answer：Okay, so I need to find all integer solutions (a, b) for the equation (a^{2} b^{2} = a^{2} + b^{2}). Hmm, let me try to figure this out step by step. First, I should probably write down the equation again to make sure I have it right: (a^2 b^2 = a^2 + b^2). Hmm, that looks a bit complicated, but maybe I can manipulate it to make it simpler.Let me try dividing both sides by (a^2 b^2) to see if that helps. If I do that, I get:[1 = frac{1}{b^2} + frac{1}{a^2}]Wait, is that right? Let me check. If I divide both sides by (a^2 b^2), then the left side becomes (1), and the right side becomes (frac{a^2}{a^2 b^2} + frac{b^2}{a^2 b^2}), which simplifies to (frac{1}{b^2} + frac{1}{a^2}). Yeah, that seems correct.So now the equation is (1 = frac{1}{a^2} + frac{1}{b^2}). Hmm, this looks a bit more manageable. Maybe I can let (x = frac{1}{a^2}) and (y = frac{1}{b^2}), so the equation becomes (x + y = 1). But since (a) and (b) are integers, (x) and (y) must be reciprocals of perfect squares. So (x) and (y) can only take values like 1, 1/4, 1/9, 1/16, etc.So, I need to find pairs ((x, y)) such that (x + y = 1), where (x) and (y) are reciprocals of perfect squares. Let me list possible values for (x) and see if (1 - x) is also a reciprocal of a perfect square.Starting with the smallest possible value for (x), which would be 1. If (x = 1), then (y = 0). But (y = frac{1}{b^2}) can't be zero because (b) is an integer, and (b^2) can't be infinity. So, (x = 1) is not possible because it would require (b) to be undefined or infinity, which isn't an integer.Next, (x = 1/4). Then (y = 1 - 1/4 = 3/4). Is 3/4 a reciprocal of a perfect square? Let's see: 3/4 is equal to 3 divided by 4, which is 3/(2^2). But 3 isn't a perfect square, so 3/4 isn't a reciprocal of a perfect square. So, (x = 1/4) doesn't work.Moving on, (x = 1/9). Then (y = 1 - 1/9 = 8/9). Is 8/9 a reciprocal of a perfect square? 8/9 is equal to 8/(3^2). Again, 8 isn't a perfect square, so 8/9 isn't a reciprocal of a perfect square. So, (x = 1/9) doesn't work either.Next, (x = 1/16). Then (y = 1 - 1/16 = 15/16). 15/16 is 15/(4^2). 15 isn't a perfect square, so this doesn't work.Wait, maybe I should try larger (x) values? But as (x) increases, (y) decreases, so maybe I should also check smaller (x) values, but I already started from the smallest. Hmm.Alternatively, maybe I should consider negative integers for (a) and (b). Since (a^2) and (b^2) are always positive, regardless of whether (a) and (b) are positive or negative. So, maybe the signs of (a) and (b) don't affect the equation, but they could contribute to solutions.Wait, but if I consider negative integers, the equation remains the same because squaring them makes them positive. So, maybe all solutions will come in pairs where (a) and (b) can be positive or negative. So, perhaps I can just find the positive solutions and then include their negatives as well.But let me get back to the equation (1 = frac{1}{a^2} + frac{1}{b^2}). Maybe another approach is better. Let's try to rearrange the original equation.Starting again with (a^2 b^2 = a^2 + b^2). Let me bring all terms to one side:[a^2 b^2 - a^2 - b^2 = 0]Hmm, maybe factor this somehow. Let me see if I can factor it. Let's factor out (a^2) from the first two terms:[a^2(b^2 - 1) - b^2 = 0]Hmm, not sure if that helps. Alternatively, maybe add 1 to both sides:[a^2 b^2 - a^2 - b^2 + 1 = 1]Wait, that's a bit random, but maybe I can factor the left side. Let me try:Looking at (a^2 b^2 - a^2 - b^2 + 1), perhaps factor by grouping.Group (a^2 b^2 - a^2) and (-b^2 + 1):[a^2(b^2 - 1) - (b^2 - 1) = (a^2 - 1)(b^2 - 1)]Oh! That's nice. So, the left side factors into ((a^2 - 1)(b^2 - 1)). So, the equation becomes:[(a^2 - 1)(b^2 - 1) = 1]Wow, that's a big simplification. So, now we have the product of two integers, ((a^2 - 1)) and ((b^2 - 1)), equal to 1. Since both (a) and (b) are integers, (a^2 - 1) and (b^2 - 1) must also be integers. So, we need two integers whose product is 1.The integer pairs that multiply to 1 are (1, 1) and (-1, -1). So, we have two cases:1. (a^2 - 1 = 1) and (b^2 - 1 = 1)2. (a^2 - 1 = -1) and (b^2 - 1 = -1)Let's solve each case separately.**Case 1: (a^2 - 1 = 1) and (b^2 - 1 = 1)**Solving for (a):(a^2 - 1 = 1 implies a^2 = 2)But (a^2 = 2) implies that (a = sqrt{2}) or (a = -sqrt{2}), which are not integers. So, this case doesn't yield any integer solutions.**Case 2: (a^2 - 1 = -1) and (b^2 - 1 = -1)**Solving for (a):(a^2 - 1 = -1 implies a^2 = 0 implies a = 0)Similarly, solving for (b):(b^2 - 1 = -1 implies b^2 = 0 implies b = 0)So, this case gives us the solution (a = 0) and (b = 0). But wait, let me plug this back into the original equation to verify.Original equation: (a^2 b^2 = a^2 + b^2)If (a = 0) and (b = 0), then left side is (0^2 times 0^2 = 0), and right side is (0^2 + 0^2 = 0). So, 0 = 0, which is true. So, (0, 0) is a solution.But wait, earlier when I considered dividing both sides by (a^2 b^2), I assumed that (a) and (b) are non-zero because otherwise, I would be dividing by zero. So, that's why I didn't consider (0, 0) earlier. So, (0, 0) is a valid solution, but it's a special case.Now, are there any other solutions? Let me think.Wait, in the factoring step, I assumed that ((a^2 - 1)(b^2 - 1) = 1). So, the only integer solutions for this are when both factors are 1 or both are -1. But in the first case, we saw that it leads to non-integer solutions, and the second case leads to (0, 0). So, is (0, 0) the only solution?Wait, hold on. Let me think again. Maybe I missed something. Because when I factored the equation, I assumed that both (a^2 - 1) and (b^2 - 1) are integers, which they are, but maybe there are other factorizations? Wait, no, because 1 only factors into 1*1 or (-1)*(-1) in integers. So, that should cover all possibilities.But let me check if there are other possibilities. Suppose (a^2 - 1 = 1) and (b^2 - 1 = 1), but as we saw, that leads to (a^2 = 2), which isn't integer. Similarly, if one of them is 1 and the other is 1, same result.Alternatively, if (a^2 - 1 = -1) and (b^2 - 1 = -1), that gives (a = 0) and (b = 0). So, that's the only solution.Wait, but let me test some small integer values to see if there are any other solutions.Let me try (a = 1). Then, plugging into the original equation:(1^2 b^2 = 1^2 + b^2 implies b^2 = 1 + b^2 implies 0 = 1). That's impossible, so (a = 1) is not a solution.Similarly, (a = -1) would give the same result because of the squares.What about (a = 2)? Then:(4 b^2 = 4 + b^2 implies 4b^2 - b^2 = 4 implies 3b^2 = 4 implies b^2 = 4/3). Not an integer.Similarly, (a = -2) gives the same result.What about (a = 0)? Then, original equation becomes (0 = 0 + b^2 implies b^2 = 0 implies b = 0). So, that's the solution we already found.What about (b = 1)? Then, equation becomes (a^2 = a^2 + 1 implies 0 = 1). Not possible.Similarly, (b = -1) gives the same.What about (b = 2)? Then, equation becomes (a^2 * 4 = a^2 + 4 implies 4a^2 - a^2 = 4 implies 3a^2 = 4 implies a^2 = 4/3). Not integer.Same for (b = -2).So, seems like the only integer solution is (0, 0). But wait, let me check (a = 0) and (b = 0) again. Plugging back in, it works.But hold on, let me think again about the factoring step. I had:(a^2 b^2 - a^2 - b^2 = 0), which I rewrote as ((a^2 - 1)(b^2 - 1) = 1). Is that correct?Let me expand ((a^2 - 1)(b^2 - 1)):(a^2 b^2 - a^2 - b^2 + 1). So, yes, that's correct. So, bringing the 1 to the other side, we have:(a^2 b^2 - a^2 - b^2 + 1 = 1 implies (a^2 - 1)(b^2 - 1) = 1). So, that's correct.So, since the product is 1, the only integer solutions are when both factors are 1 or both are -1. So, as we saw, only (0, 0) is a solution.Wait, but is that the only solution? Let me think about whether (a) or (b) can be zero without the other being zero.Suppose (a = 0), then the original equation becomes (0 = 0 + b^2), so (b^2 = 0), so (b = 0). Similarly, if (b = 0), then (a = 0). So, no, you can't have one of them zero without the other being zero.So, that suggests that (0, 0) is the only solution.But wait, let me think again. Maybe I missed something in the factoring.Wait, another way to approach this is to consider the original equation:(a^2 b^2 = a^2 + b^2)Let me rearrange it as:(a^2 b^2 - a^2 - b^2 = 0)Then, add 1 to both sides:(a^2 b^2 - a^2 - b^2 + 1 = 1)Which factors as:((a^2 - 1)(b^2 - 1) = 1)So, that's correct. So, the product of two integers is 1, so each must be 1 or -1.So, as we saw, if (a^2 - 1 = 1), then (a^2 = 2), which is not integer. If (a^2 -1 = -1), then (a^2 = 0), so (a = 0). Similarly for (b). So, only (0, 0) is a solution.Wait, but let me think about whether (a) or (b) can be 1 or -1. If (a = 1), then the equation becomes (1 * b^2 = 1 + b^2), which simplifies to (b^2 = 1 + b^2), leading to (0 = 1), which is impossible. Similarly for (a = -1). So, (a) can't be 1 or -1.Similarly, if (b = 1) or (b = -1), same result.So, yeah, seems like (0, 0) is the only solution.But wait, let me test (a = 0) and (b = 0). Plugging into the original equation: (0 * 0 = 0 + 0), which is (0 = 0). So, that works.Is there any other way to get solutions? Maybe if (a) or (b) is very large? Let me see.Suppose (a) is very large, say (a = 1000). Then, the equation becomes (1000^2 b^2 = 1000^2 + b^2). Let's write that as:(1000000 b^2 = 1000000 + b^2)Subtract (b^2) from both sides:(999999 b^2 = 1000000)So, (b^2 = 1000000 / 999999), which is approximately 1.000001, which is not an integer. So, no solution there.Similarly, for (a) approaching infinity, (b) approaches 1, but since (b) must be integer, it can't be exactly 1 because that leads to a contradiction.So, yeah, it seems like (0, 0) is the only integer solution.Wait, but let me think again. Maybe I missed some edge cases. For example, what if one of (a) or (b) is zero and the other is non-zero? Let's see.Suppose (a = 0), then the equation becomes (0 = 0 + b^2), so (b = 0). Similarly, if (b = 0), then (a = 0). So, no, you can't have one zero and the other non-zero.So, yeah, (0, 0) is the only solution.Wait, but hold on. Let me think about negative integers. Since squaring negative integers gives positive results, maybe negative values could lead to solutions. But as we saw earlier, plugging in negative values doesn't change the equation because of the squares. So, for example, if (a = -1), it's the same as (a = 1), which doesn't work. Similarly, (a = -2) is same as (a = 2), which also doesn't work. So, negative values don't give any new solutions.Therefore, the only integer solution is (0, 0).Wait, but let me check another approach. Maybe rearrange the original equation differently.Starting with (a^2 b^2 = a^2 + b^2), let's divide both sides by (a^2 b^2) (assuming (a) and (b) are non-zero, which we already saw leads to (0,0) being the only solution, but just to confirm):[1 = frac{1}{b^2} + frac{1}{a^2}]So, (1 = frac{1}{a^2} + frac{1}{b^2}). Let me denote (x = frac{1}{a^2}) and (y = frac{1}{b^2}). Then, (x + y = 1), where (x) and (y) are positive real numbers since (a) and (b) are non-zero integers.But since (a) and (b) are integers, (x) and (y) must be reciprocals of perfect squares. So, possible values for (x) are 1, 1/4, 1/9, 1/16, etc., and same for (y).So, let's list possible pairs:1. (x = 1), then (y = 0). But (y = 0) is not possible because (b) would have to be infinity, which isn't an integer.2. (x = 1/4), then (y = 3/4). But 3/4 isn't a reciprocal of a perfect square.3. (x = 1/9), then (y = 8/9). 8/9 isn't a reciprocal of a perfect square.4. (x = 1/16), then (y = 15/16). Not a reciprocal of a perfect square.5. (x = 1/25), then (y = 24/25). Not a reciprocal of a perfect square.Continuing this, it seems like none of these will result in (y) being a reciprocal of a perfect square. So, this approach also leads us to conclude that there are no solutions with (a) and (b) non-zero. Therefore, the only solution is (0, 0).Wait, but let me think again. Maybe I can consider the equation (a^2 b^2 = a^2 + b^2) and see if there's a way to express it differently. For example, maybe write it as:(a^2 b^2 - a^2 - b^2 = 0)Then, add 1 to both sides:(a^2 b^2 - a^2 - b^2 + 1 = 1)Which factors as:((a^2 - 1)(b^2 - 1) = 1)As we did before. So, same conclusion.Alternatively, maybe consider the equation as a quadratic in terms of (a^2). Let me try that.Starting with (a^2 b^2 = a^2 + b^2), let's rearrange:(a^2 b^2 - a^2 = b^2)Factor out (a^2):(a^2(b^2 - 1) = b^2)Then, solve for (a^2):(a^2 = frac{b^2}{b^2 - 1})Hmm, so (a^2) must be equal to (frac{b^2}{b^2 - 1}). Since (a^2) is an integer, (frac{b^2}{b^2 - 1}) must also be an integer. Let's denote (k = b^2 - 1), so (b^2 = k + 1). Then, (a^2 = frac{k + 1}{k} = 1 + frac{1}{k}).So, (a^2 = 1 + frac{1}{k}). Since (a^2) is an integer, (frac{1}{k}) must be an integer. Therefore, (k) must be a divisor of 1. So, (k = 1) or (k = -1).Case 1: (k = 1)Then, (b^2 = k + 1 = 2). So, (b = sqrt{2}) or (b = -sqrt{2}), which aren't integers.Case 2: (k = -1)Then, (b^2 = k + 1 = 0). So, (b = 0). Then, plugging back into the equation for (a^2):(a^2 = 1 + frac{1}{-1} = 1 - 1 = 0). So, (a = 0).So, again, we get the solution (a = 0), (b = 0).Therefore, this approach also confirms that (0, 0) is the only integer solution.Wait, but let me think about another angle. Maybe consider the equation as symmetric in (a) and (b). So, any solution ((a, b)) would imply ((b, a)) is also a solution. But since we've only found (0, 0), which is symmetric, that's consistent.Alternatively, maybe consider substituting specific values for (a) and (b) to see if any other solutions pop up. Let me try (a = 2), (b = 2):Left side: (4 * 4 = 16)Right side: (4 + 4 = 8)16 ≠ 8, so no.What about (a = 1), (b = 0):Left side: (1 * 0 = 0)Right side: (1 + 0 = 1)0 ≠ 1, so no.Similarly, (a = 0), (b = 1):Left side: (0 * 1 = 0)Right side: (0 + 1 = 1)0 ≠ 1, so no.What about (a = 2), (b = 0):Left side: (4 * 0 = 0)Right side: (4 + 0 = 4)0 ≠ 4, so no.So, seems like no other solutions.Wait, another thought: maybe consider the equation as (a^2 b^2 - a^2 - b^2 = 0), and then add 1 to both sides to get ((a^2 - 1)(b^2 - 1) = 1). So, as before, same conclusion.Alternatively, maybe consider the equation as (a^2 b^2 - a^2 - b^2 = 0), and then factor it as (a^2(b^2 - 1) - b^2 = 0), but that doesn't seem helpful.Alternatively, maybe write it as (a^2(b^2 - 1) = b^2), so (a^2 = frac{b^2}{b^2 - 1}). As before, same result.So, yeah, I think I've covered all possible approaches, and each time, the only solution is (0, 0).Therefore, the only integer solution is ((0, 0)).

answer：Alright, so I've got this geometry problem here. Let me try to parse it step by step.We have triangle ABC, with angle bisectors AD and BE intersecting at point I. So, I is the incenter of triangle ABC because it's where the angle bisectors meet. That makes sense.It's given that the area of triangle ABI is equal to the area of quadrilateral CDIE. So, [ABI] = [CDIE]. I need to find the maximum possible value of angle ACB.Hmm. Let me try to visualize this. Triangle ABC, with AD and BE as angle bisectors. Point I is the incenter. So, the incenter is equidistant from all sides, and it's the center of the incircle.Now, the areas: triangle ABI and quadrilateral CDIE. So, let me think about how these areas relate to each other.First, let me note that the incenter divides the angle bisectors in a specific ratio related to the sides of the triangle. But I don't remember the exact formula, so maybe I can derive it or use mass point geometry.Alternatively, maybe I can use area ratios since the areas are given to be equal.Let me recall that in a triangle, the inradius times the semiperimeter gives the area. But not sure if that's directly applicable here.Wait, maybe I can express the areas [ABI] and [CDIE] in terms of other areas in the triangle.Let me consider the entire triangle ABC. The incenter I divides it into three smaller triangles: ABI, BCI, and ACI. So, [ABC] = [ABI] + [BCI] + [ACI].But in this problem, we have [ABI] equal to [CDIE]. Hmm, CDIE is a quadrilateral. So, maybe I need to express [CDIE] in terms of other areas.Quadrilateral CDIE is made up of two triangles: CDI and CIE. So, [CDIE] = [CDI] + [CIE].So, [ABI] = [CDI] + [CIE].But [ABI] is one of the three smaller triangles, and [CDI] and [CIE] are parts of the other two smaller triangles, BCI and ACI.Wait, let me think. If I is the incenter, then it's equidistant from all sides. So, the heights from I to each side are equal, which is the inradius.Therefore, the areas of the smaller triangles ABI, BCI, and ACI can be expressed as (1/2)*side*inradius.So, [ABI] = (1/2)*AB*r, [BCI] = (1/2)*BC*r, and [ACI] = (1/2)*AC*r, where r is the inradius.So, the area of the whole triangle ABC is (1/2)*(AB + BC + AC)*r = r*s, where s is the semiperimeter.But in this problem, [ABI] = [CDIE]. Let me write that down:[ABI] = [CDIE]Which is:(1/2)*AB*r = [CDI] + [CIE]But [CDI] is part of [BCI], and [CIE] is part of [ACI]. So, perhaps I can express [CDI] and [CIE] in terms of the areas of BCI and ACI.Wait, but how? Maybe I need to consider the ratios in which point I divides the angle bisectors.Alternatively, maybe I can use mass point geometry or coordinate geometry.Wait, maybe coordinate geometry would be a good approach here. Let me try setting up coordinates for triangle ABC.Let me place point A at (0,0), point B at (c,0), and point C somewhere in the plane, say (d,e). But maybe that's too vague.Alternatively, maybe I can use barycentric coordinates with respect to triangle ABC. In barycentric coordinates, the incenter has coordinates proportional to the lengths of the sides.Wait, barycentric coordinates might complicate things, but maybe it's manageable.Alternatively, maybe I can use trigonometric relationships.Wait, let me think about the areas again.Given that [ABI] = [CDIE], and [ABI] is (1/2)*AB*r, as I mentioned earlier.But [CDIE] is [CDI] + [CIE]. Let me think about [CDI] and [CIE].Point I is the incenter, so it lies on angle bisectors AD and BE. So, maybe I can find the ratios AI/ID and BI/IE.Wait, in a triangle, the incenter divides the angle bisector in the ratio of (b + c)/a, where a, b, c are the lengths of the sides opposite to angles A, B, C respectively.Wait, let me recall the formula: The incenter divides the angle bisector from vertex A in the ratio of (b + c)/a.So, AI/ID = (b + c)/a.Similarly, BI/IE = (a + c)/b.So, maybe I can use these ratios to find the areas.Wait, if AI/ID = (b + c)/a, then AI = (b + c)/(a + b + c) * AD.Similarly, BI = (a + c)/(a + b + c) * BE.But maybe I can express the areas in terms of these ratios.Wait, [ABI] is the area of triangle ABI. Let me think about how to express this area.Since I is on AD, the area [ABI] can be expressed as a fraction of the area of triangle ABD.Wait, but I don't know where D is. D is the point where the angle bisector from A meets BC.Similarly, E is the point where the angle bisector from B meets AC.Wait, maybe I can use the angle bisector theorem.The angle bisector theorem states that BD/DC = AB/AC = c/b, if AB = c, AC = b, and BC = a.Similarly, AE/EC = AB/BC = c/a.So, BD = (c/(b + c)) * a, and DC = (b/(b + c)) * a.Similarly, AE = (c/(a + c)) * b, and EC = (a/(a + c)) * b.Hmm, okay. So, maybe I can express the areas in terms of these ratios.But wait, [ABI] is the area of triangle ABI. Let me think about how to compute that.Since I is the incenter, and it's located at the intersection of AD and BE, perhaps I can express [ABI] in terms of the areas of other triangles.Alternatively, maybe I can use the ratio of AI/ID to find the ratio of areas.Since AI/ID = (b + c)/a, then the area of ABI compared to the area of IBD would be in the same ratio, because they share the same base BI and their heights are proportional to AI and ID.Wait, is that correct?Wait, no, because ABI and IBD don't share the same base. Let me think again.Alternatively, perhaps I can consider the areas of triangles that share the same vertex.Wait, triangle ABI and triangle IBD share the vertex B, but their bases are AI and ID on the same line AD.So, the ratio of their areas would be equal to the ratio of AI to ID.So, [ABI]/[IBD] = AI/ID = (b + c)/a.Similarly, [ABI] = [(b + c)/a] * [IBD].But I'm not sure if that helps directly.Wait, maybe I can express [ABI] in terms of [ABC].Wait, [ABI] is part of [ABC], but it's not straightforward.Alternatively, since I is the incenter, maybe I can express [ABI] in terms of the inradius and the sides.Wait, earlier I thought [ABI] = (1/2)*AB*r, but is that correct?Wait, no, that's not correct because the inradius is the distance from I to BC, not necessarily to AB.Wait, actually, the inradius is the distance from I to all sides, so it's the same for all sides. So, the area of triangle ABI would be (1/2)*AB*r, because r is the height from I to AB.Similarly, [BCI] = (1/2)*BC*r, and [ACI] = (1/2)*AC*r.So, that's correct. So, [ABI] = (1/2)*AB*r, [BCI] = (1/2)*BC*r, [ACI] = (1/2)*AC*r.So, the total area [ABC] = (1/2)*(AB + BC + AC)*r = r*s, where s is the semiperimeter.So, [ABI] = (1/2)*AB*r, and [CDIE] is [CDI] + [CIE].But [CDI] is part of [BCI], and [CIE] is part of [ACI].Wait, so maybe I can express [CDI] and [CIE] in terms of [BCI] and [ACI].But how?Wait, point I is the incenter, so it's located at the intersection of the angle bisectors. So, in triangle BCI, point I is the incenter, but wait, no, I is the incenter of ABC, not necessarily of BCI.Wait, maybe I can think about the areas [CDI] and [CIE] in terms of the ratios in which I divides the angle bisectors.Wait, earlier I mentioned that AI/ID = (b + c)/a, and BI/IE = (a + c)/b.So, maybe the areas [ABI] and [CDIE] can be related through these ratios.Wait, let me think about [ABI]. It's (1/2)*AB*r.And [CDIE] is [CDI] + [CIE]. Let me try to express [CDI] and [CIE].In triangle BCI, point I is somewhere inside. So, [CDI] is the area of triangle CDI, which is part of BCI.Similarly, in triangle ACI, [CIE] is the area of triangle CIE, which is part of ACI.Wait, maybe I can express [CDI] as a fraction of [BCI], and [CIE] as a fraction of [ACI].But how?Wait, perhaps using the ratios in which I divides the angle bisectors.Wait, in triangle ABC, point I divides AD in the ratio AI/ID = (b + c)/a.Similarly, it divides BE in the ratio BI/IE = (a + c)/b.So, maybe in triangle BCI, point I divides the angle bisector from C? Wait, no, I is the incenter of ABC, so in triangle BCI, the incenter would be different.Wait, maybe I'm overcomplicating this.Alternatively, perhaps I can use mass point geometry to find the ratios.Wait, mass point assigns weights to the vertices based on the ratios of the sides.Given AI/ID = (b + c)/a, so mass at A is proportional to a, mass at D is proportional to (b + c). Similarly, for BI/IE = (a + c)/b, mass at B is proportional to b, mass at E is proportional to (a + c).But I'm not sure if that helps directly with the areas.Wait, maybe I can consider the areas in terms of the ratios.Let me denote AB = c, BC = a, AC = b.So, [ABI] = (1/2)*c*r.[CDIE] = [CDI] + [CIE].Now, [CDI] is part of [BCI], which is (1/2)*a*r.Similarly, [CIE] is part of [ACI], which is (1/2)*b*r.So, [CDI] = [BCI] - [BDI], and [CIE] = [ACI] - [AIE].Wait, but I don't know [BDI] or [AIE].Alternatively, maybe I can express [CDI] and [CIE] in terms of the ratios AI/ID and BI/IE.Wait, since AI/ID = (b + c)/a, then the ratio of areas [ABI]/[IBD] = (b + c)/a.Similarly, [ABI] = [(b + c)/a] * [IBD].But [ABI] = (1/2)*c*r, so [IBD] = [ABI] * (a)/(b + c) = (1/2)*c*r * (a)/(b + c).Similarly, in triangle BCI, [BCI] = (1/2)*a*r, and [IBD] is part of it.Wait, but [IBD] is part of triangle ABD, not BCI.Hmm, maybe I'm getting confused.Wait, perhaps I can think about the areas in terms of the ratios of the segments.Since AI/ID = (b + c)/a, then the area of ABI compared to the area of IBD is (b + c)/a.Similarly, since they share the same base BI, their areas are proportional to their heights from BI, which are AI and ID.So, [ABI]/[IBD] = AI/ID = (b + c)/a.So, [ABI] = [(b + c)/a] * [IBD].Similarly, [ABI] = (1/2)*c*r, so [IBD] = [ABI] * (a)/(b + c) = (1/2)*c*r * (a)/(b + c).Similarly, in triangle ABD, [ABD] = [ABI] + [IBD] = (1/2)*c*r + (1/2)*c*r*(a)/(b + c) = (1/2)*c*r*(1 + a/(b + c)) = (1/2)*c*r*((b + c) + a)/(b + c)) = (1/2)*c*r*(a + b + c)/(b + c).But [ABD] can also be expressed as (1/2)*AB*BD*sin(theta), where theta is the angle at B.Wait, but maybe that's not helpful.Alternatively, since D is on BC, BD = (c/(b + c)) * a, as per the angle bisector theorem.So, BD = (c/(b + c)) * a.So, the area of ABD is (1/2)*AB*BD*sin(angle at B).But AB = c, BD = (c/(b + c)) * a, and angle at B is angle ABC.So, [ABD] = (1/2)*c*(c*a/(b + c))*sin(angle ABC) = (1/2)*(c^2*a/(b + c))*sin(angle ABC).But earlier, we had [ABD] = (1/2)*c*r*(a + b + c)/(b + c).So, equating the two expressions:(1/2)*(c^2*a/(b + c))*sin(angle ABC) = (1/2)*c*r*(a + b + c)/(b + c).Simplify:(c^2*a/(b + c))*sin(angle ABC) = c*r*(a + b + c)/(b + c).Cancel out (c/(b + c)) from both sides:c*a*sin(angle ABC) = r*(a + b + c).But r = (Area of ABC)/s, where s = (a + b + c)/2.So, r = (Area of ABC)/( (a + b + c)/2 ) = 2*Area/(a + b + c).So, plugging back in:c*a*sin(angle ABC) = (2*Area/(a + b + c))*(a + b + c) = 2*Area.But Area of ABC is (1/2)*a*b*sin(angle ACB), wait no, actually, Area = (1/2)*ab*sin(C), but in our notation, sides are a, b, c opposite angles A, B, C respectively.Wait, let me clarify:In standard notation, side a is opposite angle A, side b opposite angle B, side c opposite angle C.So, Area of ABC = (1/2)*ab*sin(C) = (1/2)*bc*sin(A) = (1/2)*ac*sin(B).So, in our case, Area = (1/2)*ab*sin(C) = (1/2)*bc*sin(A) = (1/2)*ac*sin(B).So, in our earlier equation:c*a*sin(angle ABC) = 2*Area.But 2*Area = ab*sin(C) = bc*sin(A) = ac*sin(B).So, c*a*sin(B) = ac*sin(B) = 2*Area.Wait, but that's trivial because 2*Area = ac*sin(B). So, this equation is just confirming that.Hmm, so maybe that approach didn't give us new information.Let me try a different angle.Given that [ABI] = [CDIE], and [ABI] = (1/2)*c*r.[CDIE] = [CDI] + [CIE].Now, [CDI] is part of [BCI], which is (1/2)*a*r.Similarly, [CIE] is part of [ACI], which is (1/2)*b*r.So, maybe [CDI] = [BCI] - [BDI], and [CIE] = [ACI] - [AIE].But I don't know [BDI] or [AIE].Alternatively, perhaps I can express [CDI] and [CIE] in terms of the ratios AI/ID and BI/IE.Wait, since AI/ID = (b + c)/a, then the ratio of areas [ABI]/[IBD] = (b + c)/a.Similarly, [ABI] = (1/2)*c*r, so [IBD] = [ABI] * (a)/(b + c) = (1/2)*c*r*(a)/(b + c).Similarly, in triangle BCI, point I divides it into two parts: [CDI] and [IBD].Wait, no, [BCI] is (1/2)*a*r, and [IBD] is part of [ABD], not [BCI].Wait, maybe I'm mixing up the areas.Alternatively, perhaps I can consider the ratio of [CDI] to [BCI].Since I is the incenter, and D is on BC, maybe [CDI] is proportional to something.Wait, let me think about the coordinates approach again.Let me place triangle ABC in coordinate plane with point A at (0,0), point B at (c,0), and point C at coordinates (d,e).Then, the incenter I can be found using the formula:I = (a*A + b*B + c*C)/(a + b + c),where a, b, c are lengths of sides opposite to angles A, B, C.Wait, but in standard notation, a is BC, b is AC, c is AB.So, if I place A at (0,0), B at (c,0), and C at (d,e), then:Length AB = c,Length AC = sqrt(d^2 + e^2) = b,Length BC = sqrt( (d - c)^2 + e^2 ) = a.So, the incenter I has coordinates:I_x = (a*0 + b*c + c*d)/(a + b + c),I_y = (a*0 + b*0 + c*e)/(a + b + c).So, I = ( (b c + c d)/(a + b + c), (c e)/(a + b + c) ).Hmm, that seems complicated, but maybe manageable.Now, the area of triangle ABI can be computed using determinant formula.Points A(0,0), B(c,0), I( (b c + c d)/(a + b + c), (c e)/(a + b + c) ).So, [ABI] = (1/2)*| (c*(c e)/(a + b + c) - 0*(b c + c d)/(a + b + c) ) | = (1/2)*| c^2 e / (a + b + c) | = (1/2)*c^2 e / (a + b + c).Similarly, the area of quadrilateral CDIE.Points C(d,e), D, I, E.Wait, I need coordinates for D and E.Point D is on BC, and since AD is the angle bisector, by angle bisector theorem, BD/DC = AB/AC = c/b.So, BD = (c/(b + c)) * a,DC = (b/(b + c)) * a.Similarly, coordinates of D can be found by weighted average.Coordinates of B(c,0), C(d,e).So, D_x = (b*c + c*d)/(b + c),D_y = (b*0 + c*e)/(b + c) = (c e)/(b + c).Similarly, point E is on AC, and BE is the angle bisector.By angle bisector theorem, AE/EC = AB/BC = c/a.So, AE = (c/(a + c)) * b,EC = (a/(a + c)) * b.Coordinates of A(0,0), C(d,e).So, E_x = (a*0 + c*d)/(a + c) = (c d)/(a + c),E_y = (a*0 + c*e)/(a + c) = (c e)/(a + c).So, now we have coordinates for D and E.Now, quadrilateral CDIE is made up of points C(d,e), D( (b c + c d)/(b + c), (c e)/(b + c) ), I( (b c + c d)/(a + b + c), (c e)/(a + b + c) ), and E( (c d)/(a + c), (c e)/(a + c) ).To find the area of quadrilateral CDIE, we can use the shoelace formula.But this seems quite involved. Maybe there's a better way.Alternatively, perhaps I can express [CDIE] as [CDI] + [CIE].Compute [CDI] and [CIE] separately.First, [CDI]:Points C(d,e), D( (b c + c d)/(b + c), (c e)/(b + c) ), I( (b c + c d)/(a + b + c), (c e)/(a + b + c) ).Using determinant formula for area:[CDI] = (1/2)*| (D_x - C_x)*(I_y - C_y) - (D_y - C_y)*(I_x - C_x) |.Compute D_x - C_x = (b c + c d)/(b + c) - d = (b c + c d - d(b + c))/(b + c) = (b c + c d - b d - c d)/(b + c) = (b c - b d)/(b + c) = b(c - d)/(b + c).Similarly, I_y - C_y = (c e)/(a + b + c) - e = e*(c/(a + b + c) - 1) = e*( (c - a - b - c)/(a + b + c) ) = e*(- (a + b))/(a + b + c).Similarly, D_y - C_y = (c e)/(b + c) - e = e*(c/(b + c) - 1) = e*( (c - b - c)/(b + c) ) = e*(-b)/(b + c).I_x - C_x = (b c + c d)/(a + b + c) - d = (b c + c d - d(a + b + c))/(a + b + c) = (b c + c d - a d - b d - c d)/(a + b + c) = (b c - a d - b d)/(a + b + c).So, putting it all together:[CDI] = (1/2)*| [b(c - d)/(b + c)] * [ -e(a + b)/(a + b + c) ] - [ -e b/(b + c) ] * [ (b c - a d - b d)/(a + b + c) ] |.Simplify term by term:First term: [b(c - d)/(b + c)] * [ -e(a + b)/(a + b + c) ] = -b e (c - d)(a + b)/[(b + c)(a + b + c)].Second term: [ -e b/(b + c) ] * [ (b c - a d - b d)/(a + b + c) ] = -e b (b c - a d - b d)/[(b + c)(a + b + c)].So, [CDI] = (1/2)*| -b e (c - d)(a + b)/[(b + c)(a + b + c)] - (-e b (b c - a d - b d)/[(b + c)(a + b + c)]) |.Simplify the expression inside the absolute value:= (1/2)*| [ -b e (c - d)(a + b) + e b (b c - a d - b d) ] / [ (b + c)(a + b + c) ] |.Factor out e b:= (1/2)*| e b [ - (c - d)(a + b) + (b c - a d - b d) ] / [ (b + c)(a + b + c) ] |.Now, expand the terms inside the brackets:First term: - (c - d)(a + b) = -c(a + b) + d(a + b).Second term: (b c - a d - b d).So, combining:- c(a + b) + d(a + b) + b c - a d - b d.Simplify term by term:- c(a + b) + d(a + b) + b c - a d - b d.= -a c - b c + a d + b d + b c - a d - b d.Now, let's cancel terms:- a c - b c + a d + b d + b c - a d - b d.- a c remains.- b c cancels with + b c.+ a d cancels with - a d.+ b d cancels with - b d.So, we're left with -a c.So, the numerator becomes e b (-a c).Thus,[CDI] = (1/2)*| e b (-a c) / [ (b + c)(a + b + c) ] | = (1/2)*(a b c e)/( (b + c)(a + b + c) ).Similarly, now compute [CIE].Points C(d,e), I( (b c + c d)/(a + b + c), (c e)/(a + b + c) ), E( (c d)/(a + c), (c e)/(a + c) ).Using determinant formula:[CIE] = (1/2)*| (I_x - C_x)*(E_y - C_y) - (I_y - C_y)*(E_x - C_x) |.Compute I_x - C_x = (b c + c d)/(a + b + c) - d = (b c + c d - d(a + b + c))/(a + b + c) = (b c - a d - b d)/(a + b + c).E_y - C_y = (c e)/(a + c) - e = e*(c/(a + c) - 1) = e*( (c - a - c)/(a + c) ) = -a e/(a + c).I_y - C_y = (c e)/(a + b + c) - e = e*(c/(a + b + c) - 1) = e*( (c - a - b - c)/(a + b + c) ) = -e(a + b)/(a + b + c).E_x - C_x = (c d)/(a + c) - d = d*(c/(a + c) - 1) = d*( (c - a - c)/(a + c) ) = -a d/(a + c).So, putting it all together:[CIE] = (1/2)*| [ (b c - a d - b d)/(a + b + c) ] * [ -a e/(a + c) ] - [ -e(a + b)/(a + b + c) ] * [ -a d/(a + c) ] |.Simplify term by term:First term: [ (b c - a d - b d)/(a + b + c) ] * [ -a e/(a + c) ] = -a e (b c - a d - b d)/[ (a + b + c)(a + c) ].Second term: [ -e(a + b)/(a + b + c) ] * [ -a d/(a + c) ] = e(a + b) a d/[ (a + b + c)(a + c) ].So, [CIE] = (1/2)*| -a e (b c - a d - b d) + e(a + b) a d | / [ (a + b + c)(a + c) ].Factor out e a:= (1/2)*| e a [ - (b c - a d - b d) + (a + b) d ] | / [ (a + b + c)(a + c) ].Simplify inside the brackets:- (b c - a d - b d) + (a + b) d = -b c + a d + b d + a d + b d.Wait, let me expand:- (b c - a d - b d) = -b c + a d + b d.Then, + (a + b) d = +a d + b d.So, combining:- b c + a d + b d + a d + b d = -b c + 2 a d + 2 b d.So, numerator becomes e a (-b c + 2 a d + 2 b d).Thus,[CIE] = (1/2)*| e a (-b c + 2 a d + 2 b d) | / [ (a + b + c)(a + c) ].Hmm, this is getting quite complicated. Maybe there's a better approach.Wait, earlier I computed [ABI] = (1/2)*c^2 e / (a + b + c).And [CDIE] = [CDI] + [CIE] = (1/2)*(a b c e)/( (b + c)(a + b + c) ) + (1/2)*| e a (-b c + 2 a d + 2 b d) | / [ (a + b + c)(a + c) ].But this seems too messy. Maybe I should relate the areas in terms of the sides and angles.Wait, another approach: Since [ABI] = [CDIE], and [ABI] = (1/2)*c*r, [CDIE] is [CDI] + [CIE].But [CDI] is part of [BCI], and [CIE] is part of [ACI].So, [CDIE] = [CDI] + [CIE] = [BCI] - [BDI] + [ACI] - [AIE].But [BCI] = (1/2)*a*r, [ACI] = (1/2)*b*r.So, [CDIE] = (1/2)*a*r + (1/2)*b*r - ([BDI] + [AIE]).But [ABI] = [CDIE], so:(1/2)*c*r = (1/2)*(a + b)*r - ([BDI] + [AIE]).Thus,[BDI] + [AIE] = (1/2)*(a + b)*r - (1/2)*c*r = (1/2)*(a + b - c)*r.Hmm, interesting.But I don't know [BDI] and [AIE].Wait, but maybe I can relate [BDI] and [AIE] to other areas.Wait, [BDI] is part of [ABD], which is (1/2)*AB*BD*sin(angle at B).Similarly, [AIE] is part of [ABE], which is (1/2)*AB*AE*sin(angle at A).But I'm not sure.Wait, maybe I can express [BDI] and [AIE] in terms of the ratios AI/ID and BI/IE.Earlier, we had AI/ID = (b + c)/a, so [ABI]/[IBD] = (b + c)/a.Similarly, [ABI] = (1/2)*c*r, so [IBD] = [ABI]*(a)/(b + c) = (1/2)*c*r*(a)/(b + c).Similarly, for [AIE], since BI/IE = (a + c)/b, then [ABI]/[AIE] = (a + c)/b.Wait, is that correct?Wait, point I divides BE in the ratio BI/IE = (a + c)/b.So, the areas [ABI] and [AIE] share the same base AI, but their heights from AI are proportional to BI and IE.Thus, [ABI]/[AIE] = BI/IE = (a + c)/b.So, [ABI] = (a + c)/b * [AIE].Thus, [AIE] = [ABI]*(b)/(a + c) = (1/2)*c*r*(b)/(a + c).So, [BDI] = (1/2)*c*r*(a)/(b + c),[AIE] = (1/2)*c*r*(b)/(a + c).Thus, [BDI] + [AIE] = (1/2)*c*r*(a)/(b + c) + (1/2)*c*r*(b)/(a + c).We had earlier:[BDI] + [AIE] = (1/2)*(a + b - c)*r.So,(1/2)*c*r*(a)/(b + c) + (1/2)*c*r*(b)/(a + c) = (1/2)*(a + b - c)*r.Divide both sides by (1/2)*r:c*(a)/(b + c) + c*(b)/(a + c) = (a + b - c).So,(c a)/(b + c) + (c b)/(a + c) = a + b - c.Let me write this equation:(c a)/(b + c) + (c b)/(a + c) = a + b - c.Let me denote this as equation (1).Now, I need to solve this equation for the sides a, b, c.But since in a triangle, the sides must satisfy triangle inequalities, and we're looking for the maximum angle ACB, which is angle C.So, perhaps I can express this equation in terms of angles.Using the Law of Sines, we have:a / sin A = b / sin B = c / sin C = 2R,where R is the circumradius.So, let me denote:a = 2R sin A,b = 2R sin B,c = 2R sin C.So, substituting into equation (1):( (2R sin C)(2R sin A) ) / (2R sin B + 2R sin C) + ( (2R sin C)(2R sin B) ) / (2R sin A + 2R sin C) = 2R sin A + 2R sin B - 2R sin C.Simplify:(4 R^2 sin C sin A)/(2R (sin B + sin C)) + (4 R^2 sin C sin B)/(2R (sin A + sin C)) = 2R (sin A + sin B - sin C).Cancel out 2R from denominators:(2 R sin C sin A)/(sin B + sin C) + (2 R sin C sin B)/(sin A + sin C) = 2R (sin A + sin B - sin C).Divide both sides by 2R:( sin C sin A )/(sin B + sin C) + ( sin C sin B )/(sin A + sin C) = sin A + sin B - sin C.Let me denote S_A = sin A, S_B = sin B, S_C = sin C.So, the equation becomes:( S_C S_A )/(S_B + S_C) + ( S_C S_B )/(S_A + S_C) = S_A + S_B - S_C.Let me write this as:[ S_A S_C ] / (S_B + S_C) + [ S_B S_C ] / (S_A + S_C) = S_A + S_B - S_C.Let me combine the two terms on the left:= S_C [ S_A / (S_B + S_C) + S_B / (S_A + S_C) ].Let me compute the expression inside the brackets:S_A / (S_B + S_C) + S_B / (S_A + S_C).To combine these fractions, find a common denominator:= [ S_A (S_A + S_C) + S_B (S_B + S_C) ] / [ (S_B + S_C)(S_A + S_C) ].Expand numerator:= [ S_A^2 + S_A S_C + S_B^2 + S_B S_C ] / [ (S_B + S_C)(S_A + S_C) ].Factor numerator:= [ S_A^2 + S_B^2 + S_C(S_A + S_B) ] / [ (S_B + S_C)(S_A + S_C) ].So, the left side becomes:S_C * [ S_A^2 + S_B^2 + S_C(S_A + S_B) ] / [ (S_B + S_C)(S_A + S_C) ].Set equal to right side:S_C * [ S_A^2 + S_B^2 + S_C(S_A + S_B) ] / [ (S_B + S_C)(S_A + S_C) ] = S_A + S_B - S_C.Multiply both sides by [ (S_B + S_C)(S_A + S_C) ]:S_C [ S_A^2 + S_B^2 + S_C(S_A + S_B) ] = (S_A + S_B - S_C)(S_B + S_C)(S_A + S_C).This is getting quite complicated, but let's try to expand both sides.Left side:S_C [ S_A^2 + S_B^2 + S_A S_C + S_B S_C ].= S_C S_A^2 + S_C S_B^2 + S_A S_C^2 + S_B S_C^2.Right side:(S_A + S_B - S_C)(S_B + S_C)(S_A + S_C).First, compute (S_B + S_C)(S_A + S_C):= S_B S_A + S_B S_C + S_C S_A + S_C^2.= S_A S_B + S_B S_C + S_A S_C + S_C^2.Now, multiply by (S_A + S_B - S_C):= (S_A + S_B - S_C)(S_A S_B + S_B S_C + S_A S_C + S_C^2).Let me denote this as:= (S_A + S_B - S_C)(S_A S_B + S_B S_C + S_A S_C + S_C^2).Let me expand term by term:First, multiply S_A by each term:= S_A*(S_A S_B) + S_A*(S_B S_C) + S_A*(S_A S_C) + S_A*(S_C^2).= S_A^2 S_B + S_A S_B S_C + S_A^2 S_C + S_A S_C^2.Next, multiply S_B by each term:= S_B*(S_A S_B) + S_B*(S_B S_C) + S_B*(S_A S_C) + S_B*(S_C^2).= S_A S_B^2 + S_B^2 S_C + S_A S_B S_C + S_B S_C^2.Next, multiply (-S_C) by each term:= -S_C*(S_A S_B) - S_C*(S_B S_C) - S_C*(S_A S_C) - S_C*(S_C^2).= -S_A S_B S_C - S_B S_C^2 - S_A S_C^2 - S_C^3.Now, combine all these terms:= [S_A^2 S_B + S_A S_B S_C + S_A^2 S_C + S_A S_C^2] + [S_A S_B^2 + S_B^2 S_C + S_A S_B S_C + S_B S_C^2] + [ -S_A S_B S_C - S_B S_C^2 - S_A S_C^2 - S_C^3 ].Let me collect like terms:1. Terms with S_A^2 S_B: 1 term.2. Terms with S_A S_B S_C: 1 + 1 -1 = 1 term.3. Terms with S_A^2 S_C: 1 term.4. Terms with S_A S_C^2: 1 -1 = 0.5. Terms with S_A S_B^2: 1 term.6. Terms with S_B^2 S_C: 1 term.7. Terms with S_B S_C^2: 1 -1 = 0.8. Terms with -S_C^3: 1 term.So, combining:= S_A^2 S_B + S_A S_B S_C + S_A^2 S_C + S_A S_B^2 + S_B^2 S_C - S_C^3.So, right side simplifies to:S_A^2 S_B + S_A S_B S_C + S_A^2 S_C + S_A S_B^2 + S_B^2 S_C - S_C^3.Now, let's write both sides:Left side: S_C S_A^2 + S_C S_B^2 + S_A S_C^2 + S_B S_C^2.Right side: S_A^2 S_B + S_A S_B S_C + S_A^2 S_C + S_A S_B^2 + S_B^2 S_C - S_C^3.Bring all terms to left side:Left - Right = 0.So,S_C S_A^2 + S_C S_B^2 + S_A S_C^2 + S_B S_C^2 - S_A^2 S_B - S_A S_B S_C - S_A^2 S_C - S_A S_B^2 - S_B^2 S_C + S_C^3 = 0.Let me rearrange terms:Group terms with S_A^2:S_C S_A^2 - S_A^2 S_B - S_A^2 S_C = S_A^2 (S_C - S_B - S_C) = - S_A^2 S_B.Terms with S_B^2:S_C S_B^2 - S_A S_B^2 - S_B^2 S_C = S_B^2 (S_C - S_A - S_C) = - S_B^2 S_A.Terms with S_A S_B S_C:- S_A S_B S_C.Terms with S_A S_C^2 and S_B S_C^2:S_A S_C^2 + S_B S_C^2.Terms with S_C^3:+ S_C^3.So, overall:- S_A^2 S_B - S_B^2 S_A - S_A S_B S_C + S_A S_C^2 + S_B S_C^2 + S_C^3 = 0.Factor terms:Factor - S_A S_B from first three terms:- S_A S_B (S_A + S_B + S_C) + S_C^2 (S_A + S_B) + S_C^3 = 0.Wait, let me see:- S_A^2 S_B - S_B^2 S_A - S_A S_B S_C = - S_A S_B (S_A + S_B + S_C).Then, S_A S_C^2 + S_B S_C^2 + S_C^3 = S_C^2 (S_A + S_B + S_C).So, overall:- S_A S_B (S_A + S_B + S_C) + S_C^2 (S_A + S_B + S_C) = 0.Factor out (S_A + S_B + S_C):(S_A + S_B + S_C)( - S_A S_B + S_C^2 ) = 0.Since S_A + S_B + S_C ≠ 0 (as angles are between 0 and π, so sines are positive), we have:- S_A S_B + S_C^2 = 0.Thus,S_C^2 = S_A S_B.So,sin^2 C = sin A sin B.Hmm, interesting. So, we have sin^2 C = sin A sin B.But in triangle ABC, angles A + B + C = π.So, perhaps we can express sin A sin B in terms of sin^2 C.Let me recall that in a triangle, A + B = π - C.So, using the identity:sin A sin B = [cos(A - B) - cos(A + B)] / 2.Since A + B = π - C,sin A sin B = [cos(A - B) - cos(π - C)] / 2 = [cos(A - B) + cos C] / 2.So,sin^2 C = [cos(A - B) + cos C] / 2.Multiply both sides by 2:2 sin^2 C = cos(A - B) + cos C.But 2 sin^2 C = 1 - cos 2C.So,1 - cos 2C = cos(A - B) + cos C.Rearrange:cos(A - B) = 1 - cos 2C - cos C.Hmm, not sure if helpful.Alternatively, let me use another identity.We have sin^2 C = sin A sin B.But in triangle, A + B = π - C.So, let me set A = π - C - B.Thus,sin A = sin(π - C - B) = sin(C + B).Wait, no, sin(π - x) = sin x, so sin A = sin(C + B).Wait, but A = π - B - C, so sin A = sin(B + C).So,sin A = sin(B + C) = sin B cos C + cos B sin C.So,sin A sin B = [sin B cos C + cos B sin C] sin B = sin^2 B cos C + sin B cos B sin C.So, sin^2 C = sin^2 B cos C + sin B cos B sin C.Let me write this as:sin^2 C - sin^2 B cos C - sin B cos B sin C = 0.Factor sin C:sin C (sin C - sin B cos B) - sin^2 B cos C = 0.Hmm, not sure.Alternatively, let me divide both sides by sin C (assuming sin C ≠ 0, which it isn't in a triangle):sin C = sin B cos C + cos B sin B.Wait, that is:sin C = sin B (cos C + cos B).Wait, that seems interesting.But in triangle, C = π - A - B.So, sin C = sin(A + B) = sin A cos B + cos A sin B.So,sin A cos B + cos A sin B = sin B (cos C + cos B).Hmm, let me substitute cos C.Since C = π - A - B,cos C = -cos(A + B) = sin A sin B - cos A cos B.So,sin C = sin B ( (sin A sin B - cos A cos B) + cos B ).= sin B ( sin A sin B - cos A cos B + cos B ).= sin B ( sin A sin B + cos B (1 - cos A) ).Hmm, not sure.Alternatively, let me use the identity:sin C = sin(A + B) = sin A cos B + cos A sin B.So, from earlier:sin C = sin B (cos C + cos B).Thus,sin A cos B + cos A sin B = sin B cos C + sin B cos B.Bring all terms to left:sin A cos B + cos A sin B - sin B cos C - sin B cos B = 0.Factor sin B:sin A cos B + sin B (cos A - cos C - cos B) = 0.Hmm, not helpful.Alternatively, let me use the fact that sin^2 C = sin A sin B.And in triangle, A + B + C = π.Let me set variables:Let me denote angle C = γ, so angles A + B = π - γ.Let me denote angle A = α, angle B = β, so α + β = π - γ.We have sin^2 γ = sin α sin β.We need to find the maximum possible γ.So, express sin α sin β in terms of γ.Since α + β = π - γ,sin α sin β = [cos(α - β) - cos(α + β)] / 2 = [cos(α - β) - cos(π - γ)] / 2 = [cos(α - β) + cos γ] / 2.So,sin^2 γ = [cos(α - β) + cos γ] / 2.Multiply both sides by 2:2 sin^2 γ = cos(α - β) + cos γ.But 2 sin^2 γ = 1 - cos 2γ.So,1 - cos 2γ = cos(α - β) + cos γ.Rearrange:cos(α - β) = 1 - cos 2γ - cos γ.But I need to find the maximum γ.Let me consider that the maximum γ occurs when α = β, because if α ≠ β, then cos(α - β) < 1, which would make the right side smaller, possibly allowing γ to be larger.Wait, actually, if α = β, then cos(α - β) = 1, which would give:1 = 1 - cos 2γ - cos γ.Thus,0 = - cos 2γ - cos γ.So,cos 2γ + cos γ = 0.Using identity cos 2γ = 2 cos^2 γ - 1,So,2 cos^2 γ - 1 + cos γ = 0.Let me write this as:2 cos^2 γ + cos γ - 1 = 0.Let me solve for cos γ:Let x = cos γ,2x^2 + x - 1 = 0.Solutions:x = [ -1 ± sqrt(1 + 8) ] / 4 = [ -1 ± 3 ] / 4.So,x = (2)/4 = 1/2, or x = (-4)/4 = -1.But cos γ = -1 would imply γ = π, which is impossible in a triangle.So, cos γ = 1/2, so γ = π/3 or 60 degrees.So, when α = β, we get γ = 60 degrees.But is this the maximum?Wait, suppose α ≠ β, then cos(α - β) < 1, so the right side becomes 1 - cos 2γ - cos γ < 1 - cos 2γ - cos γ when α = β.Wait, actually, when α ≠ β, cos(α - β) < 1, so 1 - cos 2γ - cos γ = cos(α - β) < 1, so 1 - cos 2γ - cos γ < 1.Which implies that - cos 2γ - cos γ < 0,So,cos 2γ + cos γ > 0.But when γ increases, cos γ decreases, and cos 2γ = 2 cos^2 γ - 1 also decreases.So, it's possible that for larger γ, cos 2γ + cos γ becomes negative, which would not satisfy the equation.Wait, but in the case when α ≠ β, we have:cos(α - β) = 1 - cos 2γ - cos γ.If cos(α - β) < 1, then 1 - cos 2γ - cos γ < 1,Which implies - cos 2γ - cos γ < 0,So,cos 2γ + cos γ > 0.So, for the equation to hold, cos 2γ + cos γ must be greater than or equal to -1 (since cos(α - β) ≥ -1).But in the case when α = β, we have cos(α - β) = 1, which gives the maximum possible value on the left side.Thus, when α = β, we get the maximum γ, which is 60 degrees.Wait, but is that the case?Wait, let me think again.If α ≠ β, then cos(α - β) < 1, so 1 - cos 2γ - cos γ = cos(α - β) < 1,Which implies that 1 - cos 2γ - cos γ < 1,So,- cos 2γ - cos γ < 0,So,cos 2γ + cos γ > 0.So, for the equation to hold, we need cos 2γ + cos γ > 0.But when γ increases beyond 60 degrees, cos γ decreases, and cos 2γ decreases as well.Let me compute cos 2γ + cos γ for γ = 60 degrees:cos 120 + cos 60 = (-1/2) + (1/2) = 0.For γ = 90 degrees:cos 180 + cos 90 = (-1) + 0 = -1 < 0.So, for γ > 60 degrees, cos 2γ + cos γ becomes negative, which would make the equation impossible because cos(α - β) cannot be less than -1.Wait, but if cos(α - β) = 1 - cos 2γ - cos γ,and if 1 - cos 2γ - cos γ < -1,then cos(α - β) < -1, which is impossible because cosine is bounded below by -1.Thus, the equation can only hold when 1 - cos 2γ - cos γ ≥ -1,Which implies,- cos 2γ - cos γ ≥ -2,Which is always true because cos 2γ + cos γ ≤ 2.But more importantly, for the equation to hold, 1 - cos 2γ - cos γ must be ≥ -1,Which is,- cos 2γ - cos γ ≥ -2,Which is,cos 2γ + cos γ ≤ 2,Which is always true because cos 2γ + cos γ ≤ 2.But the critical point is when 1 - cos 2γ - cos γ = -1,Which gives,cos 2γ + cos γ = 2,Which only occurs when cos 2γ = 1 and cos γ = 1,Which implies γ = 0, which is impossible.So, the maximum γ occurs when cos(α - β) is maximized, which is when α = β, giving γ = 60 degrees.Thus, the maximum possible angle ACB is 60 degrees.Wait, but let me check with γ = 90 degrees.If γ = 90 degrees, then sin^2 γ = 1.And sin A sin B = sin A sin (90 - A) = sin A cos A = (1/2) sin 2A.The maximum of sin 2A is 1, so sin A sin B ≤ 1/2.But sin^2 γ = 1 > 1/2, so it's impossible.Thus, γ cannot be 90 degrees.Similarly, for γ = 60 degrees, sin^2 γ = 3/4.And sin A sin B = sin A sin (120 - A).Let me compute sin A sin (120 - A).Using identity:sin A sin (120 - A) = [cos(A - (120 - A)) - cos(A + (120 - A))]/2 = [cos(2A - 120) - cos 120]/2.= [cos(2A - 120) - (-1/2)] / 2 = [cos(2A - 120) + 1/2]/2.The maximum of cos(2A - 120) is 1, so maximum of sin A sin (120 - A) is (1 + 1/2)/2 = 3/4.Which equals sin^2 60 = 3/4.Thus, when A = B = 60 degrees, sin A sin B = 3/4, which equals sin^2 C.Thus, the maximum angle C is 60 degrees.Therefore, the maximum possible value of angle ACB is 60 degrees, or π/3 radians.

answer：Okay, so I need to find the cosine of the angle between vectors AB and AC. Hmm, let me think about how to approach this. I remember that the cosine of the angle between two vectors can be found using the dot product formula. The formula is something like the dot product of the two vectors divided by the product of their magnitudes. Let me write that down to make sure I have it right.So, if I have two vectors, say vector u and vector v, then the cosine of the angle θ between them is:cosθ = (u · v) / (|u| |v|)Right, that seems familiar. So, I need to find vectors AB and AC first, then compute their dot product, and then divide that by the product of their magnitudes. That should give me the cosine of the angle between them.Alright, let's start by finding vectors AB and AC. To find vector AB, I subtract the coordinates of point A from point B. Similarly, for vector AC, I subtract the coordinates of point A from point C.Given points:- A is at (0, 0, 4)- B is at (-3, -6, 1)- C is at (-5, -10, -1)So, vector AB = B - A = (-3 - 0, -6 - 0, 1 - 4) = (-3, -6, -3)Similarly, vector AC = C - A = (-5 - 0, -10 - 0, -1 - 4) = (-5, -10, -5)Wait, let me double-check that subtraction. For AB, x-coordinate is -3 - 0 = -3, y-coordinate is -6 - 0 = -6, z-coordinate is 1 - 4 = -3. Yeah, that looks right. For AC, x is -5 - 0 = -5, y is -10 - 0 = -10, z is -1 - 4 = -5. Okay, so vectors AB and AC are (-3, -6, -3) and (-5, -10, -5) respectively.Now, I need to compute the dot product of AB and AC. The dot product is calculated by multiplying corresponding components and then adding them up.So, AB · AC = (-3)*(-5) + (-6)*(-10) + (-3)*(-5)Let me compute each term:First term: (-3)*(-5) = 15Second term: (-6)*(-10) = 60Third term: (-3)*(-5) = 15Adding them up: 15 + 60 + 15 = 90So, the dot product is 90.Next, I need to find the magnitudes of vectors AB and AC. The magnitude of a vector is found by taking the square root of the sum of the squares of its components.First, let's find |AB|:|AB| = sqrt[ (-3)^2 + (-6)^2 + (-3)^2 ]Calculating each square:(-3)^2 = 9(-6)^2 = 36(-3)^2 = 9Adding them up: 9 + 36 + 9 = 54So, |AB| = sqrt(54). Hmm, sqrt(54) can be simplified. Since 54 = 9*6, sqrt(54) = 3*sqrt(6). So, |AB| = 3√6.Now, let's find |AC|:|AC| = sqrt[ (-5)^2 + (-10)^2 + (-5)^2 ]Calculating each square:(-5)^2 = 25(-10)^2 = 100(-5)^2 = 25Adding them up: 25 + 100 + 25 = 150So, |AC| = sqrt(150). Similarly, sqrt(150) can be simplified. 150 = 25*6, so sqrt(150) = 5*sqrt(6). Therefore, |AC| = 5√6.Alright, so now I have the dot product and the magnitudes. Let's plug them into the formula:cosθ = (AB · AC) / (|AB| |AC|) = 90 / (3√6 * 5√6)Let me compute the denominator first. 3√6 multiplied by 5√6 is:3*5 = 15√6*√6 = 6So, 15*6 = 90So, the denominator is 90.Therefore, cosθ = 90 / 90 = 1.Wait, that can't be right. If the cosine of the angle is 1, that means the angle is 0 degrees, meaning the vectors are in the same direction. But looking at vectors AB and AC, they are (-3, -6, -3) and (-5, -10, -5). Hmm, actually, if I look at these vectors, they seem to be scalar multiples of each other.Let me check: If I multiply vector AB by (5/3), do I get vector AC?Multiplying AB by 5/3:(-3)*(5/3) = -5(-6)*(5/3) = -10(-3)*(5/3) = -5Yes! So, vector AC is (5/3) times vector AB. That means they are scalar multiples, so they point in the same direction. Therefore, the angle between them is indeed 0 degrees, and the cosine is 1.Wait, but let me make sure I didn't make a mistake in my calculations. Let me go through each step again.First, vectors AB and AC:AB = (-3, -6, -3)AC = (-5, -10, -5)Dot product: (-3)*(-5) + (-6)*(-10) + (-3)*(-5) = 15 + 60 + 15 = 90. That seems correct.Magnitudes:|AB| = sqrt[9 + 36 + 9] = sqrt[54] = 3√6|AC| = sqrt[25 + 100 + 25] = sqrt[150] = 5√6So, |AB| |AC| = 3√6 * 5√6 = 15*6 = 90Therefore, cosθ = 90 / 90 = 1. So, yes, that's correct.So, the cosine of the angle between vectors AB and AC is 1.But wait, just to make sure, is there another way to think about this? Maybe by looking at the coordinates or something else?Alternatively, since AC is a scalar multiple of AB, they must be colinear, meaning the angle between them is 0 degrees, so cosine is 1. That makes sense.Alternatively, if I didn't notice that they are scalar multiples, I could have gone through the calculations as I did, which also led me to the same conclusion.So, yeah, I think that's solid. The cosine is 1.**Final Answer**The cosine of the angle between vectors overrightarrow{A B} and overrightarrow{A C} is boxed{1}.

answer：Okay, so I have this problem about the pressure of wind on a sail. It says that the pressure (P) varies jointly as the area (A) of the sail and the square of the velocity (V) of the wind. Hmm, joint variation. I remember that joint variation means that P is proportional to both A and V squared. So, in mathematical terms, that should be P = k * A * V², where k is the constant of proportionality.Alright, the problem gives me some specific information. It says that the pressure on a square foot is 1 pound when the velocity is 16 miles per hour. So, let me write that down. When A = 1 square foot and V = 16 mph, P = 1 pound. I can use this information to find the constant k.Plugging those values into the equation: 1 = k * 1 * (16)². Let me compute 16 squared. 16 * 16 is 256. So, 1 = k * 256. To solve for k, I divide both sides by 256. So, k = 1 / 256. Got that.Now, the question is asking for the velocity when the pressure on a square yard is 36 pounds. Hmm, okay. So, this time, the area is a square yard. Wait, I need to make sure about the units because earlier it was a square foot. I think 1 yard is 3 feet, so a square yard is 9 square feet. So, A = 9 square feet.And the pressure P is 36 pounds. So, I can plug these into the equation P = k * A * V². I already know k is 1/256, A is 9, and P is 36. Let me write that out: 36 = (1/256) * 9 * V².I need to solve for V. Let me first compute (1/256) * 9. That's 9/256. So, 36 = (9/256) * V². To solve for V², I can multiply both sides by 256/9. Let me do that:V² = 36 * (256/9).Simplify that. 36 divided by 9 is 4, so 4 * 256. 4 * 256 is 1024. So, V² = 1024. To find V, I take the square root of 1024. Hmm, what's the square root of 1024? I know that 32 squared is 1024 because 30 squared is 900, and 32 squared is 1024. So, V = 32.Wait, let me double-check my steps because 32 seems a bit high, but maybe it's correct. So, starting from the beginning: P varies jointly as A and V squared, so P = kAV². Given that when A is 1 square foot and V is 16 mph, P is 1 pound. So, 1 = k * 1 * 16², which gives k = 1/256. Then, when A is 9 square feet (since 1 square yard is 9 square feet) and P is 36 pounds, plugging into the equation: 36 = (1/256)*9*V². Multiply both sides by 256/9: V² = 36*(256/9) = 4*256 = 1024. Square root of 1024 is 32. So, V is 32 mph.Looking at the options, C is 32 mph. So, that should be the answer.But wait, let me just think again about the units. The initial pressure was given per square foot, and the second pressure is per square yard. Since 1 square yard is 9 square feet, the area is 9 times larger. So, if the pressure is 36 pounds on 9 square feet, that's 4 pounds per square foot. So, is that consistent?Wait, so initially, at 16 mph, the pressure was 1 pound per square foot. Now, it's 4 pounds per square foot. Since pressure varies with the square of velocity, so if pressure increases by a factor of 4, velocity should increase by a factor of 2. Because (V2/V1)² = P2/P1. So, (V2/16)² = 4/1, so V2/16 = 2, so V2 = 32. Yeah, that makes sense. So, 32 mph is correct.I think that's solid. So, the answer is 32 mph, which is option C.**Final Answer**boxed{C}

answer：Okay, so I have this problem where there's a big cube, 13x13x13, and all its surfaces are painted red. Then it's cut into 1x1x1 small cubes. After that, all the faces of these small cubes that weren't painted red are painted blue. The question is asking by what factor the total blue area is greater than the total red area.Hmm, let me try to visualize this. So, the big cube is painted entirely red on the outside, then it's broken down into small cubes. Each small cube can have 0, 1, 2, or 3 red faces, depending on their position in the big cube. The ones on the corners will have 3 red faces, the ones on the edges (but not corners) will have 2 red faces, and the ones on the faces (but not edges or corners) will have 1 red face. The ones inside the big cube, not on any face, won't have any red faces.So, for each small cube, the number of red faces depends on its position. Then, the blue faces are the ones that weren't painted red. Since each small cube has 6 faces, the number of blue faces per small cube would be 6 minus the number of red faces.Therefore, the total red area is the sum of all red faces across all small cubes, and the total blue area is the sum of all blue faces across all small cubes. Then, we need to find the ratio of blue area to red area.Let me break it down step by step.First, let's find the total number of small cubes. Since the big cube is 13x13x13, the total number of small cubes is 13^3, which is 2197.Now, each small cube can be categorized based on its position:1. Corner cubes: These are the cubes at the corners of the big cube. Each corner cube has 3 red faces. A cube has 8 corners, so there are 8 corner cubes.2. Edge cubes: These are the cubes along the edges but not at the corners. Each edge of the big cube has (13 - 2) = 11 such cubes because we exclude the two corners. There are 12 edges on a cube, so the number of edge cubes is 12 * 11 = 132. Each edge cube has 2 red faces.3. Face cubes: These are the cubes on each face but not on the edges. Each face has (13 - 2)^2 = 121 such cubes. There are 6 faces, so the number of face cubes is 6 * 121 = 726. Each face cube has 1 red face.4. Inner cubes: These are the cubes completely inside the big cube, not on any face. The number of inner cubes is (13 - 2)^3 = 11^3 = 1331. Each inner cube has 0 red faces.Let me verify that the total number of small cubes adds up:Corners: 8Edges: 132Faces: 726Inner: 1331Total: 8 + 132 + 726 + 1331 = 2197, which matches 13^3. Good.Now, let's compute the total red area.Each corner cube contributes 3 red faces. So, total red faces from corners: 8 * 3 = 24.Each edge cube contributes 2 red faces. So, total red faces from edges: 132 * 2 = 264.Each face cube contributes 1 red face. So, total red faces from faces: 726 * 1 = 726.Inner cubes contribute 0 red faces.Total red faces: 24 + 264 + 726 = 1014.But wait, each face of a small cube is 1x1, so each face has an area of 1. Therefore, the total red area is 1014 * 1 = 1014.Now, let's compute the total blue area.Each small cube has 6 faces. So, the total number of faces across all small cubes is 2197 * 6 = 13182.But, each face is shared between two small cubes, except for the outer faces of the big cube. Wait, no, actually, when the big cube is painted, the outer faces are all red, and then when it's cut into small cubes, each small cube's face that was on the outside is red, and the ones inside are blue.Wait, perhaps another approach is better.Each small cube has some red faces and some blue faces. The total number of red faces is 1014, as computed. The total number of blue faces is the total number of faces minus the red faces.But wait, each small cube has 6 faces, so the total number of faces is 2197 * 6 = 13182. But each face is either red or blue. However, in the big cube, the outer faces are all red, and the inner faces are blue.Wait, actually, when the big cube is painted red on all its surfaces, then cut into small cubes, each small cube's faces that were on the surface of the big cube are red, and the ones that were internal (i.e., adjacent to another small cube) are blue.But actually, when you paint all the outer faces red, and then cut into small cubes, each small cube's faces that were on the outside are red, and the ones that were inside the big cube are blue.But in reality, each face of a small cube is either red or blue, depending on whether it was on the surface of the big cube or not.So, the total red area is equal to the surface area of the big cube, which is 6 * (13)^2 = 6 * 169 = 1014. That's consistent with what I computed earlier.The total blue area is the total number of small cube faces minus the red area. But wait, each small cube has 6 faces, so total faces are 2197 * 6 = 13182. But each face is either red or blue. However, the red faces are only the ones on the surface of the big cube, which is 1014.Wait, but actually, when you cut the big cube into small cubes, the internal faces (the ones that were inside the big cube) are now blue. So, the total blue area is equal to the total number of internal faces.But the total number of internal faces can be computed as follows: the big cube has 13x13x13 small cubes. Each internal face is shared between two small cubes. So, the number of internal faces is equal to the total number of faces minus the number of external faces.Wait, maybe another way: the total number of faces in all small cubes is 2197 * 6 = 13182. The number of external faces (red) is 1014. Therefore, the number of internal faces (blue) is 13182 - 1014 = 12168. But wait, each internal face is shared between two small cubes, so the actual number of unique internal faces is 12168 / 2 = 6084.But wait, no. When we talk about the total blue area, we are considering all the blue faces on all the small cubes. So, each internal face is blue on both sides, but in terms of the total blue area, it's two blue faces. So, actually, the total blue area is 12168, because each internal face contributes two blue faces.Wait, I'm getting confused. Let me think again.Each small cube has 6 faces. The total number of faces across all small cubes is 2197 * 6 = 13182. However, each face of the big cube is painted red, so the number of red faces is equal to the surface area of the big cube, which is 6 * 13^2 = 1014. So, the number of blue faces is 13182 - 1014 = 12168.But wait, that can't be right because each internal face is shared between two small cubes, so each internal face contributes two blue faces. So, the total number of blue faces is 12168, which is correct because each internal face is counted twice in the total face count.Therefore, the total blue area is 12168, and the total red area is 1014.So, the factor by which blue area is greater than red area is 12168 / 1014.Let me compute that.First, let's see if 1014 divides into 12168.Divide 12168 by 1014:1014 * 12 = 12168.Yes, because 1000 * 12 = 12000, and 14 * 12 = 168, so 12000 + 168 = 12168.Therefore, 12168 / 1014 = 12.So, the total blue area is 12 times greater than the total red area.Wait, that seems high. Let me verify my steps again.Total red area: 6 * 13^2 = 6 * 169 = 1014. Correct.Total number of small cube faces: 2197 * 6 = 13182. Correct.Total blue area: 13182 - 1014 = 12168. Correct.Factor: 12168 / 1014 = 12. Correct.So, yes, the blue area is 12 times greater than the red area.But wait, intuitively, the big cube has a lot of internal faces, so it's plausible that the blue area is significantly larger.Alternatively, another approach: Each small cube contributes some blue faces. For each small cube, the number of blue faces is 6 minus the number of red faces.So, total blue area is sum over all small cubes of (6 - red faces). Which is equal to 6 * total small cubes - total red faces.Which is 6 * 2197 - 1014 = 13182 - 1014 = 12168. Same result.So, the ratio is 12168 / 1014 = 12.Therefore, the factor is 12.**Final Answer**The total area of the blue faces is greater than the total area of the red faces by a factor of boxed{12}.

answer：Alright, so I have this problem here where I need to find the coefficient of the term containing ( x ) in the expansion of the binomial ( left( sqrt{x} - frac{m}{sqrt{x}} right)^6 ). But before I can tackle that, I need to figure out what ( m ) is. The problem gives me ( m ) as an integral: ( m = int_{0}^{frac{pi}{2}} sqrt{2} sinleft(x + frac{pi}{4}right) dx ). Hmm, okay, so I need to compute this integral first.Let me write that integral down again to make sure I have it right: ( m = sqrt{2} int_{0}^{frac{pi}{2}} sinleft(x + frac{pi}{4}right) dx ). I think I can solve this integral using substitution. Let me set ( u = x + frac{pi}{4} ). Then, ( du = dx ), right? So, when ( x = 0 ), ( u = 0 + frac{pi}{4} = frac{pi}{4} ), and when ( x = frac{pi}{2} ), ( u = frac{pi}{2} + frac{pi}{4} = frac{3pi}{4} ). So, the integral becomes ( sqrt{2} int_{frac{pi}{4}}^{frac{3pi}{4}} sin(u) du ).I remember that the integral of ( sin(u) ) is ( -cos(u) ). So, evaluating from ( frac{pi}{4} ) to ( frac{3pi}{4} ), we get:( sqrt{2} left[ -cosleft( frac{3pi}{4} right) + cosleft( frac{pi}{4} right) right] ).Let me compute each cosine term. ( cosleft( frac{pi}{4} right) ) is ( frac{sqrt{2}}{2} ), and ( cosleft( frac{3pi}{4} right) ) is ( -frac{sqrt{2}}{2} ). Plugging these in:( sqrt{2} left[ -left( -frac{sqrt{2}}{2} right) + frac{sqrt{2}}{2} right] = sqrt{2} left[ frac{sqrt{2}}{2} + frac{sqrt{2}}{2} right] ).Simplifying inside the brackets: ( frac{sqrt{2}}{2} + frac{sqrt{2}}{2} = sqrt{2} ). So now we have:( sqrt{2} times sqrt{2} = 2 ). So, ( m = 2 ). Got that part down.Now, moving on to the binomial expansion. The expression is ( left( sqrt{x} - frac{m}{sqrt{x}} right)^6 ). Since we found ( m = 2 ), this becomes ( left( sqrt{x} - frac{2}{sqrt{x}} right)^6 ).I need to find the coefficient of the term containing ( x ) in this expansion. Let me recall the binomial theorem. The expansion of ( (a + b)^n ) is ( sum_{k=0}^{n} binom{n}{k} a^{n - k} b^{k} ). In this case, ( a = sqrt{x} ) and ( b = -frac{2}{sqrt{x}} ), and ( n = 6 ).So, each term in the expansion will be ( binom{6}{k} (sqrt{x})^{6 - k} left( -frac{2}{sqrt{x}} right)^k ). Let me simplify this term.First, let's compute ( (sqrt{x})^{6 - k} ). That's ( x^{frac{6 - k}{2}} ). Then, ( left( -frac{2}{sqrt{x}} right)^k ) is ( (-2)^k times x^{-frac{k}{2}} ). So, putting it all together, the term becomes:( binom{6}{k} (-2)^k x^{frac{6 - k}{2} - frac{k}{2}} ).Simplify the exponent: ( frac{6 - k - k}{2} = frac{6 - 2k}{2} = 3 - k ). So, each term is ( binom{6}{k} (-2)^k x^{3 - k} ).We need the term where the exponent of ( x ) is 1. So, set ( 3 - k = 1 ). Solving for ( k ), we get ( k = 2 ).So, the term corresponding to ( k = 2 ) will have ( x^1 ). Let's compute the coefficient for ( k = 2 ):Coefficient = ( binom{6}{2} (-2)^2 ).Compute ( binom{6}{2} ): that's ( frac{6!}{2!4!} = frac{720}{2 times 24} = frac{720}{48} = 15 ).Compute ( (-2)^2 ): that's 4.Multiply them together: 15 times 4 = 60.So, the coefficient of the term containing ( x ) is 60.Wait, let me double-check my steps to make sure I didn't make a mistake. First, the integral: substitution seems correct, limits adjusted properly, evaluated the cosine terms correctly, ended up with 2. Then, binomial expansion: correctly identified ( a ) and ( b ), applied the binomial theorem, simplified the exponent correctly, found that ( k = 2 ) gives ( x^1 ), computed the binomial coefficient and the power of -2 correctly, and multiplied them. Seems solid.Is there a chance I messed up the exponent? Let me see: ( (sqrt{x})^{6 - k} = x^{frac{6 - k}{2}} ), and ( left( frac{1}{sqrt{x}} right)^k = x^{-frac{k}{2}} ). So, when multiplied, exponents add: ( frac{6 - k}{2} - frac{k}{2} = frac{6 - 2k}{2} = 3 - k ). So, yes, that's correct.Alternatively, maybe I can think about the general term in the expansion. The general term is ( binom{6}{k} (sqrt{x})^{6 - k} left( -frac{2}{sqrt{x}} right)^k ). So, combining the exponents: ( (sqrt{x})^{6 - k} times left( frac{1}{sqrt{x}} right)^k = x^{frac{6 - k}{2}} times x^{-frac{k}{2}} = x^{frac{6 - 2k}{2}} = x^{3 - k} ). So, same result.Therefore, when ( 3 - k = 1 ), ( k = 2 ). So, the term is ( binom{6}{2} (-2)^2 x^1 ). Calculated as 15 * 4 = 60. So, yes, 60 is the coefficient.I think that's solid. So, my final answer is 60.**Final Answer**The coefficient of the term containing ( x ) is boxed{60}.