 5. Possible D's left would be 3, 8, but if A is 12, then D can be 3 or 8. So 14 + 3 = 17 (not divisible by 5), 14 + 8 = 22 (not divisible by 5). So neither works. Therefore, A cannot be 12. Because if A is 12, then Carl must be 14, but then D would have to be 3 or 8, but neither of those sums with 14 is divisible by 5. Therefore, A cannot be 12. So A must be either 3 or 8.Similarly, if A is 14: Wait, but A has to be younger than Carl. If A is 14, then Carl would have to be older than 14, but the maximum age is 14. So that's impossible. So A cannot be 14. Therefore, possible A: 3, 8.So let's check A = 3 first.Case 1: A = 3.Then from clue 2, A + D must be divisible by 5. So 3 + D ≡ 0 mod 5. So D ≡ 2 mod 5. The possible ages are 8, 12, 14. Which of these are ≡ 2 mod 5?8 mod 5 = 3, 12 mod 5 = 2, 14 mod 5 = 4. So D must be 12. So D = 12.So Dan is 12. Then remaining ages for Bob and Carl are 8 and 14. Since Alan is 3, Carl must be older than Alan. So Carl can be 8 or 14. But Carl must be older than Alan, so Carl can be either 8 or 14. But we also have to check clue 3: C + D must be divisible by 5. Carl is either 8 or 14, D is 12.So if Carl is 8: 8 + 12 = 20, which is divisible by 5. That works.If Carl is 14: 14 + 12 = 26, which is not divisible by 5. So that doesn't work. Therefore, Carl must be 8, and then Bob is 14.So in this case, the assignments would be: A = 3, C = 8, D = 12, B = 14.Then sum of Alan and Bob is 3 + 14 = 17. Hmm, but 17 is one of the options (option C). But let's check the other case where A = 8.Case 2: A = 8.From clue 2, A + D must be divisible by 5. So 8 + D ≡ 0 mod 5. So D ≡ 2 mod 5 (since 8 mod 5 = 3, so 3 + D ≡ 0 mod 5 → D ≡ 2 mod 5). The possible ages for D (since A is 8, the remaining ages are 3, 12, 14). Which of these are ≡ 2 mod 5?3 mod 5 = 3, 12 mod 5 = 2, 14 mod 5 = 4. So D must be 12 again. So D = 12.Then remaining ages for Bob and Carl are 3 and 14. From clue 1: Alan is younger than Carl. Alan is 8, so Carl must be older than 8. So Carl must be 14. Then Bob is 3.Now check clue 3: C + D = 14 + 12 = 26. 26 divided by 5 is 5.2, so not divisible by 5. So that's a problem. 26 isn't divisible by 5, so this contradicts clue 3. Therefore, A can't be 8. So only Case 1 is valid.Therefore, in Case 1, Alan is 3, Bob is 14. So 3 + 14 = 17. So answer is C?Wait, but the options given are A 26, B 22, C 17, D 15, E 11. So 17 is option C.But let me double-check.Wait, in Case 1: Assign A = 3, D = 12. Then remaining ages are 8 and 14. Since Carl must be older than Alan (3), Carl can be 8 or 14. Then C + D must be divisible by 5. If Carl is 8: 8 + 12 = 20, which is divisible by 5. So that works. Therefore, Carl is 8, Bob is 14. So sum of Alan and Bob: 3 + 14 = 17.Alternatively, if Carl is 14: 14 + 12 = 26, which is not divisible by 5, so invalid. So yes, Carl must be 8. Therefore, that's correct.Alternatively, let me check if there are other possibilities. Suppose A is 3, D is 12, then Carl is 8, Bob is 14. Then sum A + B is 17. Seems right.Alternatively, could there be another assignment? Let's see. Suppose in Case 2, A is 8, but then Dan has to be 12. Then Carl is 14, but C + D = 26, which is not divisible by 5. So invalid. So only Case 1 works. Therefore, the answer is 17, option C.Wait, but wait a minute. The problem statement says "the sum of the ages of Alan and Dan is divisible by 5" and "the sum of the ages of Carl and Dan is divisible by 5." So both sums need to be divisible by 5. In Case 1, A=3, D=12. So 3 + 12 = 15 (divisible by 5). C=8, so 8 + 12 = 20 (divisible by 5). So both are okay. Then Bob is 14. So Alan + Bob is 3 +14=17. Correct.Alternatively, let's check if there is another possibility where A=3 and D=12. Then Carl must be 8. So Bob is 14. So yes, that's the only way.Alternatively, if A=3 and D= something else? But from clue 2, A + D must be divisible by 5. So if A=3, D has to be 12. Since 3 +8=11, 3+12=15, 3+14=17. Only 15 is divisible by 5, so D=12.Similarly, if A=8, D=12. Then C must be 14, but that fails. So yes, only Case 1 works.Therefore, answer is C, 17. So I think the answer is boxed{C}Final Answerboxed{C}</think>Given the problem, we need to determine the sum of the ages of Alan and Bob based on the clues provided. The ages of the four cousins (Alan, Bob, Carl, and Dan) are 3, 8, 12, and 14. The clues are:1. Alan is younger than Carl.2. The sum of the ages of Alan and Dan is divisible by 5.3. The sum of the ages of Carl and Dan is divisible by 5.We start by analyzing the possible ages for Alan (A) and Dan (D) such that their sum is divisible by 5. - If Alan is 3, then Dan must be 12 (since 3 + 12 = 15, which is divisible by 5).- If Alan is 8, then Dan must be 12 (since 8 + 12 = 20, which is divisible by 5). However, we need to check if this leads to a valid solution for Carl's age.Next, we check the remaining ages for Carl and Bob, ensuring Carl is older than Alan and the sum of Carl and Dan's ages is divisible by 5.Case 1: Alan is 3 and Dan is 12- Remaining ages are 8 and 14 for Bob and Carl.- Carl must be older than Alan (3), so Carl could be 8 or 14.- Checking the sum of Carl and Dan's ages: - If Carl is 8, then 8 + 12 = 20 (divisible by 5). This works, leaving Bob as 14.- Therefore, Alan is 3, Bob is 14, Carl is 8, and Dan is 12. The sum of Alan and Bob's ages is 3 + 14 = 17.Case 2: Alan is 8 and Dan is 12- Remaining ages are 3 and 14 for Bob and Carl.- Carl must be older than Alan (8), so Carl must be 14.- Checking the sum of Carl and Dan's ages: - 14 + 12 = 26 (not divisible by 5). This does not work.Thus, the only valid solution is from Case 1, where the sum of Alan and Bob's ages is 17.[boxed{C}],<think>Okay, let's see here. I need to figure out the ages of four cousins: Alan, Bob, Carl, and Dan. Their ages are 3, 8, 12, and 14, but not necessarily in that order. The problem gives a few clues, and I need to use those to determine the sum of Alan and Bob's ages. The answer choices are A) 26, B) 22, C) 17, D) 15, and E) 11. Alright, let's break this down step by step.First, let's list the given information:1. Alan is younger than Carl. So, Alan's age < Carl's age.2. The sum of the ages of Alan and Dan is divisible by 5.3. The sum of the ages of Carl and Dan is also divisible by 5.The possible ages are 3, 8, 12, and 14. Each cousin has a unique age from this list. I need to assign these ages to Alan, Bob, Carl, and Dan based on the given clues.Let me start by noting that the sum of two numbers being divisible by 5 means that their sum modulo 5 is 0. So, (Alan + Dan) mod 5 = 0 and (Carl + Dan) mod 5 = 0. Therefore, both Alan + Dan and Carl + Dan are multiples of 5. Since Alan and Carl are different people, their ages are different. Also, Alan is younger than Carl. So, if I can figure out Dan's age, maybe that will help me determine Alan and Carl's ages.Let me consider the possible ages for Dan first. Dan could be 3, 8, 12, or 14. Let's test each possibility.Case 1: Dan is 3 years old.If Dan is 3, then Alan's age must be such that Alan + 3 is divisible by 5. The possible ages for Alan are 8, 12, or 14 (since 3 is already taken by Dan). Let's check:- Alan = 8: 8 + 3 = 11. 11 divided by 5 is 2.2, not a whole number. Not divisible.- Alan = 12: 12 + 3 = 15. 15/5 = 3. That works. So Alan could be 12. But then Carl must be older than Alan. If Alan is 12, Carl would have to be 14. Then check Carl + Dan: 14 + 3 = 17. 17/5 = 3.4. Not divisible by 5. So that doesn't work. Therefore, Dan can't be 3.Case 2: Dan is 8 years old.Then, Alan's age + 8 must be divisible by 5. Possible ages for Alan: 3, 12, 14.- Alan = 3: 3 + 8 = 11. Not divisible by 5.- Alan = 12: 12 + 8 = 20. 20/5 = 4. That works. Then Carl must be older than Alan (12), so Carl would be 14. Then Carl + Dan = 14 + 8 = 22. 22/5 = 4.4. Not divisible by 5. Doesn't work.- Alan = 14: 14 + 8 = 22. 22/5 = 4.4. Not divisible. So Dan can't be 8.Case 3: Dan is 12 years old.Then, Alan's age + 12 must be divisible by 5. Possible Alan ages: 3, 8, 14.- Alan = 3: 3 + 12 = 15. 15/5 = 3. Good. Then Carl must be older than Alan (3), so Carl can be 8, 12, or 14. But Dan is already 12, so Carl can be 8 or 14. Then check Carl + Dan (Carl + 12). If Carl is 8: 8 + 12 = 20, divisible by 5. That's good. If Carl is 14: 14 + 12 = 26, which is not divisible by 5 (26/5=5.2). So Carl must be 8. But then Carl's age (8) is less than Alan's age (3)? Wait, no, Alan is 3, Carl is 8. But wait, Alan is supposed to be younger than Carl. 3 is younger than 8, so that works. Wait, hold on. Let me confirm. If Alan is 3, Carl is 8. Then the sum of Alan (3) and Dan (12) is 15, which is divisible by 5. The sum of Carl (8) and Dan (12) is 20, which is also divisible by 5. So that works. Then Bob would be the remaining age, which is 14. So the ages would be:Alan: 3Bob: 14Carl: 8Dan: 12But hold on, Carl is 8, which is older than Alan (3), which is okay. But wait, the ages are 3,8,12,14. So assigning 3 to Alan, 8 to Carl, 12 to Dan, and 14 to Bob. Then the sum of Alan and Bob would be 3 + 14 = 17. Hmm, 17 is one of the options (C). But let's check if there are other possibilities.Wait, let me check if Alan could be 8. Wait, in this case, Dan is 12, so Alan could be 8. Then 8 + 12 = 20, which is divisible by 5. Then Carl has to be older than Alan (8), so Carl could be 12 or 14. But Dan is already 12, so Carl would have to be 14. Then Carl + Dan = 14 + 12 = 26. 26/5 = 5.2, not divisible by 5. So that's invalid. So only if Alan is 3 and Carl is 8, but wait, Carl is 8 and Dan is 12, so Carl's age (8) plus Dan's age (12) is 20, which is divisible by 5. That works. Then Bob is 14. So the sum of Alan (3) and Bob (14) is 17. So that would be answer C. But let me check the other case for Dan's age.Case 4: Dan is 14 years old.Then, Alan's age + 14 must be divisible by 5. Possible Alan ages: 3, 8, 12.- Alan = 3: 3 + 14 = 17. 17/5 = 3.4. Not divisible.- Alan = 8: 8 + 14 = 22. 22/5 = 4.4. Not divisible.- Alan = 12: 12 + 14 = 26. 26/5 = 5.2. Not divisible. So none of these work. So Dan can't be 14.Therefore, the only valid case is when Dan is 12. Then Alan is 3, Carl is 8, Dan is 12, and Bob is 14. The sum of Alan and Bob is 3 + 14 = 17, which is option C. But wait, let me check again. If Carl is 8, that's older than Alan (3), which satisfies the first condition. The sum of Alan and Dan is 3 + 12 = 15, divisible by 5. The sum of Carl and Dan is 8 + 12 = 20, which is divisible by 5. So all conditions are met. Then the remaining age, Bob, is 14. So the answer is 3 + 14 = 17, which is C. But let me double-check if there's another possible assignment. Wait, if Dan is 12, and Alan is 3, Carl is 8, but is there another way? For example, suppose Dan is 12, Alan is 8. Then 8 + 12 = 20, divisible by 5. Then Carl must be older than Alan (8), so Carl could be 12 or 14. But Dan is 12, so Carl would be 14. Then Carl (14) + Dan (12) = 26, which is not divisible by 5. So that doesn't work. Therefore, only Alan being 3 works here.Alternatively, if Dan is 12, and Alan is 12, but no, each cousin is a different age. So Alan can't be 12 if Dan is 12. So the only possible assignment is Alan:3, Carl:8, Dan:12, Bob:14. So sum is 17. That's answer C.Wait, but in the answer choices, C is 17. But the options were A)26, B)22, C)17, D)15, E)11. So 17 is an option. Therefore, the answer is C.But just to be thorough, let me check if there's another possible Dan's age. Wait, in case 3 when Dan is 12, we found a valid assignment. Are there any other cases? For example, if Dan is 3, 8, or 14, we saw they don't work. So the only possible solution is when Dan is 12. Therefore, the answer must be 17, which is C.But wait, let me check if Carl could be 14. Suppose Dan is 12, Alan is 3, Carl is 14. Then Carl is older than Alan, which is okay. Then sum of Carl and Dan is 14 + 12 = 26, which is not divisible by 5. So that's invalid. Therefore, Carl has to be 8. So the only valid assignment is Alan:3, Carl:8, Dan:12, Bob:14. Thus, the answer is 3 + 14 = 17. So I think that's correct.Final Answerboxed{C}</think>Given the ages of four cousins Alan, Bob, Carl, and Dan as 3, 8, 12, and 14, we need to determine the sum of the ages of Alan and Bob based on the following clues:1. Alan is younger than Carl.2. The sum of the ages of Alan and Dan is divisible by 5.3. The sum of the ages of Carl and Dan is divisible by 5.We start by considering the possible ages for Dan and checking the conditions:- Dan is 3 years old: This does not work because the sum of Alan and Dan's ages would not be divisible by 5.- Dan is 8 years old: This does not work because the sum of Alan and Dan's ages would not be divisible by 5.- Dan is 12 years old: - Possible ages for Alan are 3, 8, or 14. - If Alan is 3, then the sum of Alan and Dan (3 + 12 = 15) is divisible by 5. - Carl must be older than Alan (3), so Carl could be 8, 12, or 14. Since Dan is 12, Carl can be 8 or 14. - Checking the sum of Carl and Dan: If Carl is 8, then 8 + 12 = 20 (divisible by 5). If Carl is 14, then 14 + 12 = 26 (not divisible by 5). Therefore, Carl must be 8. - The remaining age for Bob is 14. Thus, the sum of Alan (3) and Bob (14) is 17.- Dan is 14 years old: This does not work because the sum of Alan and Dan's ages would not be divisible by 5.The valid assignment is Alan: 3, Carl: 8, Dan: 12, and Bob: 14. The sum of the ages of Alan and Bob is (3 + 14 = 17).[boxed{C}]]

🔑:9. C The ages of the four cousins are 3,8,12 and 14. When these are added in pairs, we obtain 3+8=11,3+12=15,3+14=17,8+12=20,8+14=22 and 12+14=26. Only two of these, 15 and 20, are divisible by 5 . However, we are told that the sum of the ages of Alan and Dan and the sum of the ages of Carl and Dan are both divisible by 5 . Hence, since Dan's age appears in both sums that are divisible by 5 , his age is 12 . Since Alan is younger than Carl, Alan's age is 3 and Carl's age is 8 . Hence Bob's age is 14 . Therefore the sum of the ages of Alan and Bob is 3+14=17.