answer：Okay, so I have this problem about a rectangular prism that has two extra vertices on one of its top faces, making a total of 10 vertices. I need to figure out how many distinct triangles can be formed by connecting three different vertices. The problem mentions that two triangles are distinct if they have different locations in space. First, I remember that a standard rectangular prism has 8 vertices. So, adding two more vertices on one of its top faces brings the total to 10. That makes sense. Now, to find the number of distinct triangles, I think I need to calculate how many ways I can choose three vertices out of these ten. I recall that the number of ways to choose k elements from a set of n elements is given by the combination formula, which is:[binom{n}{k} = frac{n!}{k!(n - k)!}]In this case, n is 10 and k is 3 because we're choosing three vertices to form a triangle. So, plugging in the numbers:[binom{10}{3} = frac{10!}{3!(10 - 3)!} = frac{10 times 9 times 8 times 7!}{3 times 2 times 1 times 7!}]Wait, I see that the 7! cancels out from the numerator and the denominator, so it simplifies to:[frac{10 times 9 times 8}{3 times 2 times 1} = frac{720}{6} = 120]So, at first glance, it seems like there are 120 distinct triangles that can be formed. But hold on, I need to make sure that every set of three vertices actually forms a triangle. I remember that three points are collinear if they lie on a straight line, and in that case, they don't form a triangle. So, I need to subtract the number of collinear triplets from the total number of combinations. But wait, in a rectangular prism, how many sets of three collinear points are there? Let me think. In a standard rectangular prism, there are edges, face diagonals, and space diagonals. The edges are straight lines, face diagonals are diagonals on the faces, and space diagonals go through the interior of the prism. But in this case, we have two additional vertices on one of the top faces. So, I need to consider if these additional vertices create any new lines where three points lie on the same line. Let me visualize the top face of the prism. Normally, a rectangular face has four vertices. With two additional vertices, it now has six vertices on that top face. So, these two extra points are probably placed somewhere on the edges or the face. If the two additional vertices are placed on the edges of the top face, then each edge now has three points instead of two. That means on each of those edges, there are three collinear points. So, if an edge originally had two vertices, adding one more vertex on that edge would make three collinear points. Similarly, if both additional vertices are on the same edge, that edge would have four points, but the problem says two additional vertices on one of the top faces, not necessarily on the same edge. Wait, the problem says two additional vertices on one of its top faces, making a total of 10 vertices. So, originally, the prism had 8 vertices, and now it has 10. So, the top face, which originally had 4 vertices, now has 6. So, the top face now has 6 vertices. How are these arranged? If it's a rectangle, adding two more vertices could mean adding one on each of two different edges, or both on the same edge. If they are on different edges, then each of those edges now has three points. If they are on the same edge, then that edge has four points. But the problem doesn't specify where the additional vertices are placed, just that they are on one of the top faces. So, I might need to consider both possibilities. However, for the sake of this problem, I think it's safe to assume that the two additional vertices are placed on two different edges of the top face, making each of those edges have three points. So, on the top face, there are now six vertices: the original four plus two more on two different edges. Now, how many sets of three collinear points are there? On each edge that has three points, the number of ways to choose three collinear points is:[binom{3}{3} = 1]So, for each edge with three points, there is one set of three collinear points. If the two additional vertices are on two different edges, then there are two edges with three points each, leading to two sets of three collinear points. Additionally, the original edges on the top face had two points each, so choosing three points from them isn't possible because there are only two. But wait, actually, the top face now has six vertices. Are there any other lines on the top face where three points lie on the same line? If the top face is a rectangle, the only lines are the edges and the diagonals. The diagonals of the top face would have four points each if the two additional vertices are placed on the diagonals. But the problem says the additional vertices are on the top face, not necessarily on the diagonals. So, assuming the two additional vertices are on the edges, the diagonals still have only two points each, so no three collinear points on the diagonals. Therefore, the only sets of three collinear points are on the two edges with three points each, giving us two sets. So, the total number of triangles would be the total number of combinations minus the number of collinear triplets:[120 - 2 = 118]Wait, but I'm not sure if that's all. Let me think again. If the two additional vertices are on the same edge, then that edge has four points, and the number of collinear triplets on that edge would be:[binom{4}{3} = 4]And the other edges on the top face still have two points each, so no collinear triplets there. But the problem doesn't specify where the additional vertices are placed, just that they are on one of the top faces. So, depending on their placement, the number of collinear triplets could be different. Hmm, this is a bit confusing. Maybe I need to consider both cases. Case 1: The two additional vertices are on different edges of the top face. In this case, each of those edges has three points, leading to two sets of three collinear points. So, total collinear triplets: 2.Case 2: The two additional vertices are on the same edge of the top face. In this case, that edge has four points, leading to four sets of three collinear points. So, total collinear triplets: 4.But since the problem doesn't specify, I think I need to consider the maximum number of collinear triplets to be safe. Wait, no, actually, the problem says "two additional vertices on one of its top faces." It doesn't specify if they are on the same edge or different edges. So, perhaps I need to consider both possibilities and see which one makes sense.Alternatively, maybe the two additional vertices are placed in such a way that they don't lie on the same line as any other vertex. But that seems unlikely because they are on the top face, which is a plane.Wait, actually, in a rectangular prism, the top face is a rectangle, which is a planar figure. So, any three points on the same line on the top face would be collinear.So, if the two additional vertices are placed on the same edge, then that edge has four points, leading to four collinear triplets. If they are placed on different edges, then each of those edges has three points, leading to two collinear triplets.But the problem doesn't specify, so perhaps I need to assume the minimal number of collinear triplets, which would be two.Alternatively, maybe the two additional vertices are placed such that they don't create any new lines with three collinear points. But that seems impossible because adding two points on the top face would likely create at least one line with three points.Wait, actually, if the two additional vertices are placed on the same edge, then that edge has four points, leading to four collinear triplets. If they are placed on different edges, then each of those edges has three points, leading to two collinear triplets.But the problem doesn't specify, so perhaps I need to consider the maximum number of collinear triplets to subtract, which would be four.But I'm not sure. Maybe I should look for another approach.Alternatively, perhaps the two additional vertices are placed on the top face but not on the edges, but somewhere on the face itself. In that case, they might not lie on any existing lines with other vertices, except possibly the diagonals.But if they are placed on the face but not on the edges, then the lines connecting them to other vertices might not have three points. Wait, but the problem says "two additional vertices on one of its top faces." It doesn't specify if they are on the edges or inside the face. So, maybe they are placed on the face but not on the edges, meaning that they don't create any new lines with three collinear points.In that case, the only collinear triplets would be the original edges of the prism, which have two points each, so no collinear triplets.But that seems contradictory because the top face now has six vertices, so there must be some lines with three points.Wait, no, if the two additional vertices are placed on the face but not on the edges, then the lines connecting them to other vertices might not have three points. For example, if I place two points on the top face, not on the edges, then the lines connecting them to other vertices might only have two points each, so no three collinear points.But that seems unlikely because the top face is a rectangle, and any line on the face would either be an edge or a diagonal. If the two additional vertices are placed on the face but not on the edges, then they might lie on the diagonals.Wait, if they are placed on the diagonals, then the diagonals would have three points each, leading to collinear triplets.But the problem doesn't specify where the additional vertices are placed, so I'm stuck.Maybe I need to make an assumption. Let's assume that the two additional vertices are placed on two different edges of the top face, making each of those edges have three points. Therefore, there are two sets of three collinear points.So, total number of triangles would be:[binom{10}{3} - 2 = 120 - 2 = 118]But I'm not entirely sure. Alternatively, if the two additional vertices are placed on the same edge, leading to four collinear triplets, then the number of triangles would be:[120 - 4 = 116]But since the problem doesn't specify, I think the answer is 120, assuming that all sets of three vertices form a triangle. However, I'm not sure if that's correct because there might be collinear triplets.Wait, in a standard rectangular prism with 8 vertices, the number of triangles is:[binom{8}{3} - text{number of collinear triplets}]In a rectangular prism, the only collinear triplets are along the edges, face diagonals, and space diagonals. But in a rectangular prism, the edges have two points each, so no collinear triplets. The face diagonals have two points each, so no collinear triplets. The space diagonals have two points each, so no collinear triplets. Therefore, in a standard rectangular prism, all combinations of three vertices form a triangle, so the number of triangles is:[binom{8}{3} = 56]But in this case, we have 10 vertices, so the number of triangles would be:[binom{10}{3} - text{number of collinear triplets}]But since we added two vertices on the top face, we need to check if that creates any collinear triplets.If the two additional vertices are on the same edge, then that edge has four points, leading to four collinear triplets. If they are on different edges, then each of those edges has three points, leading to two collinear triplets.But the problem doesn't specify, so perhaps the answer is 120, assuming no collinear triplets. However, that seems incorrect because adding vertices on the same face would likely create collinear triplets.Alternatively, maybe the two additional vertices are placed in such a way that they don't create any new lines with three collinear points. But that seems unlikely.Wait, maybe the two additional vertices are placed on the top face but not on the edges, so they don't lie on any existing lines with other vertices. In that case, there would be no new collinear triplets, so the number of triangles would still be 120.But I'm not sure. I think the problem expects me to assume that all combinations are valid, so the answer is 120.But earlier, I thought that adding two vertices on the top face would create collinear triplets, so I need to subtract those.Wait, maybe the two additional vertices are placed on the same edge, leading to four collinear triplets. So, the number of triangles would be 120 - 4 = 116.But I'm not sure. I think I need to clarify.Alternatively, maybe the two additional vertices are placed on the top face but not on the edges, so they don't create any new lines with three collinear points. Therefore, the number of triangles is 120.But I'm not sure. I think the answer is 120, but I'm not entirely confident because of the possibility of collinear triplets.

answer：Okay, so I have this problem where I need to find the minimum value of M such that for any partition of the interval [0, 4], the sum of the absolute differences of the function f(x) = x² - 2x + 1 between consecutive points in the partition is less than or equal to M. First, I should probably simplify the function f(x). Let me see, f(x) = x² - 2x + 1. Hmm, that looks like a quadratic equation. Maybe I can complete the square or factor it. Let me try completing the square. So, f(x) = x² - 2x + 1. To complete the square, I take the coefficient of x, which is -2, divide it by 2 to get -1, and then square it to get 1. So, f(x) = (x - 1)². Oh, that's much simpler! So, f(x) is just a parabola shifted to the right by 1 unit, opening upwards. Now, since f(x) is a parabola, it has a minimum at x = 1. That means on the interval [0, 1], the function is decreasing, and on [1, 4], it's increasing. That might be important because the behavior of the function changes at x = 1.The problem is asking about the sum of absolute differences of f(x) over any partition of [0, 4]. So, for any partition 0 = x₀ < x₁ < ... < xₙ = 4, I need to compute the sum Σ|f(x_i) - f(x_{i-1})| from i=1 to n, and find the smallest M such that this sum is always less than or equal to M, regardless of the partition.I think this sum is related to the total variation of the function over the interval [0, 4]. The total variation is the supremum of such sums over all possible partitions. So, maybe M is the total variation of f(x) on [0, 4].Since f(x) is differentiable everywhere, the total variation can be calculated as the integral of the absolute value of its derivative over the interval. Let me recall that formula: Total variation = ∫₀⁴ |f’(x)| dx.So, first, let's compute f’(x). f(x) = (x - 1)², so f’(x) = 2(x - 1). Therefore, |f’(x)| = |2(x - 1)|.Now, let's compute the integral of |2(x - 1)| from 0 to 4. To do this, I need to consider where the expression inside the absolute value changes sign. That happens at x = 1. So, I can split the integral into two parts: from 0 to 1 and from 1 to 4.For x in [0, 1], (x - 1) is negative, so |2(x - 1)| = 2(1 - x). For x in [1, 4], (x - 1) is positive, so |2(x - 1)| = 2(x - 1).Therefore, the total variation is:∫₀¹ 2(1 - x) dx + ∫₁⁴ 2(x - 1) dx.Let me compute each integral separately.First integral: ∫₀¹ 2(1 - x) dx.Let me expand that: 2 ∫₀¹ (1 - x) dx = 2 [∫₀¹ 1 dx - ∫₀¹ x dx] = 2 [ (x)₀¹ - (x²/2)₀¹ ] = 2 [ (1 - 0) - (1²/2 - 0) ] = 2 [1 - 1/2] = 2*(1/2) = 1.Second integral: ∫₁⁴ 2(x - 1) dx.Similarly, expand: 2 ∫₁⁴ (x - 1) dx = 2 [∫₁⁴ x dx - ∫₁⁴ 1 dx] = 2 [ (x²/2)₁⁴ - (x)₁⁴ ].Compute each part:(x²/2)₁⁴ = (16/2) - (1/2) = 8 - 0.5 = 7.5.(x)₁⁴ = 4 - 1 = 3.So, 2 [7.5 - 3] = 2*(4.5) = 9.Therefore, the total variation is 1 + 9 = 10.So, the total variation of f(x) over [0, 4] is 10. That means the supremum of the sum Σ|f(x_i) - f(x_{i-1})| over all partitions is 10. Therefore, the minimal M that satisfies the condition is 10.Wait, just to make sure, let me think about this again. The total variation is indeed the supremum of such sums, so any partition will give a sum less than or equal to the total variation. So, M must be at least the total variation, and since we can approach it arbitrarily closely with finer partitions, M cannot be smaller than 10. Hence, M = 10 is the minimal value.I think that makes sense. So, the minimum M is 10.**Final Answer**The minimum value of ( M ) is boxed{10}.

answer：First, I'll define the variables for the costs of a sandwich, coffee, and pie as ( s ), ( c ), and ( p ) respectively.Next, I'll set up two equations based on the given luncheon costs:[2s + 5c + 2p = 6.25 quad text{(Equation 1)}][5s + 8c + 3p = 12.10 quad text{(Equation 2)}]To eliminate one variable, I'll multiply Equation 1 by 3 and Equation 2 by 2:[6s + 15c + 6p = 18.75 quad text{(Equation 3)}][10s + 16c + 6p = 24.20 quad text{(Equation 4)}]Subtracting Equation 3 from Equation 4 to eliminate ( p ):[4s + c = 5.45 quad text{(Equation 5)}]From Equation 5, solve for ( c ):[c = 5.45 - 4s]Substitute ( c ) back into Equation 1:[2s + 5(5.45 - 4s) + 2p = 6.25][2s + 27.25 - 20s + 2p = 6.25][-18s + 2p = -21][9s - p = 10.5]Now, I'll solve for ( p ) and substitute back to find ( s ) and ( c ):[p = 0.05][9s - 0.05 = 10.5][9s = 10.55][s = 1.1722][c = 5.45 - 4 times 1.1722 approx 0.322]Finally, calculate the total cost for one sandwich, one coffee, and one pie:[s + c + p = 1.1722 + 0.322 + 0.05 = 1.5442]The cost is approximately 1.55.

answer：Okay, so I have this trigonometry problem here: I need to find the degree measure of the least positive angle θ for which cos 10° equals sin 30° plus sin θ. Hmm, let me write that down to make it clearer.The equation is:cos 10° = sin 30° + sin θAlright, first things first, I remember that sin 30° is a standard angle. Let me recall its value. Sin 30° is 1/2. So, substituting that into the equation, we get:cos 10° = 1/2 + sin θNow, I need to solve for sin θ. Let me subtract 1/2 from both sides to isolate sin θ:sin θ = cos 10° - 1/2Hmm, okay. So sin θ equals cos 10° minus 1/2. I need to find θ such that this equation holds. I wonder if I can express cos 10° in terms of sine to make it easier. I remember that cos x is equal to sin(90° - x). So, cos 10° is equal to sin(80°). Let me write that down:sin θ = sin 80° - 1/2Wait, so sin θ is equal to sin 80° minus 1/2. That seems a bit tricky. Maybe I can compute the numerical values to get a better sense.Let me calculate cos 10° first. I know that cos 0° is 1, and as the angle increases, the cosine decreases. So cos 10° should be slightly less than 1. Let me use a calculator for a more precise value. Cos 10° is approximately 0.9848.So, sin θ = 0.9848 - 0.5 = 0.4848Therefore, sin θ ≈ 0.4848Now, I need to find θ such that sin θ ≈ 0.4848. I can use the inverse sine function to find θ. Let me compute arcsin(0.4848). Using a calculator, arcsin(0.4848) is approximately 29°. Wait, is that right?Wait, hold on. Let me double-check. If sin θ ≈ 0.4848, then θ is approximately 29°, because sin 30° is 0.5, so 0.4848 is just slightly less than 0.5, so θ should be just slightly less than 30°, maybe around 29°. Hmm, but wait, earlier I thought of expressing cos 10° as sin 80°, which is a much larger angle. So, is θ 29° or 80°? That seems conflicting.Wait, maybe I made a mistake in my earlier step. Let me go back.I had:cos 10° = sin 30° + sin θWhich became:sin θ = cos 10° - sin 30°Which is:sin θ = cos 10° - 1/2Then, I converted cos 10° to sin 80°, so:sin θ = sin 80° - 1/2But sin 80° is approximately 0.9848, so sin θ ≈ 0.9848 - 0.5 = 0.4848So, sin θ ≈ 0.4848, which gives θ ≈ 29°, as I thought earlier.But wait, another thought: sine is positive in the first and second quadrants, so θ could also be 180° - 29° = 151°. But since we are looking for the least positive angle, 29° is smaller than 151°, so 29° would be the answer.But hold on, earlier I thought of expressing cos 10° as sin 80°, which led me to sin θ = sin 80° - 1/2. But that approach didn't directly help me find θ. Instead, I had to compute the numerical value.Alternatively, maybe I can use some trigonometric identities to solve this equation without resorting to numerical calculations.Let me think. The equation is:cos 10° = sin 30° + sin θI know that cos 10° can be written as sin(90° - 10°) = sin 80°, so:sin 80° = sin 30° + sin θSo, sin θ = sin 80° - sin 30°Hmm, now I have sin θ expressed as the difference of two sines. Maybe I can use the sine subtraction formula or some identity to simplify this.I recall that sin A - sin B = 2 cos[(A+B)/2] sin[(A-B)/2]. Let me apply that identity here.So, sin 80° - sin 30° = 2 cos[(80° + 30°)/2] sin[(80° - 30°)/2]Calculating the arguments:(80° + 30°)/2 = 110°/2 = 55°(80° - 30°)/2 = 50°/2 = 25°So, sin 80° - sin 30° = 2 cos 55° sin 25°Therefore, sin θ = 2 cos 55° sin 25°Hmm, okay. So sin θ equals 2 cos 55° sin 25°. Maybe I can compute this product.Let me compute 2 cos 55° sin 25°. I know that 2 sin A cos B = sin(A + B) + sin(A - B). Wait, but here it's 2 cos A sin B, which is sin(A + B) + sin(B - A). Let me verify.Yes, indeed, 2 cos A sin B = sin(A + B) + sin(B - A). So, in this case, A is 55°, B is 25°, so:2 cos 55° sin 25° = sin(55° + 25°) + sin(25° - 55°) = sin 80° + sin(-30°)But sin(-30°) is -sin 30°, which is -1/2. So:2 cos 55° sin 25° = sin 80° - 1/2Wait, but that's exactly the expression we had earlier. So, sin θ = sin 80° - 1/2, which brings us back to where we were before.So, this approach just circles back. Maybe I need a different strategy.Alternatively, perhaps I can consider writing the equation as:cos 10° - sin 30° = sin θAnd then compute the left-hand side numerically. Let me do that.We have cos 10° ≈ 0.9848 and sin 30° = 0.5, so:0.9848 - 0.5 = 0.4848So, sin θ ≈ 0.4848As I thought earlier, θ ≈ arcsin(0.4848) ≈ 29°But let me check if θ = 29° satisfies the original equation.Compute sin 30° + sin 29°:sin 30° = 0.5sin 29° ≈ 0.4848So, 0.5 + 0.4848 ≈ 0.9848Which is equal to cos 10°, so that works.But wait, another thought: sine is positive in both the first and second quadrants, so another possible solution is θ = 180° - 29° = 151°. But since we are asked for the least positive angle, 29° is smaller than 151°, so 29° is the answer.But hold on, earlier I thought of expressing cos 10° as sin 80°, which is a much larger angle. So, is there a way to relate θ to 80°?Wait, let's think about the equation again:cos 10° = sin 30° + sin θExpressed as:sin θ = cos 10° - sin 30°I can also think about this as:sin θ = sin(90° - 10°) - sin 30° = sin 80° - sin 30°Which is what I did earlier. Then, using the identity, I found that sin θ = 2 cos 55° sin 25°, which is approximately 0.4848, leading to θ ≈ 29°.Alternatively, maybe I can use another identity or approach.Wait, another idea: perhaps use the sum-to-product identities on sin 30° + sin θ.I know that sin A + sin B = 2 sin[(A + B)/2] cos[(A - B)/2]So, sin 30° + sin θ = 2 sin[(30° + θ)/2] cos[(30° - θ)/2]So, the equation becomes:cos 10° = 2 sin[(30° + θ)/2] cos[(30° - θ)/2]Hmm, that seems a bit complicated, but maybe I can set this equal to cos 10° and see if I can find θ.Alternatively, perhaps I can write cos 10° as sin(90° - 10°) = sin 80°, so:sin 80° = 2 sin[(30° + θ)/2] cos[(30° - θ)/2]Hmm, not sure if that helps directly.Alternatively, maybe I can set up an equation where both sides are expressed in terms of sine or cosine.Wait, another approach: let's consider that cos 10° is approximately 0.9848, and sin 30° is 0.5, so sin θ is approximately 0.4848, as we found earlier. So, θ is approximately 29°, as we saw.But perhaps I can find an exact value. Let me think.Wait, 0.4848 is approximately sin 29°, but 29° is not a standard angle. So, maybe the exact value is 80°, but that seems conflicting with our earlier calculation.Wait, hold on. Let me think again.If sin θ = sin 80° - 1/2, and sin 80° is approximately 0.9848, so sin θ ≈ 0.4848, which is approximately sin 29°, as we saw.But wait, is there a way to express sin θ as sin(80° - 30°)? Because 80° - 30° = 50°, but sin(50°) is approximately 0.7660, which is not 0.4848.Alternatively, maybe using some other identity.Wait, perhaps I can use the sine of a difference identity.Wait, sin(A - B) = sin A cos B - cos A sin B.But I don't see how that directly applies here.Alternatively, perhaps I can write sin θ = sin 80° - sin 30°, and then use the identity for sin A - sin B.Wait, I did that earlier, and it led me to sin θ = 2 cos 55° sin 25°, which is approximately 0.4848.So, perhaps the exact value is 25°, but 25° is less than 29°, but sin 25° is approximately 0.4226, which is less than 0.4848.Wait, so 25° is too small. Hmm.Alternatively, maybe θ is 29°, which is approximately 0.4848.But since 29° is not a standard angle, perhaps the answer is 80°, but that seems conflicting.Wait, let me think again.The original equation is:cos 10° = sin 30° + sin θExpressed as:sin θ = cos 10° - sin 30°We can write this as:sin θ = sin 80° - sin 30°Using the identity sin A - sin B = 2 cos[(A + B)/2] sin[(A - B)/2], which gives:sin θ = 2 cos[(80° + 30°)/2] sin[(80° - 30°)/2] = 2 cos 55° sin 25°So, sin θ = 2 cos 55° sin 25°Hmm, so sin θ = 2 cos 55° sin 25°But 2 cos A sin B = sin(A + B) + sin(B - A)Wait, so 2 cos 55° sin 25° = sin(55° + 25°) + sin(25° - 55°) = sin 80° + sin(-30°) = sin 80° - sin 30°Which is exactly where we started. So, this approach just circles back.Alternatively, perhaps I can compute 2 cos 55° sin 25° numerically.Compute cos 55°: approximately 0.5736Compute sin 25°: approximately 0.4226So, 2 * 0.5736 * 0.4226 ≈ 2 * 0.5736 * 0.4226 ≈ 2 * 0.2423 ≈ 0.4846Which is approximately 0.4848, as before. So, sin θ ≈ 0.4848, leading to θ ≈ 29°, as we saw.Therefore, θ is approximately 29°, which is the least positive angle satisfying the equation.But wait, let me check if θ = 80° could be a solution.If θ = 80°, then sin θ = sin 80° ≈ 0.9848So, sin 30° + sin 80° ≈ 0.5 + 0.9848 ≈ 1.4848, which is greater than cos 10° ≈ 0.9848. So, that's not equal. Therefore, θ cannot be 80°.Alternatively, maybe θ = 10°, let's check.sin 10° ≈ 0.1736So, sin 30° + sin 10° ≈ 0.5 + 0.1736 ≈ 0.6736, which is less than cos 10° ≈ 0.9848. So, that's not equal either.Wait, so θ must be somewhere between 10° and 80°, but closer to 30°, as we saw.Wait, another thought: perhaps θ is 29°, but let me check if that's exact.Wait, 29° is not a standard angle, so perhaps the answer is 80°, but that doesn't satisfy the equation, as we saw.Alternatively, maybe I made a mistake in my earlier steps.Wait, let me go back to the original equation:cos 10° = sin 30° + sin θExpressed as:sin θ = cos 10° - sin 30°I can also write cos 10° as sin(90° - 10°) = sin 80°, so:sin θ = sin 80° - sin 30°Now, using the identity sin A - sin B = 2 cos[(A + B)/2] sin[(A - B)/2], we have:sin θ = 2 cos[(80° + 30°)/2] sin[(80° - 30°)/2] = 2 cos 55° sin 25°So, sin θ = 2 cos 55° sin 25°Now, let's compute 2 cos 55° sin 25°:We know that 2 cos A sin B = sin(A + B) + sin(B - A)So, 2 cos 55° sin 25° = sin(55° + 25°) + sin(25° - 55°) = sin 80° + sin(-30°) = sin 80° - sin 30°Which brings us back to the original expression. So, this approach doesn't help us find θ directly.Alternatively, maybe I can use another identity or approach.Wait, perhaps I can write the equation as:cos 10° - sin 30° = sin θAnd then use the identity for cos A - sin B.But I don't recall a direct identity for that. Alternatively, perhaps I can express both terms in terms of sine or cosine.Wait, cos 10° can be written as sin 80°, as before, so:sin 80° - sin 30° = sin θWhich is what we had earlier.Alternatively, perhaps I can use the identity for sin A - sin B, which we did, leading to sin θ = 2 cos 55° sin 25°, which is approximately 0.4848.Therefore, θ ≈ arcsin(0.4848) ≈ 29°, as we found earlier.But let me check if θ = 29° is indeed the solution.Compute sin 30° + sin 29°:sin 30° = 0.5sin 29° ≈ 0.4848So, 0.5 + 0.4848 ≈ 0.9848, which is equal to cos 10°, so that works.Therefore, θ ≈ 29° is the solution.But wait, another thought: since sine is positive in both the first and second quadrants, another solution could be θ = 180° - 29° = 151°. But since we are asked for the least positive angle, 29° is smaller than 151°, so 29° is the answer.But let me think again: is there a way to express θ exactly without resorting to decimal approximations?Wait, perhaps not, because 29° is not a standard angle, so the answer is likely 29°, but let me check if there's a different approach.Wait, another idea: perhaps use the cosine of 10° and express it in terms of sine, then set up an equation.We have cos 10° = sin 30° + sin θExpressed as:sin θ = cos 10° - sin 30°Now, let me consider that cos 10° is approximately 0.9848, and sin 30° is 0.5, so sin θ ≈ 0.4848.Therefore, θ ≈ arcsin(0.4848) ≈ 29°, as before.Alternatively, perhaps I can use the inverse sine function to find θ.But since 0.4848 is not a standard value, I think the answer is approximately 29°, but let me check if there's a way to express it exactly.Wait, another thought: perhaps θ is 25°, but sin 25° ≈ 0.4226, which is less than 0.4848.Alternatively, θ is 30°, but sin 30° = 0.5, which is greater than 0.4848.So, θ must be between 25° and 30°, closer to 30°, as 0.4848 is closer to 0.5.Therefore, θ ≈ 29°, as we found earlier.But wait, another approach: perhaps use the sum formula for sine.Wait, let me think: if I have sin θ = cos 10° - sin 30°, can I write this as sin θ = sin(90° - 10°) - sin 30° = sin 80° - sin 30°, which we did earlier.Alternatively, perhaps I can write this as sin θ = sin 80° - sin 30°, and then use the identity for sin A - sin B, which we did, leading to sin θ = 2 cos 55° sin 25°, which is approximately 0.4848.Therefore, θ ≈ 29°, as before.So, after considering various approaches and identities, it seems that θ is approximately 29°, which is the least positive angle satisfying the equation.But wait, let me double-check my calculations to ensure I didn't make any mistakes.Starting with the original equation:cos 10° = sin 30° + sin θSubtract sin 30° from both sides:sin θ = cos 10° - sin 30°Compute cos 10° ≈ 0.9848 and sin 30° = 0.5, so:sin θ ≈ 0.9848 - 0.5 = 0.4848Compute θ ≈ arcsin(0.4848) ≈ 29°Check if sin 30° + sin 29° ≈ 0.5 + 0.4848 ≈ 0.9848, which is equal to cos 10°, so it works.Therefore, θ ≈ 29° is the correct solution.But wait, another thought: since we're dealing with trigonometric functions, sometimes there can be multiple solutions within the range of 0° to 360°, but since we're asked for the least positive angle, 29° is the smallest positive solution.Therefore, the least positive angle θ is approximately 29°, but let me check if there's a way to express it exactly.Wait, perhaps not, because 29° is not a standard angle, so the answer is 29°, but let me see if I can find an exact value.Wait, another idea: perhaps use the fact that sin θ = 2 cos 55° sin 25°, and see if that can be simplified.But 2 cos 55° sin 25° is equal to sin(55° + 25°) + sin(25° - 55°) = sin 80° + sin(-30°) = sin 80° - sin 30°, which is where we started.Therefore, I think we've exhausted all possible approaches, and the conclusion is that θ ≈ 29°, which is the least positive angle satisfying the equation.But wait, another thought: perhaps I can use the identity for sin(A) - sin(B) again, but I think we've already done that.Alternatively, perhaps I can consider that sin θ = cos 10° - sin 30°, and then use the identity for cos A - sin B.But I don't recall a specific identity for that, so perhaps it's better to stick with the numerical approach.Therefore, after careful consideration and multiple approaches, I conclude that the least positive angle θ is approximately 29°, which is the solution to the equation.

answer：Okay, so I have this problem here: In the complex plane, the distance from the point corresponding to the complex number 3 divided by (2 minus i) squared to the origin is blank. I need to find that distance. Hmm, okay, let's break this down step by step.First, I remember that in the complex plane, any complex number can be represented as a point with a real part and an imaginary part. The distance from this point to the origin is essentially the modulus or the absolute value of the complex number. So, if I can find the modulus of the given complex number, that should give me the distance.The complex number in question is 3 divided by (2 minus i) squared. Let me write that down: 3 / (2 - i)^2. To find its modulus, I might need to simplify this expression first or maybe find the modulus directly. I'm not sure which is easier, so I'll try both approaches.Let me recall that for any complex number z = a + bi, the modulus is sqrt(a^2 + b^2). Also, the modulus of a quotient is the quotient of the moduli, and the modulus of a power is the power of the modulus. So, modulus of (3 / (2 - i)^2) is equal to modulus of 3 divided by modulus of (2 - i)^2.Calculating modulus of 3 is straightforward because 3 is a real number, so its modulus is just 3. Now, modulus of (2 - i)^2. Hmm, modulus of a square is the square of the modulus, so modulus of (2 - i)^2 is (modulus of (2 - i))^2.What's the modulus of (2 - i)? That's sqrt(2^2 + (-1)^2) = sqrt(4 + 1) = sqrt(5). Therefore, modulus of (2 - i)^2 is (sqrt(5))^2, which is 5.So, putting it all together, modulus of 3 / (2 - i)^2 is 3 / 5. So, the distance from the point to the origin is 3/5.Wait, that seems straightforward, but maybe I should verify by actually computing the complex number and then finding its modulus to make sure I didn't make a mistake.Alright, let's compute 3 / (2 - i)^2 step by step. First, compute (2 - i)^2. Expanding that, (2 - i)^2 is 2^2 - 2*2*i + (i)^2, which is 4 - 4i + (-1), since i squared is -1. So, 4 - 4i -1 is 3 - 4i.So, (2 - i)^2 is 3 - 4i. Therefore, 3 / (3 - 4i). Now, to simplify this, I can multiply the numerator and the denominator by the complex conjugate of the denominator to eliminate the imaginary part in the denominator.The complex conjugate of 3 - 4i is 3 + 4i. So, multiplying numerator and denominator by 3 + 4i:Numerator: 3 * (3 + 4i) = 9 + 12i.Denominator: (3 - 4i)(3 + 4i) = 9 + 12i -12i -16i^2. The middle terms cancel out, and i^2 is -1, so it becomes 9 - (-16) = 9 + 16 = 25.So, the simplified form is (9 + 12i)/25, which is 9/25 + (12/25)i. So, the complex number is 9/25 + 12/25 i.Now, to find the modulus of this, we take sqrt((9/25)^2 + (12/25)^2). Let's compute that:(9/25)^2 = 81/625.(12/25)^2 = 144/625.Adding them together: 81/625 + 144/625 = 225/625.Simplify that: 225 divided by 625 is 9/25, but wait, no, 225/625 is 9/25? Wait, 225 times 25 is 5625, and 625 times 9 is 5625, so yes, 225/625 is 9/25. Wait, no, that's not right. 225 divided by 625 is actually 9/25 because 225 divided by 25 is 9 and 625 divided by 25 is 25. So, 225/625 is 9/25.Wait, but 9/25 is 0.36, but earlier I had modulus as 3/5, which is 0.6. Hmm, that's conflicting. So, which one is correct?Wait, hold on, I think I made a mistake in the calculation. Let me go back.So, modulus squared is (9/25)^2 + (12/25)^2 = 81/625 + 144/625 = 225/625. So, modulus is sqrt(225/625) = sqrt(225)/sqrt(625) = 15/25 = 3/5.Ah, okay, so that's where I went wrong earlier. I thought 225/625 simplifies to 9/25, but actually, sqrt(225/625) is 15/25, which simplifies to 3/5. So, both methods give me the same result, which is 3/5.So, the distance from the point corresponding to the complex number 3/(2 - i)^2 to the origin is 3/5.Wait, just to make sure, let me recap:1. Calculated modulus directly: modulus of 3 is 3, modulus of (2 - i)^2 is 5, so 3/5.2. Expanded (2 - i)^2 to get 3 - 4i, then divided 3 by that, multiplied by the conjugate to get 9/25 + 12/25 i, then modulus is sqrt( (9/25)^2 + (12/25)^2 ) = sqrt(225/625) = 15/25 = 3/5.Yes, both methods lead to the same answer, so I think 3/5 is correct.I was confused for a moment when I saw 225/625 simplifies to 9/25, but that's modulus squared. Taking the square root gives 15/25, which is 3/5. So, that's consistent.Therefore, I'm confident that the distance is 3/5.**Final Answer**The distance from the point corresponding to the complex number frac{3}{(2-i)^{2}} to the origin is boxed{dfrac{3}{5}}.

answer：Okay, so I have this problem where I need to show that for any continuous real-valued functions ( f ) and ( g ) defined on the interval ([0,1]), the variance of their product ( fg ) is bounded by twice the variance of ( f ) times the square of the maximum of ( g ) plus twice the variance of ( g ) times the square of the maximum of ( f ). That is, I need to prove:[text{Var}(fg) leq 2text{Var}(f)M(g)^2 + 2text{Var}(g)M(f)^2]First, let me recall the definitions given:- The mean of a function ( f ) is ( mu(f) = int_0^1 f(x) , dx ).- The variance of ( f ) is ( text{Var}(f) = int_0^1 (f(x) - mu(f))^2 , dx ).- The maximum of ( f ) is ( M(f) = max_{0 leq x leq 1} |f(x)| ).So, I need to work with these definitions to relate the variance of the product ( fg ) to the variances and maxima of ( f ) and ( g ).Let me start by expanding ( text{Var}(fg) ). Using the definition:[text{Var}(fg) = int_0^1 (f(x)g(x) - mu(fg))^2 , dx]Where ( mu(fg) = int_0^1 f(x)g(x) , dx ).Hmm, this seems a bit complicated. Maybe I can find a way to express ( text{Var}(fg) ) in terms of ( f ) and ( g ) separately. I remember that variance can sometimes be expressed using double integrals, so maybe that's a way to go.Wait, I recall that for any function ( h ), the variance can also be written as:[text{Var}(h) = frac{1}{2} int_0^1 int_0^1 (h(x) - h(y))^2 , dx , dy]Is that correct? Let me verify. If I expand ( (h(x) - h(y))^2 ), I get ( h(x)^2 - 2h(x)h(y) + h(y)^2 ). Integrating over ( x ) and ( y ), the cross term becomes ( -2(int h(x) , dx)(int h(y) , dy) ), which is ( -2mu(h)^2 ). The other terms give ( 2int h(x)^2 , dx ). So, putting it all together:[frac{1}{2} int_0^1 int_0^1 (h(x) - h(y))^2 , dx , dy = int_0^1 h(x)^2 , dx - mu(h)^2 = text{Var}(h)]Yes, that works. So, I can write:[text{Var}(fg) = frac{1}{2} int_0^1 int_0^1 (f(x)g(x) - f(y)g(y))^2 , dx , dy]Now, I need to manipulate this expression to relate it to ( text{Var}(f) ) and ( text{Var}(g) ). Maybe I can use an inequality to bound ( (f(x)g(x) - f(y)g(y))^2 ).Let me think about how to bound ( (ab - cd)^2 ) in terms of ( (a - c)^2 ) and ( (b - d)^2 ). Maybe I can use the Cauchy-Schwarz inequality or some other algebraic manipulation.Wait, here's an idea. Let me consider the difference ( f(x)g(x) - f(y)g(y) ). This can be written as:[f(x)g(x) - f(y)g(y) = f(x)g(x) - f(x)g(y) + f(x)g(y) - f(y)g(y) = f(x)(g(x) - g(y)) + g(y)(f(x) - f(y))]So, it's the sum of two terms. Maybe I can square this and use the fact that ( (a + b)^2 leq 2a^2 + 2b^2 ) by the Cauchy-Schwarz inequality.Applying this, we get:[(f(x)g(x) - f(y)g(y))^2 leq 2f(x)^2(g(x) - g(y))^2 + 2g(y)^2(f(x) - f(y))^2]Hmm, that seems useful. Now, since ( M(f) ) is the maximum of ( |f(x)| ) over ([0,1]), we have ( |f(x)| leq M(f) ) for all ( x ). Similarly, ( |g(y)| leq M(g) ) for all ( y ).Therefore, ( f(x)^2 leq M(f)^2 ) and ( g(y)^2 leq M(g)^2 ). Substituting these into the inequality, we get:[(f(x)g(x) - f(y)g(y))^2 leq 2M(f)^2(g(x) - g(y))^2 + 2M(g)^2(f(x) - f(y))^2]Great, now I can substitute this back into the expression for ( text{Var}(fg) ):[text{Var}(fg) leq frac{1}{2} int_0^1 int_0^1 left[ 2M(f)^2(g(x) - g(y))^2 + 2M(g)^2(f(x) - f(y))^2 right] , dx , dy]Simplifying the constants:[text{Var}(fg) leq frac{1}{2} times 2 left[ M(f)^2 int_0^1 int_0^1 (g(x) - g(y))^2 , dx , dy + M(g)^2 int_0^1 int_0^1 (f(x) - f(y))^2 , dx , dy right]]The 2's cancel out:[text{Var}(fg) leq M(f)^2 int_0^1 int_0^1 (g(x) - g(y))^2 , dx , dy + M(g)^2 int_0^1 int_0^1 (f(x) - f(y))^2 , dx , dy]But wait, from earlier, I know that:[text{Var}(h) = frac{1}{2} int_0^1 int_0^1 (h(x) - h(y))^2 , dx , dy]So, the double integrals are just twice the variances:[int_0^1 int_0^1 (h(x) - h(y))^2 , dx , dy = 2 text{Var}(h)]Therefore, substituting back:[text{Var}(fg) leq M(f)^2 times 2 text{Var}(g) + M(g)^2 times 2 text{Var}(f)]Which simplifies to:[text{Var}(fg) leq 2 M(f)^2 text{Var}(g) + 2 M(g)^2 text{Var}(f)]And that's exactly what I needed to prove! So, the key steps were expressing the variance as a double integral, expanding the product ( fg ), applying the Cauchy-Schwarz inequality to bound the squared difference, and then using the definitions of variance and maximum to relate everything back to ( text{Var}(f) ), ( text{Var}(g) ), ( M(f) ), and ( M(g) ).I think this makes sense. Let me just recap to make sure I didn't skip any steps or make any mistakes. I started by expressing the variance of ( fg ) as a double integral, then expanded the product, applied an inequality to bound the squared difference, used the maximums to simplify, and then related the double integrals back to variances. It all seems to flow logically, so I feel confident about this proof.