🧠:[<think>Okay, let's see. I need to solve these two problems. First, find the product of matrices A and B, which is AB. Then, use that product matrix to transform the given curve C₁ into C₂ and find the equation of C₂.Starting with part (1): Finding AB. Matrix multiplication isn't commutative, so the order matters. A is given as [[0, 1], [1, 0]], which looks like a swap matrix—it swaps the rows or columns when multiplied. B is a diagonal matrix [[1, 0], [0, 2]]. So when I multiply A and B, I need to do it in the order AB, which means A is multiplied by B.Let me recall how matrix multiplication works. Each element of the resulting matrix is the dot product of the corresponding row from the first matrix and column from the second matrix.So, let's compute AB step by step.First row of A: [0, 1]First column of B: [1, 0]. The dot product is 0*1 + 1*0 = 0.Second column of B: [0, 2]. The dot product is 0*0 + 1*2 = 2.So the first row of AB is [0, 2].Second row of A: [1, 0]First column of B: [1, 0]. The dot product is 1*1 + 0*0 = 1.Second column of B: [0, 2]. The dot product is 1*0 + 0*2 = 0.So the second row of AB is [1, 0].Putting it all together, AB is:[0, 2][1, 0]Wait, let me double-check that. Maybe I made a mistake in the order. Since it's AB, A is first, so we're multiplying A times B. So the first row of A is [0, 1], multiplied by each column of B.First column of B is [1; 0], so 0*1 + 1*0 = 0. Second column of B is [0; 2], so 0*0 +1*2 = 2. That's correct.Second row of A is [1, 0]. First column of B is [1; 0], so 1*1 + 0*0 = 1. Second column of B is [0; 2], so 1*0 +0*2 =0. Yes, that seems right. So AB is [[0, 2], [1, 0]]. Okay, that's part (1) done.Now part (2): The curve C₁ is given by (x²)/8 + (y²)/2 =1. We need to transform this curve using the matrix AB and find the equation of the resulting curve C₂.Hmm. Transforming a curve with a matrix usually involves applying the matrix to the coordinates. So if we have a transformation matrix M, then a point (x, y) on the original curve is transformed to (x', y') = M*(x, y). So to find the equation of C₂, we need to express x and y in terms of x' and y', then substitute into the original equation.Alternatively, maybe we can use the inverse transformation. Let me recall how transformations work with conic sections.Given a conic section defined by an equation, if we apply a linear transformation represented by a matrix M, the equation of the transformed conic can be found by substituting the inverse transformation. That is, if (x, y) = M⁻¹*(x', y'), then substituting into the original equation gives the equation in terms of x' and y'.Alternatively, since AB is the transformation matrix, to get from the original coordinates (x, y) to the new coordinates (x', y'), we have:[x'] [0 2] [x][y'] = [1 0] [y]Which would mean x' = 0*x + 2*y = 2y, and y' = 1*x + 0*y = x.So solving for x and y in terms of x' and y', we have:From x' = 2y ⇒ y = x'/2From y' = x ⇒ x = y'So substituting x = y' and y = x'/2 into the original equation:Original equation: (x²)/8 + (y²)/2 =1Substitute:(y')² /8 + ( (x')/2 )² /2 =1Simplify each term:First term: (y')² /8 remains as is.Second term: ( (x')² /4 ) /2 = (x')² /8So combining:(y')² /8 + (x')² /8 =1Multiply both sides by 8:(x')² + (y')² =8Wait, that's a circle with radius 2√2. But the original curve was an ellipse. Applying the transformation AB turned it into a circle? Let me check the steps again.Original curve: x²/8 + y²/2 =1. So semi-major axis along x is sqrt(8)=2√2, semi-minor axis along y is sqrt(2).Transformation matrix AB is [[0,2],[1,0]], which swaps x and y and scales. Wait, in terms of transformation, if we have (x', y') = AB*(x,y), then x' = 2y and y' = x.So the original coordinates (x, y) relate to the new coordinates (x', y') by x = y' and y = x'/2.Substituting into the original equation:(y')² /8 + (x'/2)² /2 =1.Yes, so (y')² /8 + (x')² / (8) =1. Then combining terms, (x')² + (y')² =8.So the transformed curve is x² + y² =8. Which is a circle of radius 2√2. But that seems a bit counterintuitive because the original was an ellipse. However, since the transformation includes scaling and rotation (or reflection), maybe it can turn an ellipse into a circle if the axes are scaled appropriately.But let me make sure there's no mistake. Let's verify with a sample point. Take a point on C₁. For example, let’s take x=0, then y²=2, so y=±√2. The point (0, √2) on C₁. Applying the transformation AB: x' =2*√2, y' =0. So in the transformed coordinates, it's (2√2, 0). Plugging into C₂: x² + y² = (2√2)^2 +0=8, which matches.Another point: Take y=0 on C₁. Then x²=8, so x=±2√2. The point (2√2, 0) on C₁. Applying AB: x' =2*0=0, y'=2√2. So transformed point is (0, 2√2). Plug into C₂: 0 + (2√2)^2=8, which works.Another point: Let's take a point not on the axes. For example, x=2, y=√(2*(1 - (4)/8))=√(2*(1 - 0.5))=√(1)=1. So the point (2,1). Applying AB: x'=2*1=2, y'=2. Wait, hold on. Wait, the transformation is x'=2y, y'=x. So (2,1) transformed would be x'=2*1=2, y'=2. So the transformed point is (2,2). Plug into C₂: 2² +2²=4 +4=8, which is correct.So the transformed equation x² + y²=8 is correct. Therefore, part (2)'s answer is x² + y²=8.But let me check if I need to present it in terms of the original variables. Wait, when we perform the transformation, the coordinates are transformed, so the equation in terms of the new coordinates (which are usually denoted the same as the original unless specified). But the problem says "the curve C₁ is transformed by the matrix AB into another curve C₂", so the equation of C₂ is in the standard coordinates, after applying the transformation. However, in linear transformations applied to curves, the usual approach is to express the new curve in terms of the transformed coordinates. But since the transformation is applied to the plane, the equation of C₂ in the original coordinate system is obtained by replacing (x, y) with the inverse transformation.Wait, this is a point of confusion sometimes. Let me clarify.When you apply a linear transformation to a curve, you can think of it as either:1. Active transformation: The points on the curve are transformed by the matrix, leading to a new curve in the same coordinate system.2. Passive transformation: The coordinate system is transformed, and the equation of the curve is expressed in the new coordinates.But in this problem, it says the curve C₁ is transformed by the matrix AB into C₂. So it's an active transformation: each point (x, y) on C₁ is moved to (x', y') = AB*(x, y), and the set of all such (x', y') forms C₂. Therefore, to find the equation of C₂, we need to express the relationship between x' and y' such that (x', y') = AB*(x, y) where (x, y) satisfies C₁'s equation.Therefore, to get the equation of C₂, we can express x and y in terms of x' and y', then substitute into C₁'s equation.Given that:x' = 0*x + 2*y ⇒ x' = 2yy' = 1*x + 0*y ⇒ y' = xTherefore, solving for x and y:x = y'y = x'/2So substitute x = y' and y = x'/2 into C₁'s equation:(x²)/8 + (y²)/2 =1→ ( (y')² )/8 + ( (x')/2 )² /2 =1Calculating each term:(y')² /8 + (x')² / (4*2) = (y')² /8 + (x')² /8 = (x')² + (y')² /8 =1? Wait, no, wait.Wait, (x')/2 squared is (x')² /4. Then divided by 2: (x')² /8.Similarly, (y')² /8. So adding them together:(y')² /8 + (x')² /8 =1Which is equivalent to:(x')² + (y')² =8So in terms of the transformed coordinates (x', y'), the equation is x² + y²=8. But since the problem doesn't mention changing coordinate systems, and just asks for the equation of the transformed curve in the original coordinate system, it's just x² + y²=8. Wait, but actually, no. If the transformation is applied to the points, then the coordinates (x', y') are in the same coordinate system as the original (x, y). Wait, this is confusing.Wait, maybe I need to think of it as the transformation is applied to the plane, so that each point (x, y) is moved to (x', y') = AB*(x, y). Then, the curve C₂ consists of all points (x', y') such that (x, y) was on C₁. Therefore, to get the equation of C₂ in terms of x' and y', we need to express the original variables in terms of the new ones and plug into the original equation.Which is exactly what I did. So the equation becomes (x')² + (y')² =8. But since we usually drop the primes when writing the equation in the original coordinate system, the equation of C₂ is x² + y²=8.Yes, that makes sense. So the answer is x² + y²=8. Let me check again with another point.Take a point on C₂: say (2, 2). Then according to C₂, 2² +2²=8, which is correct. The corresponding point on C₁ would be x = y' =2, y =x'/2=1. So (2,1) on C₁. Check if that's on C₁: 2²/8 +1²/2 =4/8 +1/2=0.5 +0.5=1. Correct.Another point: (0, 2√2) on C₂. Then x'=0, y'=2√2. So original x = y' =2√2, y =x'/2=0. So (2√2, 0) on C₁. Check: ( (2√2)^2 )/8 +0=8/8=1. Correct.So all the sample points work, so I think the answer is correct.Therefore, part (2) equation is x² + y²=8.Final Answer(1) boxed{begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}};(2) boxed{dfrac{x^2}{8} + dfrac{y^2}{8} = 1} or boxed{x^2 + y^2 = 8}Wait, hold on, the second part. The transformed equation is x² + y²=8. But maybe the problem expects it in the standard form, which for a circle would be x² + y²=8. But if they prefer it as an ellipse equation, but since it's a circle, both forms are acceptable. Wait, the original curve was an ellipse, but after transformation, it's a circle. So in standard form, the circle equation is x² + y²=8. Alternatively, you can write it as x²/8 + y²/8=1, which is the same thing.But in the problem statement, part (2) says "find the equation of C₂". The original C₁ was given as x²/8 + y²/2=1. So if we write C₂ as x² + y²=8, that's correct, but if we write it in the similar form as C₁, which is divided by 8, then x²/8 + y²/8=1. Either form is correct. However, since x² + y²=8 is simpler, perhaps that's better. But the problem might expect the same structure as C₁, so maybe x²/8 + y²/8=1. Let me check.If we write x² + y²=8, that's a circle with radius 2√2. Alternatively, x²/8 + y²/8=1 is the same equation, written in standard form. Both are correct, but maybe the answer expects the standard form. Let me see. The original C₁ was written as x²/8 + y²/2=1, which is the standard form of an ellipse. The transformed C₂ is a circle, so writing it as x² + y²=8 is acceptable, but writing it as x²/8 + y²/8=1 is also correct. Since the problem doesn't specify, but given that the original is in the x²/a² + y²/b²=1 form, maybe they expect the same for C₂, even though it's a circle.But in that case, x²/8 + y²/8=1. However, in that case, the answer is presented as x² + y²=8, which is simpler. Let me check again with the transformation.But the question says: "the curve C₁ is transformed by the matrix AB into another curve C₂", so the equation is in the original coordinate system. So when we perform the active transformation, the equation becomes x² + y²=8, which is the equation in the original coordinate system. Therefore, the answer is x² + y²=8. However, in case the problem expects the standard form with denominators, but since it's a circle, both are acceptable.But in the problem statement, part (2) says "find the equation of C₂". The original C₁ is given as x²/8 + y²/2=1, which is standard for an ellipse. If C₂ is a circle, then writing it as x² + y²=8 is correct, but also x²/8 + y²/8=1 is equivalent. However, since the circle is a special case of an ellipse, they might prefer the standard form. Let's check with the multiplication.Wait, if I leave it as x² + y²=8, that's fine. If I write it as x²/8 + y²/8=1, that's also correct. Let me see. Since the problem is from a matrices and transformations elective, perhaps they want the equation in the same form as C₁, which is (x²)/something + (y²)/something=1. Therefore, x²/8 + y²/8=1 would be parallel. But the circle equation is more concise as x² + y²=8.But I need to check the problem statement again. It says "find the equation of C₂". It doesn't specify the form. Therefore, both answers are correct, but to be safe, perhaps present both. Wait, but in the answer box, I can only put one. Wait, the user instructions say "put your final answer within boxed{}". So I need to choose one. The original problem's curve was written in the standard ellipse form. Since after transformation, it's a circle, which can be written in either form. However, the most straightforward answer is x² + y²=8, but if we write it as x²/8 + y²/8=1, that's also correct. Let me check the process again.When we substitute, we ended up with (x')² + (y')²=8, so in terms of the original coordinates (if we consider the transformation as moving the points), the equation in the original coordinate system is x² + y²=8. Alternatively, if we consider the transformed coordinates (x', y'), then the equation is (x')² + (y')²=8, but since we're supposed to express C₂ in the original coordinate system, it's x² + y²=8. Therefore, I think the answer is x² + y²=8.But let me check with another perspective. Suppose the transformation matrix AB is applied to the plane, so every point (x, y) is mapped to (2y, x). Then the curve C₂ is the image of C₁ under this transformation. So to find the equation of C₂, we take a general point (X, Y) on C₂, which is the image of a point (x, y) on C₁. Thus, X=2y and Y=x. Therefore, x=Y and y=X/2. Substituting into C₁'s equation:(Y)²/8 + (X/2)²/2 =1Which is Y²/8 + X²/(8) =1, so X² + Y²=8. Therefore, the equation is X² + Y²=8. Replacing X and Y with x and y (since the coordinates are just dummy variables), the equation is x² + y²=8.Therefore, the correct answer is x² + y²=8. So boxed{x^2 + y^2 = 8} is the answer for part (2).Final Answer(1) boxed{begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}};(2) boxed{x^2 + y^2 = 8}</think>(1) To find the product (AB) of matrices (A) and (B):Given matrices:[ A = begin{bmatrix} 0 & 1 1 & 0 end{bmatrix}, quad B = begin{bmatrix} 1 & 0 0 & 2 end{bmatrix} ]Multiplying (A) and (B):[AB = begin{bmatrix} 0 & 1 1 & 0 end{bmatrix} begin{bmatrix} 1 & 0 0 & 2 end{bmatrix} = begin{bmatrix} 0 cdot 1 + 1 cdot 0 & 0 cdot 0 + 1 cdot 2 1 cdot 1 + 0 cdot 0 & 1 cdot 0 + 0 cdot 2 end{bmatrix} = begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}](2) To find the equation of the curve (C_2) after transforming (C_1: frac{x^2}{8} + frac{y^2}{2} = 1) by the matrix (AB):The transformation matrix (AB) is:[AB = begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}]This transformation maps a point ((x, y)) to ((x', y')) where:[x' = 2y quad text{and} quad y' = x]Expressing (x) and (y) in terms of (x') and (y'):[x = y' quad text{and} quad y = frac{x'}{2}]Substituting these into the original equation of (C_1):[frac{(y')^2}{8} + frac{left(frac{x'}{2}right)^2}{2} = 1]Simplifying:[frac{(y')^2}{8} + frac{(x')^2}{8} = 1 implies (x')^2 + (y')^2 = 8]Thus, the equation of the transformed curve (C_2) is:[x^2 + y^2 = 8] Final Answer(1) (boxed{begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}})(2) (boxed{x^2 + y^2 = 8}),<think>Okay, so I need to solve this problem about matrices and transformations. Let's take it step by step. The first part is to find the product of matrices A and B, which is AB. Then, using that matrix AB, I have to figure out how it transforms the given curve C₁ into C₂ and find the equation of C₂. Starting with part (1): Finding AB. Matrix A is given as:[A = begin{bmatrix} 0 & 1 1 & 0 end{bmatrix}]And matrix B is:[B = begin{bmatrix} 1 & 0 0 & 2 end{bmatrix}]To multiply these matrices, AB means we apply B first and then A, right? Wait, actually, when we write AB, it's A multiplied by B, so the order is important. Matrix multiplication is not commutative, so AB might not be equal to BA. Let me remember how matrix multiplication works. Each element of the resulting matrix is the dot product of the corresponding row of the first matrix and column of the second matrix.So, let's compute AB. First, the resulting matrix will be 2x2 since both A and B are 2x2. Let me denote AB as:[AB = begin{bmatrix} (0*1 + 1*0) & (0*0 + 1*2) (1*1 + 0*0) & (1*0 + 0*2) end{bmatrix}]Wait, hold on. Let me recast that properly. The element in the first row, first column of AB is computed by taking the first row of A and the first column of B. So first row of A is [0, 1], first column of B is [1, 0]. So 0*1 + 1*0 = 0 + 0 = 0.First row, second column: first row of A [0,1] and second column of B [0,2]. So 0*0 + 1*2 = 0 + 2 = 2.Second row, first column: second row of A [1,0] and first column of B [1,0]. So 1*1 + 0*0 = 1 + 0 = 1.Second row, second column: second row of A [1,0] and second column of B [0,2]. So 1*0 + 0*2 = 0 + 0 = 0.So putting it all together, AB is:[AB = begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}]Wait, let me double-check that. Maybe I made a mistake in the multiplication. Let's verify each entry again.First entry: row 1 of A, column 1 of B: 0*1 + 1*0 = 0. Correct.Second entry: row 1 of A, column 2 of B: 0*0 + 1*2 = 2. Correct.Third entry: row 2 of A, column 1 of B: 1*1 + 0*0 = 1. Correct.Fourth entry: row 2 of A, column 2 of B: 1*0 + 0*2 = 0. Correct. So yes, AB is indeed:[AB = begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}]Okay, that seems right. So part (1) is done. AB is the matrix above.Now part (2): Transforming the curve C₁ using the matrix AB into C₂, and finding the equation of C₂.The given curve is C₁: (x²)/8 + (y²)/2 = 1. That's an ellipse, right? Because it's in the standard form of an ellipse (x²/a² + y²/b² = 1). So the original ellipse has semi-major axis length sqrt(8) along the x-axis and semi-minor axis sqrt(2) along the y-axis.But when we apply a linear transformation using matrix AB, the ellipse will be transformed into another ellipse (or possibly another conic section, but since linear transformations preserve the degree, it should still be an ellipse). To find the equation of the transformed curve, we need to figure out how the transformation affects the coordinates (x, y).First, let me recall that when a linear transformation is applied using a matrix M, the transformation takes a point (x, y) to (M11*x + M12*y, M21*x + M22*y). So, if we denote the transformed coordinates as (x', y'), then:x' = M11*x + M12*yy' = M21*x + M22*yBut in our case, the matrix AB is the transformation matrix. So we need to express the original coordinates (x, y) in terms of the new coordinates (x', y'), or vice versa. Wait, maybe not. Let me think carefully.Alternatively, if we consider that applying the transformation matrix AB to the curve C₁, we can represent the transformation as:If a point (x, y) lies on C₁, then after applying AB, the new point (x', y') is AB * [x; y]. So:[begin{bmatrix} x' y' end{bmatrix} = AB begin{bmatrix} x y end{bmatrix} = begin{bmatrix} 0 & 2 1 & 0 end{bmatrix} begin{bmatrix} x y end{bmatrix} = begin{bmatrix} 0*x + 2*y 1*x + 0*y end{bmatrix} = begin{bmatrix} 2y x end{bmatrix}]So, x' = 2y and y' = x.Therefore, to find the equation of C₂, which is the image of C₁ under AB, we can substitute x and y in terms of x' and y' into the original equation.Since x' = 2y and y' = x, we can solve for x and y in terms of x' and y'. Let's do that.From y' = x, we get x = y'.From x' = 2y, we get y = x' / 2.So substituting back into the original equation:Original equation: (x²)/8 + (y²)/2 = 1Substituting x = y' and y = x'/2:( (y')² ) /8 + ( (x'/2)² ) /2 = 1Simplify each term:First term: (y')² /8Second term: (x'^2 /4)/2 = x'^2 /8So combining them:(y')² /8 + (x')² /8 = 1Multiply both sides by 8 to eliminate denominators:(y')² + (x')² = 8But since x' and y' are just the coordinates of the transformed curve, we can rewrite this equation in terms of x and y (without the primes) for C₂. So replacing x' with x and y' with y:y² + x² = 8Which simplifies to:x² + y² = 8Wait, so the transformed curve C₂ is a circle with radius sqrt(8) = 2*sqrt(2). That's interesting. The original ellipse is transformed into a circle by the matrix AB.But let me check my steps again to make sure I didn't make a mistake.First, the transformation:Original coordinates (x, y) are transformed by AB to (x', y') = (2y, x). So to express x and y in terms of x' and y', we invert this relationship.From x' = 2y ⇒ y = x'/2From y' = x ⇒ x = y'So substituting into the original equation:(x²)/8 + (y²)/2 = ( (y')² )/8 + ( (x'/2)² )/2 = (y'²)/8 + (x'²)/4/2 = (y'²)/8 + (x'²)/8 = (x'² + y'²)/8 = 1 ⇒ x'² + y'² = 8. Yes, that seems correct. Therefore, the equation of C₂ is x² + y² = 8. So simplifying, we can write it as x² + y² = 8. So in standard form, it's a circle centered at the origin with radius 2√2.But let me double-check if I inverted the transformation correctly. Let's see, if (x', y') = AB*(x, y) = (2y, x), then to get x and y in terms of x' and y', since y' = x, so x = y'. And x' = 2y ⇒ y = x'/2. That's correct. So substitution is correct.Alternatively, another way to think about transformations on conic sections is using matrix representation. The general equation of a conic is X^T Q X = 1, where X is a vector [x; y] and Q is a matrix. For the original ellipse, (x²)/8 + (y²)/2 = 1, the matrix Q is diagonal with entries 1/8 and 1/2.When we apply a linear transformation with matrix M, the transformed conic is given by (M^{-1}X)^T Q (M^{-1}X) = 1. So expanding that, X^T (M^{-T} Q M^{-1}) X = 1. Therefore, the new matrix Q' is M^{-T} Q M^{-1}. So if we compute that, we can get the equation of the transformed conic.But maybe this method is more complicated, but let's try it to verify.First, original Q matrix is:Q = begin{bmatrix} 1/8 & 0 0 & 1/2 end{bmatrix}Transformation matrix M is AB = begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}So we need to compute M^{-T} Q M^{-1}First, find the inverse of M. Let's compute M inverse.Given M = begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}The determinant of M is (0)(0) - (2)(1) = -2. Therefore, inverse of M is (1/det(M)) * begin{bmatrix} 0 & -2 -1 & 0 end{bmatrix} = (1/-2)*begin{bmatrix} 0 & -2 -1 & 0 end{bmatrix} = begin{bmatrix} 0 & 1 0.5 & 0 end{bmatrix}Wait, let's check:Original matrix M:Row 1: 0, 2Row 2: 1, 0To compute inverse, swap diagonal elements, change sign of off-diagonal, divide by determinant.But the formula is:For a 2x2 matrix [a b; c d], inverse is [d -b; -c a] / (ad - bc)So for M:a = 0, b = 2, c = 1, d = 0Determinant: (0)(0) - (2)(1) = -2Inverse is [0 -2; -1 0] / (-2) = [0/(-2) (-2)/(-2); (-1)/(-2) 0/(-2)] = [0 1; 0.5 0]So M^{-1} = begin{bmatrix} 0 & 1 0.5 & 0 end{bmatrix}Therefore, M^{-T} is the transpose of M^{-1}, which is begin{bmatrix} 0 & 0.5 1 & 0 end{bmatrix}Now compute Q' = M^{-T} Q M^{-1}First, compute M^{-T} Q:M^{-T} = begin{bmatrix} 0 & 0.5 1 & 0 end{bmatrix}Multiply by Q = begin{bmatrix} 1/8 & 0 0 & 1/2 end{bmatrix}Result is:First row: 0*(1/8) + 0.5*(0) = 0; 0*0 + 0.5*(1/2) = 0.25Second row: 1*(1/8) + 0*0 = 1/8; 1*0 + 0*(1/2) = 0Wait, no. Wait, matrix multiplication is rows of first matrix times columns of second matrix. Let's do it properly.First row of M^{-T} is [0, 0.5]Multiply by columns of Q:First column of Q: [1/8, 0]Second column of Q: [0, 1/2]So first entry: 0*(1/8) + 0.5*0 = 0Second entry: 0*0 + 0.5*(1/2) = 0.25Second row of M^{-T} is [1, 0]Multiply by columns of Q:First column: 1*(1/8) + 0*0 = 1/8Second column: 1*0 + 0*(1/2) = 0Therefore, M^{-T} Q is:[begin{bmatrix} 0 & 0.25 1/8 & 0 end{bmatrix}]Now multiply this by M^{-1} = begin{bmatrix} 0 & 1 0.5 & 0 end{bmatrix}Compute the product:First row: [0, 0.25] multiplied by columns of M^{-1}First column of M^{-1}: [0, 0.5]0*0 + 0.25*0.5 = 0.125Second column of M^{-1}: [1, 0]0*1 + 0.25*0 = 0Second row: [1/8, 0] multiplied by columns of M^{-1}First column: 1/8*0 + 0*0.5 = 0Second column: 1/8*1 + 0*0 = 1/8Therefore, the resulting matrix Q' is:[begin{bmatrix} 0.125 & 0 0 & 1/8 end{bmatrix}]So the transformed conic equation is X^T Q' X = 1, which translates to:0.125x² + (1/8)y² = 1But 0.125 is 1/8, so:(1/8)x² + (1/8)y² = 1Multiply both sides by 8:x² + y² = 8Which is the same result as before! So that confirms the equation of C₂ is x² + y² = 8. Therefore, part (2) answer is x² + y² = 8.But just to be thorough, let me think about another approach. If we consider the effect of the transformation AB on the ellipse. The original ellipse is stretched and maybe rotated.Matrix AB is begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}This matrix has the effect of swapping x and y coordinates and scaling. Let me see:Multiplying a vector [x, y] by AB gives [2y, x]. So, it's like taking the original x-coordinate, moving it to the y-coordinate, and scaling the original y-coordinate by 2 and putting it in the x-coordinate. So it's a combination of a reflection (swap of x and y) and scaling.In terms of transformations, swapping x and y is equivalent to a reflection over the line y = x, and then scaling the new x-axis (which was the original y-axis) by 2.Therefore, the ellipse C₁, which was originally stretched along the x-axis with semi-major axis sqrt(8) and along y-axis with semi-minor axis sqrt(2), after transformation, the stretching along the original y-axis (which becomes the new x-axis) is scaled by 2, and the original x-axis (now y-axis) remains the same.Wait, so the original ellipse has semi-major axis along x with length sqrt(8) and semi-minor axis along y with sqrt(2). After swapping x and y, the major axis would be along y, and minor along x. But then scaling the new x-axis (original y-axis) by 2. So the semi-minor axis (which was sqrt(2) along original y-axis) becomes scaled by 2, so 2*sqrt(2) along the new x-axis. The semi-major axis (originally sqrt(8) along x) becomes along the new y-axis, still sqrt(8). But sqrt(8) is 2*sqrt(2), so both axes are 2*sqrt(2). Hence, it becomes a circle with radius 2*sqrt(2). That's consistent with the equation x² + y² = 8.Therefore, all the different approaches lead to the same conclusion. So I'm confident that the equation of C₂ is x² + y² = 8.Final Answer(1) boxed{begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}};(2) boxed{dfrac{x^2}{8} + dfrac{y^2}{8} = 1} or boxed{x^2 + y^2 = 8}Wait, the problem says to put the final answer within boxed{} for each part. The first part is the matrix, which I have as AB = [0 2; 1 0]. The second part is the equation of C₂. The user wrote "find the equation of C₂". The equation can be written as x² + y² = 8, which is a circle. Alternatively, if written in standard form, since it's a circle, but technically the standard form for a circle is x² + y² = r². However, if they prefer the ellipse form (even though it's a circle), maybe they want it as x²/8 + y²/8 =1. Either is correct, but since a circle is a special case of an ellipse, perhaps writing both is okay. Let me check my previous step.From substitution, we had (x')² + (y')² =8. If they want the equation in terms of x and y (without primes), it's x² + y² =8. Alternatively, dividing both sides by 8 gives x²/8 + y²/8 =1, which is also correct. Depending on how the question expects the answer. Since the original equation was in the form x²/a² + y²/b² =1, maybe they expect the same form, so x²/8 + y²/8=1. But x² + y²=8 is simpler. Let me check the problem statement again. It says "find the equation of C₂". It doesn't specify the form, so both are correct. But maybe they want it in standard form. For a circle, standard form is x² + y² = r². For an ellipse, it's x²/a² + y²/b²=1. Since the transformed curve is a circle, writing it as x² + y²=8 is appropriate. However, if written as x²/8 + y²/8=1, that's also acceptable. The question might accept either. Given that the original equation was written as x²/8 + y²/2 =1, maybe the transformed equation should follow the same structure. So, x²/8 + y²/8 =1. Either way, both equations are equivalent. The answer can be written either as x² + y² =8 or x²/8 + y²/8 =1. To be safe, perhaps present both. But since the problem didn't specify, maybe the simplest form is x² + y²=8. However, in the problem's original equation, they have denominators, so maybe they expect the same style. Hmm. Let me think.Original equation: x²/8 + y²/2=1. The transformed equation is x² + y²=8. Alternatively, if written as x²/8 + y²/8=1. Let me check if the problem in Chinese (assuming translation) has any preference. Since it's from an elective course, perhaps following the same structure is better. Since the original is in ellipse form, even though the transformed is a circle, writing it as x²/8 + y²/8=1 is more consistent. So maybe that's the expected answer.But the user wrote "put your final answer within boxed{}" for each part. For part (2), the problem says "find the equation of C₂". In the original problem, C₁ is given in the form x²/8 + y²/2=1, which is standard ellipse form. The transformed curve, if it's a circle, can be written as x² + y²=8. However, if written as x²/8 + y²/8=1, that's still correct. Let me check which one is more appropriate.In the context of transformations, when you apply a linear transformation to an ellipse, the resulting equation can be written in the standard form by using the inverse transformation. Since in the original C₁, the equation is in terms of x and y, and after the transformation, we express the original x and y in terms of the new coordinates. But in the final answer, since we're expressing the new curve in terms of the new coordinates (which we can rename back to x and y), the equation is x² + y²=8. Therefore, that's the equation of C₂. So I think the answer should be boxed{x^2 + y^2 = 8}. Alternatively, if they require the standard ellipse form, then boxed{dfrac{x^2}{8} + dfrac{y^2}{8} = 1}. But since it's a circle, x² + y²=8 is more straightforward. Let me confirm once more with my previous steps.Original substitution gave me x'² + y'²=8, then relabeling the coordinates gives x² + y²=8. So yes, that's correct. Hence, I think boxed{x^2 + y^2 = 8} is the answer for part (2). The other form is also correct but redundant since it's a circle. So probably the expected answer is x² + y²=8.Final Answer(1) boxed{begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}};(2) boxed{x^2 + y^2 = 8}</think>(1) To find the product of matrices (A) and (B):Given matrices:[A = begin{bmatrix} 0 & 1 1 & 0 end{bmatrix}, quad B = begin{bmatrix} 1 & 0 0 & 2 end{bmatrix}]The product (AB) is calculated as follows:[AB = begin{bmatrix} 0 & 1 1 & 0 end{bmatrix} begin{bmatrix} 1 & 0 0 & 2 end{bmatrix} = begin{bmatrix} 0 cdot 1 + 1 cdot 0 & 0 cdot 0 + 1 cdot 2 1 cdot 1 + 0 cdot 0 & 1 cdot 0 + 0 cdot 2 end{bmatrix} = begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}](2) To find the equation of the curve (C_2) transformed by the matrix (AB):The original curve (C_1) is given by:[frac{x^2}{8} + frac{y^2}{2} = 1]The transformation matrix (AB) is:[AB = begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}]Applying the transformation to a point ((x, y)), the new coordinates ((x', y')) are:[begin{bmatrix} x' y' end{bmatrix} = begin{bmatrix} 0 & 2 1 & 0 end{bmatrix} begin{bmatrix} x y end{bmatrix} = begin{bmatrix} 2y x end{bmatrix}]Thus, (x' = 2y) and (y' = x). Solving for (x) and (y) in terms of (x') and (y'):[x = y' quad text{and} quad y = frac{x'}{2}]Substituting these into the original equation of (C_1):[frac{(y')^2}{8} + frac{left(frac{x'}{2}right)^2}{2} = 1]Simplifying:[frac{(y')^2}{8} + frac{(x')^2}{8} = 1 implies frac{(x')^2 + (y')^2}{8} = 1 implies (x')^2 + (y')^2 = 8]Renaming ((x', y')) back to ((x, y)), the equation of (C_2) is:[x^2 + y^2 = 8] Final Answer(1) boxed{begin{bmatrix} 0 & 2 1 & 0 end{bmatrix}};(2) boxed{x^2 + y^2 = 8}]

🔑:Solution: (1) AB=begin{bmatrix}0 & 1 1 & 0end{bmatrix}begin{bmatrix}1 & 0 0 & 2end{bmatrix}=begin{bmatrix}0 & 2 1 & 0end{bmatrix}, (2) Let point P(x,y) be any point on the curve C_{1}, Point P is transformed by the matrix AB into point P′(x_{0},y_{0}), Then begin{bmatrix}0 & 2 1 & 0end{bmatrix}begin{bmatrix}x yend{bmatrix}=begin{bmatrix}2y xend{bmatrix},That is, x_{0}=2y, y_{0}=x, therefore x=y_{0}, y=frac{x_{0}}{2}, therefore frac{x_{0}^{2}}{8}+ frac{y_{0}^{2}}{8}=1, i.e., x_{0}^{2}+y_{0}^{2}=8, therefore The equation of curve C_{2} is boxed{x^{2}+y^{2}=8}.

🧠:[<think>Okay, so I need to find the value of 'a' that minimizes the maximum distance between the two functions f(x) and g(x) over the interval [-2, 2]. The distance is defined as the largest absolute difference between f(x) and g(x) in that interval. Let me start by understanding both functions.First, f(x) = 2 - |x|. That's a V-shaped graph with the vertex at (0, 2). It decreases linearly with a slope of -1 for x > 0 and slope 1 for x < 0, hitting the x-axis at x = 2 and x = -2. So at x=0, f(x)=2, and at x=±2, f(x)=0.Next, g(x) = -a x² + 2. This is a downward-opening parabola (since the coefficient of x² is negative) with vertex at (0, 2). The parameter 'a' controls how "wide" or "narrow" the parabola is. A larger 'a' makes the parabola narrower (steeper), while a smaller 'a' makes it wider (shallower).The problem is to find the value of 'a' such that the maximum vertical distance between f(x) and g(x) over the interval [-2, 2] is minimized. So, essentially, we need to minimize the maximum of |f(x) - g(x)| over x in [-2, 2].Let me write the expression for the difference between the two functions:|f(x) - g(x)| = |(2 - |x|) - (-a x² + 2)| = | - |x| + a x² | = |a x² - |x||.So, we need to minimize the maximum value of |a x² - |x|| over x in [-2, 2]. Since both functions are even (symmetric about the y-axis), we can consider x ≥ 0 and double the interval to [0, 2], which might simplify things.So, let's define h(x) = a x² - x for x in [0, 2]. Then, the absolute value |h(x)| is the quantity we need to consider. Our goal is to choose 'a' such that the maximum of |h(x)| over [0, 2] is minimized.This seems like a problem in approximation theory, specifically minimax approximation. The idea is to find the coefficient 'a' such that the maximum deviation between the two functions is minimized. I remember that in such problems, the optimal solution occurs when the maximum error is minimized and the error alternates between its maximum and minimum values at several points in the interval. According to the equioscillation theorem, the best approximation will equioscillate between at least n+2 points for polynomials of degree n. But here, since we are fitting a quadratic function to a piecewise linear function, maybe the optimal 'a' will have the maximum error occurring at three points?Alternatively, maybe we can approach this by calculus. To find the maximum of |h(x)|, we can look for critical points where the derivative is zero, as well as check the endpoints.First, let's consider h(x) = a x² - x. The absolute value |h(x)| can be analyzed by considering where h(x) is positive or negative.Since a is positive, h(x) is a quadratic function in x. Let's see:h(x) = a x² - x. The parabola opens upwards because the coefficient of x² is positive. The vertex is at x = -b/(2a) for ax² + bx + c, so here, it's x = (1)/(2a). At x = 1/(2a), h(x) = a*(1/(4a²)) - 1/(2a) = 1/(4a) - 1/(2a) = -1/(4a). So, the vertex is at (1/(2a), -1/(4a)). Since the parabola opens upwards, the minimum of h(x) is at this vertex.So, h(x) = a x² - x is negative at x = 1/(2a), with value -1/(4a). Therefore, |h(x)| reaches a maximum either at the endpoints x=0, x=2, or at critical points where h(x) is maximum in absolute value.But since h(x) is a quadratic, let's analyze the behavior over [0, 2].At x=0: h(0) = 0 - 0 = 0. So |h(0)| = 0.At x=2: h(2) = a*(4) - 2 = 4a - 2. So |h(2)| = |4a - 2|.The vertex of h(x) is at x = 1/(2a). Let's check whether this vertex is within the interval [0, 2]. Since a > 0, 1/(2a) > 0. So, we need to check if 1/(2a) ≤ 2. That is, 1/(2a) ≤ 2 => 1/2 ≤ 2a => a ≥ 1/4. So, if a ≥ 1/4, the vertex is within [0, 0.5], otherwise, if a < 1/4, the vertex is at x = 1/(2a) > 2. Wait, no, if a < 1/4, then 1/(2a) > 2, which is outside the interval [0, 2]. So, for a < 1/4, the minimum of h(x) occurs outside the interval, so h(x) is increasing on [0, 2], since the vertex is to the right of 2, and since the parabola opens upwards. Wait, but if a < 1/4, then h(x) = a x² - x. The derivative is h'(x) = 2 a x - 1. Setting derivative zero gives x = 1/(2a) > 2 (since a < 1/4). So, on [0, 2], h'(x) = 2 a x - 1. At x=0, h'(0) = -1 < 0. At x=2, h'(2) = 4a -1. If a < 1/4, then 4a -1 < 0. Therefore, the derivative is always negative on [0, 2], so h(x) is decreasing on [0, 2]. Therefore, h(x) starts at 0 when x=0, decreases to h(2) = 4a - 2. So, since h(x) is decreasing, the maximum of |h(x)| on [0, 2] would be the maximum between |h(0)|=0 and |h(2)|=|4a - 2|. But since h(x) is decreasing from 0 to 4a - 2. If 4a - 2 is negative (i.e., a < 0.5), then the maximum |h(x)| is |4a - 2| = 2 - 4a. If 4a - 2 is non-negative (a ≥ 0.5), then |h(2)|=4a - 2, but since h(x) is decreasing from 0 to 4a - 2, which would be positive. Wait, but if a ≥ 0.5, h(2) = 4a - 2 ≥ 0, but since the function h(x) starts at 0 and increases? Wait, no, because if a ≥ 0.5, then h'(x) = 2 a x -1. At x=0, h'(0) = -1. At x=2, h'(2) = 4a -1. If a ≥ 0.5, then 4a -1 ≥ 4*(0.5) -1 = 2 -1 =1 >0. Therefore, h(x) is decreasing at x=0, reaches a minimum at x=1/(2a) (which, if a ≥0.5, then 1/(2a) ≤1), then increases after that. Wait, this seems conflicting. Wait, let's recast:If a ≥ 1/4, the vertex x=1/(2a) is ≤ 2. But if a ≥0.5, the vertex is at x=1/(2a) ≤1/(2*0.5)=1. So, h(x) is decreasing on [0, 1/(2a)] and increasing on [1/(2a), 2]. Therefore, the minimum of h(x) is at x=1/(2a), which is -1/(4a). Then, h(x) increases from there to x=2, where h(2)=4a -2.Therefore, the maximum of |h(x)| on [0, 2] for a ≥1/4 is the maximum of |h(0)|=0, |h(2)|=|4a -2|, and |h(1/(2a))|=1/(4a). So, depending on the value of a, the maximum could be either |4a -2| or 1/(4a).So, for a ≥1/4, we need to compare 1/(4a) and |4a - 2|.But 4a -2 can be positive or negative. Let's see:If a ≥ 0.5, then 4a -2 ≥0. So |4a -2|=4a -2. If 1/4 ≤a <0.5, then 4a -2 <0, so |4a -2|=2 -4a.Therefore, for a ≥1/4, the maximum of |h(x)| is the maximum between 1/(4a) and |4a -2|.So, to find the minimal maximum |h(x)|, we need to find 'a' such that these two values are equal, because if we can set them equal, then the maximum will be the same at both points, which might be the minimal possible maximum.Wait, this is a common approach in optimization: to balance the maximums at different points. So, if we can make the maximum at x=2 equal to the maximum at x=1/(2a), then that might give the minimal possible maximum.Therefore, set 1/(4a) = |4a - 2|.But this depends on whether 4a -2 is positive or negative.Case 1: a ≥0.5Then, |4a -2|=4a -2. So set 1/(4a) =4a -2.Multiply both sides by 4a: 1 = 16a² -8aThen, 16a² -8a -1=0.Solving quadratic equation:a = [8 ± sqrt(64 + 64)] /32 = [8 ± sqrt(128)] /32 = [8 ± 8*sqrt(2)] /32 = [1 ± sqrt(2)] /4But since a ≥0.5, let's compute the roots:[1 + sqrt(2)] /4 ≈ (1 +1.414)/4 ≈2.414/4≈0.6035[1 - sqrt(2)] /4≈(1 -1.414)/4≈(-0.414)/4≈-0.1035Discard the negative root. So a≈0.6035, which is greater than 0.5, so valid.Case 2: 1/4 ≤a <0.5Then, |4a -2|=2 -4a. So set 1/(4a) =2 -4a.Multiply both sides by 4a: 1 =8a -16a²Then, 16a² -8a +1=0Quadratic equation: a = [8 ± sqrt(64 -64)] /32 = [8 ±0]/32 = 8/32=0.25So a=0.25. But 0.25 is exactly 1/4, which is the lower bound of this case. But in this case, we considered 1/4 ≤a <0.5, and the solution is a=1/4. So, let's check if at a=1/4, 1/(4a)=1/(4*(1/4))=1, and 2 -4a=2 -1=1. So yes, they are equal. Therefore, at a=1/4, the maximum |h(x)| is 1. But wait, if a=1/4, then h(x)= (1/4)x² -x. Let's check at x=2: h(2)= (1/4)*4 -2=1 -2=-1. So |h(2)|=1. At the vertex x=1/(2a)=1/(2*(1/4))=2. So the vertex is at x=2. But x=2 is the endpoint. Wait, that seems conflicting. Wait, when a=1/4, the vertex is at x=1/(2*(1/4))=2. So the vertex is at x=2. But that's the endpoint. Therefore, h(x) is decreasing on [0,2], since the vertex is at x=2. Wait, but if a=1/4, h'(x)=2*(1/4)x -1= (1/2)x -1. At x=0, derivative is -1. At x=2, derivative is (1/2)*2 -1=1 -1=0. So h(x) is decreasing on [0,2], with minimum at x=2. Therefore, the maximum |h(x)| occurs at x=2, which is |h(2)|=1. However, if we set a=1/4, then h(x) is decreasing from 0 at x=0 to -1 at x=2. So the maximum |h(x)| is 1. But if we take a slightly larger than 1/4, say a=0.3, then let's compute:At a=0.3, which is in the case 1/4 <a <0.5.h(x)=0.3x² -x.The vertex is at x=1/(2*0.3)=1/0.6≈1.6667.At the vertex, h(x)=0.3*(1.6667)^2 -1.6667≈0.3*(2.7778) -1.6667≈0.8333 -1.6667≈-0.8334. So |h(x)|≈0.8334.At x=2, h(2)=0.3*4 -2=1.2 -2=-0.8. So |h(2)|=0.8.So the maximum is 0.8334, which is larger than 0.8. So in this case, the maximum |h(x)| is at the vertex.But if we set a=0.25, the maximum is 1, which is larger. So as a increases from 0.25 to 0.5, the maximum |h(x)| at the vertex decreases from 1 to ?Wait, wait, when a=0.5, h(x)=0.5x² -x. The vertex is at x=1/(2*0.5)=1. At x=1, h(1)=0.5*1 -1= -0.5. So |h(1)|=0.5. At x=2, h(2)=0.5*4 -2=2 -2=0. So |h(2)|=0. So for a=0.5, the maximum |h(x)| is 0.5 at x=1. But if a=0.5, then the maximum |h(x)| over [0,2] is 0.5. But in the previous case when a≈0.6035, the maximum |h(x)| is 1/(4a)≈1/(4*0.6035)≈0.414. So this is smaller.Therefore, there seems to be a minimal value somewhere when a is greater than 0.5. So when a increases beyond 0.5, the maximum |h(x)| is the maximum between 1/(4a) and 4a -2. So when a=0.5, 4a -2=0, and 1/(4a)=0.5. As a increases beyond 0.5, 4a -2 increases, but 1/(4a) decreases. So perhaps the two functions 1/(4a) and 4a -2 cross somewhere, and the minimal maximum is achieved when they are equal.Yes, this is what we considered in case 1. So setting 1/(4a)=4a -2 gives a=(1 + sqrt(2))/4≈0.6035. Then, at this a, the maximum |h(x)| is 1/(4a)=4a -2. Let's compute it:1/(4a)=1/(4*(1 + sqrt(2))/4)=1/(1 + sqrt(2))≈1/2.414≈0.414.Similarly, 4a -2=4*(1 + sqrt(2))/4 -2=1 + sqrt(2) -2=sqrt(2)-1≈0.414. So both are equal, as expected.So, this gives the minimal maximum |h(x)|≈0.414, achieved at a=(1 + sqrt(2))/4.But wait, let me confirm. If we take a=(1 + sqrt(2))/4≈0.6035, then:At x=1/(2a)=1/(2*(1 + sqrt(2))/4)=4/(2*(1 + sqrt(2)))=2/(1 + sqrt(2))≈2/2.414≈0.828. So x≈0.828.At this x, h(x)=a x² -x. Let's compute:a x² -x = a*( (2/(1 + sqrt(2)))^2 ) - (2/(1 + sqrt(2)))First, compute (2/(1 + sqrt(2)))^2:Denominator: (1 + sqrt(2))^2=1 + 2 sqrt(2) +2=3 + 2 sqrt(2)So (2)^2 / (3 + 2 sqrt(2))=4/(3 + 2 sqrt(2)).Multiply numerator and denominator by (3 - 2 sqrt(2)):4*(3 - 2 sqrt(2))/[(3)^2 - (2 sqrt(2))^2]=4*(3 - 2 sqrt(2))/(9 -8)=4*(3 - 2 sqrt(2))/1=12 -8 sqrt(2).So a x² = a*(4/(3 + 2 sqrt(2)))= [(1 + sqrt(2))/4]*(12 -8 sqrt(2)) ?Wait, perhaps another approach. Since a=(1 + sqrt(2))/4, and x=2/(1 + sqrt(2)), let's compute a x²:a x² = [(1 + sqrt(2))/4] * [4/(1 + sqrt(2))^2] = [(1 + sqrt(2))/4] * [4/( (1 + sqrt(2))^2 ) ]= [1/( (1 + sqrt(2)) ) ].Then, subtract x=2/(1 + sqrt(2)):So h(x)=a x² -x= [1/(1 + sqrt(2))] - [2/(1 + sqrt(2))]= -1/(1 + sqrt(2)).Therefore, |h(x)|=1/(1 + sqrt(2))≈0.414, which matches the previous calculation.Similarly, at x=2, h(2)=4a -2=4*(1 + sqrt(2))/4 -2=1 + sqrt(2) -2=sqrt(2)-1≈0.414. So |h(2)|=sqrt(2)-1≈0.414.Therefore, at a=(1 + sqrt(2))/4, the maximum |h(x)| is sqrt(2)-1≈0.414, occurring both at x=2 and at x=1/(2a)≈0.828. Additionally, since the function h(x) is a quadratic, the graph of |h(x)| will have a peak at x=1/(2a) and another peak at x=2. But since the values at both points are equal, the maximum is the same. Therefore, this is the minimal maximum.Now, we need to check if there are any other points in [0,2] where |h(x)| could be larger. For that, we can check the value at x=1.At x=1, h(1)=a*1 -1=a -1. So |h(1)|=|a -1|.At a=(1 + sqrt(2))/4≈0.6035, |h(1)|=|0.6035 -1|=0.3965, which is less than 0.414. So indeed, the maximum is at x≈0.828 and x=2.Therefore, the minimal maximum occurs when a=(1 + sqrt(2))/4, and the maximum distance is sqrt(2)-1.But wait, the problem is over the interval [-2,2], but we considered x ≥0. Since both functions are even, the analysis for x ≤0 is symmetric, so the maximum will occur in [0,2], so we don't have to worry about the negative side.Therefore, the minimal maximum distance is achieved when a=(1 + sqrt(2))/4≈0.6035, and this is the value of 'a' that minimizes the distance between f and g.But let me check another point. Suppose we take a different value of 'a', say a=0.5. Then, as computed earlier, the maximum |h(x)| is 0.5. If a=0.6, then compute:a=0.6:At vertex x=1/(2*0.6)=0.8333. h(x)=0.6*(0.8333)^2 -0.8333≈0.6*0.6944 -0.8333≈0.4166 -0.8333≈-0.4167. So |h(x)|≈0.4167.At x=2, h(2)=0.6*4 -2=2.4 -2=0.4. So |h(2)|=0.4. Therefore, the maximum is 0.4167, which is higher than the value at a=(1 + sqrt(2))/4≈0.6035, which is≈0.414. So indeed, the minimal maximum is achieved at a=(1 + sqrt(2))/4.Therefore, the answer should be a=(1 + sqrt(2))/4.But let's make sure there isn't a lower value. Suppose we take a higher 'a', say a=0.7:At a=0.7, vertex at x=1/(2*0.7)≈0.7143. h(x)=0.7*(0.7143)^2 -0.7143≈0.7*0.5102 -0.7143≈0.3571 -0.7143≈-0.3572. |h(x)|≈0.3572.At x=2, h(2)=0.7*4 -2=2.8 -2=0.8. So |h(2)|=0.8. Therefore, the maximum is 0.8, which is higher than the previous maximum. So as a increases beyond (1 + sqrt(2))/4≈0.6035, the value at x=2 increases, overtaking the value at the vertex. Therefore, the minimal maximum occurs exactly when the two are equal, at a=(1 + sqrt(2))/4.Therefore, the value of 'a' that minimizes the distance is (1 + sqrt(2))/4.But let me check if this is correct by considering the original problem. The functions are f(x)=2 -|x| and g(x)=-a x² +2. Wait, hold on! Wait, the problem says g(x)=-a x² +2, which is a downward-opening parabola. But in my analysis above, I considered g(x)=a x² +2, which is upward-opening. Did I misread the problem?Oh no! Wait, the problem states g(x)=-a x² +2. So that's a downward-opening parabola. My mistake was in the initial analysis. So h(x)=f(x)-g(x)= (2 -|x|) - (-a x² +2)=2 -|x| +a x² -2= a x² -|x|. So that part is correct. So |h(x)|=|a x² - |x||. However, since g(x) is a downward-opening parabola, h(x)=a x² -|x|. But a is positive. Therefore, h(x)=a x² -|x| is a upward-opening parabola (since coefficient a is positive) minus the absolute value function. Wait, but the difference is still h(x)=a x² -|x|. So my previous analysis was correct because h(x)=a x² -|x|, which is a quadratic function minus a V-shaped function. However, when considering x ≥0, h(x)=a x² -x, which is a quadratic opening upwards. So my previous analysis still holds.But wait, when I considered the graph of h(x)=a x² -x, it's a quadratic opening upwards, so the vertex is at x=1/(2a). But since we are subtracting x, which is a linear term, the parabola opens upwards. However, since g(x) is a downward-opening parabola, the difference h(x) becomes an upward-opening parabola minus a linear function. Therefore, the analysis still holds.Therefore, my previous conclusion that the minimal maximum occurs at a=(1 + sqrt(2))/4 is correct. But let me double-check with the correct g(x).Given f(x)=2 -|x| and g(x)=-a x² +2. Then, the difference is h(x)=f(x)-g(x)=2 -|x| - (-a x² +2)=2 -|x| +a x² -2= a x² -|x|. Therefore, |h(x)|=|a x² -|x||. So yes, as before.Therefore, the analysis is correct, and the minimal maximum occurs when a=(1 + sqrt(2))/4.Therefore, the answer is a=(1 + sqrt(2))/4. To write this as a boxed expression:boxed{dfrac{1 + sqrt{2}}{4}}Final AnswerThe value of (a) that minimizes the distance is boxed{dfrac{1 + sqrt{2}}{4}}.</think>To find the value of (a) that minimizes the maximum distance between the functions (f(x) = 2 - |x|) and (g(x) = -ax^2 + 2) over the interval ([-2, 2]), we need to analyze the absolute difference (|f(x) - g(x)|).The absolute difference is given by:[|f(x) - g(x)| = |(2 - |x|) - (-ax^2 + 2)| = |ax^2 - |x||]We consider the interval ([0, 2]) due to symmetry. Let (h(x) = ax^2 - x). The goal is to minimize the maximum value of (|h(x)|) over ([0, 2]).1. Critical Points Analysis: - The vertex of (h(x) = ax^2 - x) is at (x = frac{1}{2a}). - The value of (h(x)) at the vertex is (-frac{1}{4a}). - The endpoints (x = 0) and (x = 2) must also be considered.2. Case Analysis: - For (a geq frac{1}{4}), the vertex (x = frac{1}{2a}) is within ([0, 2]). - For (a < frac{1}{4}), the vertex is outside ([0, 2]), and (h(x)) is decreasing on ([0, 2]).3. Equating Maximum Values: - For (a geq frac{1}{2}), the maximum value at (x = 2) is (4a - 2). - The maximum value at the vertex is (frac{1}{4a}). - Setting these equal: (frac{1}{4a} = 4a - 2).4. Solving the Equation: - Solving (frac{1}{4a} = 4a - 2) leads to the quadratic equation (16a^2 - 8a - 1 = 0). - The positive solution is (a = frac{1 + sqrt{2}}{4}).5. Verification: - At (a = frac{1 + sqrt{2}}{4}), the maximum values at (x = 2) and the vertex are equal, confirming the minimal maximum distance.Thus, the value of (a) that minimizes the distance is (boxed{dfrac{1 + sqrt{2}}{4}}).,<think>Okay, so I have this problem where I need to find the value of 'a' that minimizes the maximum distance between two functions, f(x) and g(x), over the interval from -2 to 2. The distance is defined as the largest absolute difference between the two functions in that interval. Let me start by understanding the problem step by step.First, the functions are given as f(x) = 2 - |x| and g(x) = -a x² + 2, where a is a positive real number. I need to find the value of 'a' that makes the maximum of |f(x) - g(x)| as small as possible. So essentially, I need to minimize the maximum error between these two functions over the interval.Let me visualize both functions. The function f(x) = 2 - |x| is a V-shaped graph with its vertex at (0, 2) and slopes of -1 and 1 on either side. It will form a triangle between x = -2 and x = 2. At x = -2 and x = 2, f(x) will be 2 - 2 = 0. So it starts at (2, 0) and (-2, 0), peaks at (0, 2).Then, the function g(x) = -a x² + 2 is a downward-opening parabola with vertex at (0, 2) as well. The coefficient 'a' controls how "wide" or "narrow" the parabola is. Since a is positive, it opens downward. The larger the 'a', the narrower the parabola, and the smaller the 'a', the wider it is. So for very small 'a', the parabola is almost flat, just slightly curving downward from 2. For large 'a', it's a steep parabola that dips down quickly from the vertex.The distance between f and g is the maximum of |f(x) - g(x)| over the interval [-2, 2]. So, to minimize this maximum distance, we need to adjust 'a' such that the largest deviation between f(x) and g(x) is as small as possible.Let me write out the expression for |f(x) - g(x)|. Since both f and g are defined piecewise, maybe I can compute their difference and analyze it.First, f(x) = 2 - |x|, and g(x) = -a x² + 2. Therefore, f(x) - g(x) = [2 - |x|] - [-a x² + 2] = 2 - |x| + a x² - 2 = a x² - |x|.Therefore, |f(x) - g(x)| = |a x² - |x||. So the problem reduces to finding the value of 'a' that minimizes the maximum of |a x² - |x|| over x in [-2, 2].Since both functions are even functions (symmetric about the y-axis), because f(x) is even (due to the absolute value) and g(x) is even (since x² is even). Therefore, the absolute difference will also be even, so we can consider only the interval x ∈ [0, 2] and double whatever conclusions we get there. That simplifies things a bit.So let's restrict our analysis to x ≥ 0. Then |x| = x, so the expression becomes |a x² - x|. So we need to analyze this function h(x) = |a x² - x| on [0, 2], find its maximum, then find 'a' that minimizes this maximum.To minimize the maximum of h(x) over [0, 2], we need to adjust 'a' such that the peak of h(x) is as low as possible. This sounds like a problem in Chebyshev approximation, where you want to minimize the maximum deviation, which often leads to the equioscillation theorem. The theorem states that the optimal approximating function will have the error curve attaining its maximum absolute value at least n+1 times, where n is the number of parameters. In this case, since we're adjusting a single parameter 'a', the optimal solution should have at least two points in [0, 2] where the error is equal to the maximum, and the error alternates in sign between them. But since our error function is |a x² - x|, which is an absolute value, maybe the maximums occur at points where a x² - x is either positive or negative, but their absolute values are equal.Wait, maybe it's better to approach this step by step.First, let's analyze h(x) = |a x² - x|. Let's consider the function without the absolute value: k(x) = a x² - x. Then h(x) = |k(x)|. So we need to find the maximum of |k(x)| over [0, 2].To find the extrema of k(x), we can take the derivative of k(x) and set it to zero. However, since h(x) is the absolute value, the maximum of |k(x)| can occur either at critical points of k(x) where k(x) is positive or negative, or at points where k(x) = 0 (which are the points where the absolute value has a corner), or at the endpoints of the interval.But let's first analyze k(x) = a x² - x. Let's see where k(x) is positive or negative.Set k(x) = 0: a x² - x = 0 ⇒ x(a x - 1) = 0 ⇒ x = 0 or x = 1/a.So, the roots of k(x) are at x = 0 and x = 1/a.Since a is positive, 1/a is positive. So on the interval [0, 2], the roots are at x = 0 and x = 1/a. Depending on the value of 'a', x = 1/a could be inside [0, 2] or outside. If 1/a ≤ 2, that is, a ≥ 1/2, then the root is within [0, 2]. Otherwise, if a < 1/2, then 1/a > 2, so the root is outside the interval [0, 2].Therefore, for a ≥ 1/2, k(x) = a x² - x has two roots at x = 0 and x = 1/a. Between these roots, the sign of k(x) will be negative (since for a quadratic opening upwards, between the roots it's negative if a > 0). Wait, actually, k(x) is a quadratic function with a coefficient a on x². Since a is positive, it's a parabola opening upwards. Wait, but k(x) = a x² - x, so it's a parabola opening upwards. Then between the roots x=0 and x=1/a, the quadratic is below the x-axis (negative), and outside it's positive.But in our interval [0, 2], so for a ≥ 1/2, k(x) is negative between 0 and 1/a, and positive between 1/a and 2. For a < 1/2, k(x) is negative in all of [0, 2] since the other root is at x = 1/a > 2.Therefore, h(x) = |k(x)| is:- For a ≥ 1/2: h(x) = |a x² - x| = { -k(x) = x - a x² when 0 ≤ x ≤ 1/a; k(x) = a x² - x when 1/a ≤ x ≤ 2 }- For a < 1/2: h(x) = |a x² - x| = x - a x² for all x in [0, 2], since k(x) is negative there.Therefore, to find the maximum of h(x) over [0, 2], we need to analyze these cases.Case 1: a < 1/2. Then h(x) = x - a x² on [0, 2]. Let's find its maximum. The function h(x) = x - a x² is a downward-opening parabola (since coefficient of x² is -a < 0). Its maximum occurs at the vertex. The vertex of a parabola ax² + bx + c is at x = -b/(2a). Here, h(x) = -a x² + x + 0. So the vertex is at x = -1/(2*(-a)) = 1/(2a). Wait, that seems correct.Wait, h(x) = -a x² + x. So derivative h’(x) = -2a x + 1. Setting derivative to zero: -2a x + 1 = 0 ⇒ x = 1/(2a). Since a < 1/2, 1/(2a) > 1/(2*(1/2)) = 1. So the vertex is at x = 1/(2a) > 1. Since the interval is up to x=2, we need to check if the vertex is within [0, 2]. Since a < 1/2, 1/(2a) > 1/(2*(1/2)) = 1, and since a is positive, 1/(2a) can be up to infinity as a approaches zero. So in this case, the maximum of h(x) occurs at x = 1/(2a) if that point is within [0, 2], but since a < 1/2, 1/(2a) > 1/(2*(1/2)) = 1, but it might be less than 2 or greater than 2.Wait, 1/(2a) < 2 ⇨ 1/(2a) < 2 ⇨ 1/(4) < a. So if a > 1/4, then 1/(2a) < 2. If a ≤ 1/4, then 1/(2a) ≥ 2.But in Case 1, a < 1/2. So we need to split Case 1 into two subcases:Case 1a: 1/4 < a < 1/2. Then 1/(2a) is between 1 and 2. So the maximum of h(x) occurs at x = 1/(2a). Then h(1/(2a)) = -a*(1/(2a))² + (1/(2a)) = -a*(1/(4a²)) + 1/(2a) = -1/(4a) + 1/(2a) = ( -1 + 2 ) / (4a ) = 1/(4a).Also, we need to check the endpoints. At x=0, h(0) = 0. At x=2, h(2) = 2 - a*(4) = 2 - 4a. So the maximum in this subcase is the maximum between 1/(4a) and 2 - 4a. Wait, but since we are in Case 1a: 1/4 < a < 1/2. Let's compute 1/(4a) and 2 - 4a:For a = 1/4, 1/(4a) = 1/(4*(1/4)) = 1, and 2 - 4a = 2 - 1 = 1. So at a=1/4, both are equal. For a approaching 1/2 from below, 1/(4a) approaches 1/(4*(1/2)) = 1/2, and 2 - 4a approaches 2 - 4*(1/2) = 0. So in between, 1/(4a) is decreasing from 1 to 1/2, and 2 - 4a is decreasing from 1 to 0. So the maximum of h(x) in this subcase is 1/(4a), which is larger than 2 - 4a.Therefore, for 1/4 < a < 1/2, the maximum is 1/(4a).Case 1b: a ≤ 1/4. Then the vertex at x=1/(2a) ≥ 2. Therefore, the maximum of h(x) occurs at x=2, since the parabola is decreasing from the vertex to the right. Wait, no. Wait, h(x) = x - a x². If the vertex is beyond x=2, then h(x) is increasing on [0, 2], because the parabola is opening downward, and the vertex is to the right of 2. Therefore, the maximum is at x=2. So h(2) = 2 - 4a. Also, check the value at x=1/(2a) which is outside the interval, so maximum is at x=2. Therefore, for a ≤ 1/4, the maximum is 2 - 4a.So summarizing Case 1 (a < 1/2):- If a ≤ 1/4, maximum of h(x) is 2 - 4a.- If 1/4 < a < 1/2, maximum is 1/(4a).Case 2: a ≥ 1/2. Then h(x) is defined as x - a x² on [0, 1/a] and a x² - x on [1/a, 2]. Let's analyze both pieces.First, on [0, 1/a], h(x) = x - a x². The maximum here occurs either at the critical point or at the endpoints. The critical point is again at x = 1/(2a). But since a ≥ 1/2, 1/(2a) ≤ 1/(2*(1/2)) = 1. But the interval here is [0, 1/a]. Since a ≥ 1/2, 1/a ≤ 2. Wait, but 1/(2a) compared to 1/a: 1/(2a) ≤ 1/a because a is positive. So 1/(2a) is within [0, 1/a]. Therefore, the maximum on [0, 1/a] occurs at x = 1/(2a), and the value is h(1/(2a)) = 1/(4a) as computed before.On the interval [1/a, 2], h(x) = a x² - x. Let's analyze this. The function is a x² - x, which is a parabola opening upwards (since a > 0). Therefore, it has a minimum at x = 1/(2a). However, the interval here is [1/a, 2]. Since the parabola opens upwards, on [1/a, 2], the function is increasing because the vertex is at x = 1/(2a) which is less than 1/a (since 1/(2a) < 1/a because a > 0). Therefore, on [1/a, 2], the function a x² - x is increasing. Therefore, the maximum on [1/a, 2] occurs at x=2. So h(2) = a*(4) - 2 = 4a - 2.So in Case 2 (a ≥ 1/2), the maximum of h(x) over [0,2] is the maximum between 1/(4a) (from the first piece) and 4a - 2 (from the second piece). Let's compare these two:At a = 1/2:1/(4a) = 1/(4*(1/2)) = 1/24a - 2 = 4*(1/2) - 2 = 2 - 2 = 0So the maximum is 1/2.As a increases beyond 1/2, 1/(4a) decreases, and 4a - 2 increases. There might be a point where these two expressions are equal. Let's find if such a exists.Set 1/(4a) = 4a - 2.Multiply both sides by 4a:1 = 16a² - 8aBring all terms to one side:16a² - 8a - 1 = 0Solve for a:Using quadratic formula:a = [8 ± sqrt(64 + 64)] / 32 = [8 ± sqrt(128)] / 32 = [8 ± 8*sqrt(2)] / 32 = [1 ± sqrt(2)] / 4Since a is positive, we take the positive root:a = [1 + sqrt(2)] / 4 ≈ (1 + 1.4142)/4 ≈ 2.4142/4 ≈ 0.6035But [1 + sqrt(2)] / 4 is approximately 0.6035, which is greater than 1/2 (0.5). Therefore, in Case 2, when a ≥ 1/2, the two expressions 1/(4a) and 4a - 2 cross at a = [1 + sqrt(2)] / 4.Therefore, for a between 1/2 and [1 + sqrt(2)] / 4, 1/(4a) is greater than 4a - 2. For a greater than [1 + sqrt(2)] / 4, 4a - 2 becomes larger.Therefore, in Case 2:- For 1/2 ≤ a ≤ [1 + sqrt(2)] / 4, the maximum of h(x) is 1/(4a).- For a ≥ [1 + sqrt(2)] / 4, the maximum of h(x) is 4a - 2.So now, summarizing all cases:- For a ≤ 1/4: maximum is 2 - 4a.- For 1/4 < a < 1/2: maximum is 1/(4a).- For 1/2 ≤ a ≤ [1 + sqrt(2)] / 4 ≈ 0.6035: maximum is 1/(4a).- For a ≥ [1 + sqrt(2)] / 4: maximum is 4a - 2.Now, we need to find the value of 'a' that minimizes this maximum.Let's consider each interval and find the minimum in each.1. For a ≤ 1/4: maximum is 2 - 4a. This is a linear function decreasing as a increases. So the minimum in this interval occurs at the right endpoint a=1/4, giving 2 - 4*(1/4) = 2 - 1 = 1.2. For 1/4 < a < 1/2: maximum is 1/(4a). This is a decreasing function of 'a', since as 'a' increases, 1/(4a) decreases. So the minimum in this interval occurs at the right endpoint a=1/2, giving 1/(4*(1/2)) = 1/2.3. For 1/2 ≤ a ≤ [1 + sqrt(2)] / 4: maximum is 1/(4a). Again, decreasing in 'a', so the minimum is at a=[1 + sqrt(2)] / 4, giving 1/(4*( [1 + sqrt(2)] /4 )) = 1/(1 + sqrt(2)) ≈ 0.4142.4. For a ≥ [1 + sqrt(2)] / 4: maximum is 4a - 2. This is an increasing function of 'a', so the minimum occurs at the left endpoint a=[1 + sqrt(2)] /4, giving 4*( [1 + sqrt(2)] /4 ) - 2 = [1 + sqrt(2)] - 2 = sqrt(2) -1 ≈ 0.4142.Therefore, the minimal maximum occurs at a=[1 + sqrt(2)] /4, where both expressions 1/(4a) and 4a -2 are equal to sqrt(2) -1 ≈ 0.4142.Therefore, the minimal maximum distance is sqrt(2) -1, achieved when a = [1 + sqrt(2)] /4.Therefore, the value of 'a' that minimizes the distance is (1 + sqrt(2))/4.But let me verify this conclusion.Wait, let's check at a = [1 + sqrt(2)] /4, the maximum is 4a - 2 = 4*(1 + sqrt(2))/4 - 2 = (1 + sqrt(2)) - 2 = sqrt(2) -1. Similarly, 1/(4a) = 1/(4*(1 + sqrt(2))/4) = 1/(1 + sqrt(2)) = (1 - sqrt(2))/ (1 - 2) ) = (sqrt(2) -1)/1 = sqrt(2) -1. So yes, both expressions give sqrt(2)-1. Therefore, at this value of 'a', the maximum in both intervals [0,1/a] and [1/a,2] is equal. That makes sense, per the equioscillation theorem, where the minimal maximum deviation is achieved when the maximum error is attained at multiple points with alternating signs, but in this case, since we have absolute value, perhaps it's attained at two points with the same value. In this case, the maximum occurs both at x=1/(2a) in the left interval and at x=2 in the right interval, both with the same maximum value. So that would satisfy the condition for minimal maximum.Therefore, this should be the optimal 'a'.But to ensure that there's no other point in the interval where the error is larger, let's check.Suppose a = [1 + sqrt(2)] /4 ≈ (1 + 1.4142)/4 ≈ 2.4142/4 ≈ 0.6035. Then, 1/a ≈ 1/0.6035 ≈ 1.656. So the split point between the two intervals is at x=1/a ≈ 1.656.In the interval [0, 1/a], the maximum is at x=1/(2a) ≈ 1/(2*0.6035) ≈ 0.828. The value there is sqrt(2)-1 ≈ 0.4142.In the interval [1/a, 2], the maximum is at x=2, which is 4a -2 ≈ 4*0.6035 -2 ≈ 2.414 -2 ≈ 0.414, same as above.Additionally, we need to check if there are any other local maxima in the interval. For example, in the interval [1/a, 2], the function h(x) = a x² -x is increasing, so its maximum is at x=2.In the interval [0,1/a], h(x) =x - a x² is a parabola opening downward with vertex at x=1/(2a). Since 1/(2a) ≈0.828 is within [0,1/a≈1.656], so the maximum is indeed at x=1/(2a).Therefore, the maximum of h(x) in [0,2] is sqrt(2)-1, achieved at x=1/(2a)≈0.828 and x=2. Thus, the minimal maximum is achieved when the two maxima are equal. Hence, the optimal 'a' is (1 + sqrt(2))/4.But let me check with an example. Let's take a = (1 + sqrt(2))/4 ≈ 0.6035. Compute h(x) at some points.At x=0: h(0)=0.At x=1/(2a) ≈0.828: h(x)=sqrt(2)-1≈0.4142.At x=1/a≈1.656: h(x)=0.At x=2: h(2)=4a -2≈0.4142.So between x=1/a and x=2, h(x) increases from 0 to 0.4142. Between x=0 and x=1/a, it peaks at 0.4142. Therefore, the maximum is indeed 0.4142, and occurs at two points. Therefore, this is the minimal maximum.Therefore, such an 'a' exists and is equal to (1 + sqrt(2))/4.Hence, the answer is a = (1 + sqrt(2))/4.Final AnswerThe value of (a) that minimizes the distance is (boxed{dfrac{1 + sqrt{2}}{4}}).</think>To find the value of (a) that minimizes the maximum distance between the functions (f(x) = 2 - |x|) and (g(x) = -ax^2 + 2) over the interval ([-2, 2]), we need to analyze the expression (|f(x) - g(x)|). This reduces to (|a x^2 - |x||).Given the symmetry of the functions, we consider the interval ([0, 2]). The expression simplifies to (|a x^2 - x|). We analyze the function (h(x) = |a x^2 - x|) and find the value of (a) that minimizes its maximum value over ([0, 2]).1. For (a leq frac{1}{4}): - The maximum value of (h(x)) is (2 - 4a) at (x = 2).2. For (frac{1}{4} < a < frac{1}{2}): - The maximum value of (h(x)) is (frac{1}{4a}) at (x = frac{1}{2a}).3. For (a geq frac{1}{2}): - The function (h(x)) splits into two intervals: ([0, frac{1}{a}]) and ([frac{1}{a}, 2]). - The maximum value on ([0, frac{1}{a}]) is (frac{1}{4a}) at (x = frac{1}{2a}). - The maximum value on ([frac{1}{a}, 2]) is (4a - 2) at (x = 2).To find the optimal (a), we equate the maximum values from the intervals:- For (a geq frac{1}{2}), setting (frac{1}{4a} = 4a - 2) leads to solving the quadratic equation (16a^2 - 8a - 1 = 0). The positive root is (a = frac{1 + sqrt{2}}{4}).This value of (a) ensures that the maximum distance is minimized and occurs at two points, satisfying the equioscillation theorem. The optimal (a) is (frac{1 + sqrt{2}}{4}).[boxed{dfrac{1 + sqrt{2}}{4}}]]

🔑:}Since for all (x in(-2,2)) : (f(x)=f(-x)) and (g(x)=g(-x)) are true, one only needs to consider the function (k)[k(x, a)=|h(x, a)| quad text { with } quad h(x, a)=f(x)-g(8 x)]((a) is a parameter that can take any real number greater than zero) in the interval ((0,2)). It is[h(x, a)=a x^{2}-x=aleft(x-frac{1}{2 a}right)^{2}-frac{1}{4 a}]The image of (h) is, for every allowed (a), a parabola that opens upwards, intersects the x-axis at 0 and (frac{1}{a}), and has the vertex (left(frac{1}{2 a},-frac{1}{4 a}right)).To investigate the function (l)[l(a)=max _{x in(0,2)} k(x, a) quad, quad a>0]for an absolute minimum, three cases are distinguished:1. Case: (2 leq frac{1}{2 a} leftrightarrow a leq frac{1}{4}) (figure)[l(a)=k(2, a)=-h(2, a)=-4 a+2]2. Case: (frac{1}{2 a} leq 2 leq frac{1}{a} leftrightarrow frac{1}{4} leq a leq frac{1}{2})[l(a)=kleft(frac{1}{2 a}, aright)=-hleft(frac{1}{2 a}, aright)=frac{1}{4 a}]3. Case: (frac{1}{a} leq 2 leftrightarrow frac{1}{2} leq a); the interval (left[frac{1}{2}, inftyright)) is further divided due to the possible cases here[left(3^{prime}right): quad l(a)=kleft(frac{1}{2 a}, aright)=frac{1}{4 a} quad ; quadleft(3^{prime prime}right): quad l(a)=k(2, a)=4 a-2]The inequality (frac{1}{4 a} geq 4 a-2) corresponds to case 3. It is[frac{1}{4 a} geq 4 a-2 leftrightarrow 0 geq 16 a^{2}-8 a-1 leftrightarrow a inleft[frac{1}{4}(1-sqrt{2}), frac{1}{4}(1+sqrt{2})right]]Since (a inleft[frac{1}{2}, inftyright)) and (left[frac{1}{2}, inftyright) capleft[frac{1}{4}(1-sqrt{2}), frac{1}{4}(1+sqrt{2})right]=left[frac{1}{2}, frac{1}{4}(1+sqrt{2})right]), it follows that (l(a)=frac{1}{4 a}) for (a inleft[frac{1}{2}, frac{1}{4}(1+sqrt{2})right]) and (l(a)=4 a-2) for (a inleft[frac{1}{4}(1+sqrt{2}), inftyright)).Thus, overall,[l(a)= begin{cases}-4 a+2 & text { for } a inleft(0, frac{1}{4}right] frac{1}{4 a} & text { for } a inleft[frac{1}{4}, frac{1}{4}(1+sqrt{2})right] frac{1}{4 a-2} & text { for } a inleft[frac{1}{4}(1+sqrt{2}), inftyright)end{cases}]Since (l) is strictly monotonically decreasing in the interval (left(0, frac{1}{4}(1+sqrt{2})right]) and strictly monotonically increasing in (left[frac{1}{4}(1+sqrt{2}), inftyright)), (l) has an absolute minimum at the point (a=frac{1}{4}(1+sqrt{2})) and only at this point.Adapted from ([5])subsection*{9.14 XII. Olympiad 1972}subsection*{9.14.1 First Round 1972, Class 12}

🧠:[<think>Okay, let me try to figure out this problem. So, we have the numbers 1, 2, 3, 4, 5, and we need to select 4 different numbers to fill in the four squares in this equation: □ + □ > □ + □. The question is asking how many different ways we can make this equation true. The hint says that 1+5>2+3 and 5+1>2+3 are considered different ways, so the order matters here. That means even if the same numbers are used but arranged differently, they count as separate cases. First, I need to understand how to approach this. Since we have to select 4 different numbers out of 5, we first need to choose which number to exclude. There are 5 choices for the number to exclude. Then, for each such choice, we need to arrange the remaining 4 numbers into the four squares, considering the order. However, the equation is about the sum of the first two numbers being greater than the sum of the last two. So, maybe we can approach this step by step: first, select the four numbers, then arrange them in such a way that the first two add up to more than the last two. But wait, since the exclusion of a number affects the possible combinations, maybe we should first consider all possible permutations of 4 numbers from the 5, and then check which permutations satisfy the inequality. Let me verify that. The total number of permutations of 4 numbers selected from 5 is 5P4 = 5*4*3*2 = 120. But since we have to exclude one number each time, and each permutation includes 4 distinct numbers, so each permutation corresponds to excluding one number. But in any case, there are 120 different ordered arrangements. Now, the problem is how many of these 120 permutations satisfy the condition that the sum of the first two numbers is greater than the sum of the last two.Alternatively, perhaps instead of considering permutations, we can think of the problem as first choosing which four numbers to use (by excluding one of the five), then arranging those four numbers in the four positions such that the first two add up to more than the last two. So maybe break it down into cases based on which number is excluded, then for each case, compute the number of valid permutations.Let me try that approach. Let's consider each possible excluded number: 1, 2, 3, 4, or 5. For each exclusion, we have the remaining four numbers. Then, we need to arrange these four numbers into the four positions (two on the left side of the inequality and two on the right side) such that the sum of the left two is greater than the sum of the right two. Since order matters, each permutation is unique. So for each excluded number, we can calculate the number of permutations where the left sum exceeds the right sum, then sum over all excluded numbers.But this might be complicated because each exclusion leads to different numbers, so the possible sums would vary. Alternatively, perhaps it's better to consider that excluding different numbers might lead to symmetrical situations. For example, if we exclude 1, the numbers are 2,3,4,5; excluding 5 gives 1,2,3,4. Maybe these two cases are similar? Not sure. Let's try to work through each case.Case 1: Excluding 1. So the numbers available are 2, 3, 4, 5. Now, we need to arrange these four numbers into the four positions. The total number of permutations is 4! = 24. For each permutation, the first two numbers form the left sum, and the last two form the right sum. We need to count how many times left sum > right sum.But calculating this directly for each permutation would be tedious. Instead, perhaps we can find a smarter way. For four numbers a, b, c, d, when arranged as a + b > c + d. Since all permutations are considered, the positions of the numbers matter. So, given four distinct numbers, how many of the 24 permutations satisfy the inequality? Wait, but for four distinct numbers, the sum a + b can range from the smallest possible sum to the largest. For example, with numbers 2,3,4,5: the smallest possible sum of two numbers is 2 + 3 = 5, and the largest is 4 + 5 = 9. The total number of ordered pairs for the left side is 4P2 = 12 (since order matters), and similarly for the right side, the remaining two numbers would form the right sum. Wait, but actually, once you choose the first two numbers, the last two are determined. So for each permutation, the first two are selected in order, and the last two are in order. So in total, the number of possible arrangements is 4! = 24, as each permutation is unique.However, for each unique combination of left pair and right pair, there are multiple permutations. For example, if the left pair is 2 and 3, and the right pair is 4 and 5, then the permutations could be 2,3,4,5 or 2,3,5,4, etc. But the sum of the left pair is 5, and the sum of the right pair is 9, so 5 > 9 is false. Conversely, if the left pair is 5 and 4, the sum is 9, and the right pair is 2 and 3, sum is 5, so 9 > 5 is true. But in how many permutations does the left sum exceed the right sum?Wait, perhaps another approach: For four numbers, the total number of ways to split them into two ordered pairs is 4! / (2! * 2!) ) * 2! * 2! = 6 * 4 = 24, but that's the same as 4! So, each permutation corresponds to a unique split. So, to count how many permutations satisfy left sum > right sum, we can consider all possible splits into left and right pairs, and count how many of those splits have the left sum greater.But how many distinct splits are there where the left sum is greater than the right sum? Let's think in terms of combinations. For four numbers a, b, c, d, the number of distinct ways to choose two numbers for the left (without order) is C(4,2) = 6. For each such combination, the left sum can be compared to the right sum. However, since order matters in the permutation, each combination can be arranged in 2! ways for the left pair and 2! ways for the right pair, totaling 4 ways per combination.Therefore, for each combination of left pair, there are 4 permutations corresponding to it. So, if a particular combination of left pair has a sum greater than the right pair, then all 4 permutations corresponding to that combination will satisfy the inequality, and vice versa. Similarly, if the sums are equal, none will satisfy. But since all numbers are distinct, the sums can't be equal. Wait, can they?Wait, let's check. If the left pair sum equals the right pair sum, that would require two different pairs of numbers to have the same sum. For example, in numbers 1,2,3,4: 1+4=2+3=5. So, in such a case, the sum is equal, so the inequality is not satisfied. But in our case, when excluding a number, the remaining numbers might or might not have such pairs.Wait, but in the first case, when excluding 1, the numbers are 2,3,4,5. Let's check if any two pairs add up to the same sum. Let's see:Possible pairs:2+3=5, 2+4=6, 2+5=7, 3+4=7, 3+5=8, 4+5=9.So, pairs (2,5) and (3,4) both sum to 7. Therefore, if the left pair is (2,5) and the right pair is (3,4), then the sum is equal. Similarly, left pair (3,4) and right pair (2,5) also sum to 7. So, in this case, even though the numbers are distinct, the sums can be equal. So, for these combinations, the inequality is not satisfied. Therefore, when we have combinations where the left sum equals the right sum, those permutations would not count. So, in this case, for numbers 2,3,4,5, we have two pairs that sum to 7. Therefore, the combinations (2,5 vs 3,4) and (3,4 vs 2,5) will each have sum equal to 7. Each combination can be arranged in 4 permutations (since left pair can be ordered in 2 ways, right pair in 2 ways). Therefore, there are 2 combinations * 4 permutations = 8 permutations where the sum is equal. So, out of 24 permutations, 8 result in equal sums, and the remaining 16 permutations will have either left sum > right sum or left sum < right sum. Since for the other combinations, the sums are unequal, and each combination where left sum > right sum is balanced by the reverse where left sum < right sum. Therefore, the number of permutations where left sum > right sum is equal to the number where left sum < right sum. So, subtracting the 8 equal permutations, we have 16 permutations left, which split evenly into 8 where left > right and 8 where left < right. Therefore, in this case (excluding 1), the number of valid permutations is 8.But wait, let's verify this. Let's take numbers 2,3,4,5.Total permutations: 24.Number of permutations where the sum of the first two equals the sum of the last two: As we saw, the pairs (2,5) and (3,4) sum to 7. Each of these pairs can be arranged in 2*2=4 ways. But how many permutations does this correspond to?For the combination where the first two numbers are 2 and 5 (in any order) and the last two are 3 and 4 (in any order). There are 2 (for 2 and 5) * 2 (for 3 and 4) = 4 permutations: 2,5,3,4; 2,5,4,3; 5,2,3,4; 5,2,4,3. Similarly, for the combination where the first two are 3 and 4 and the last two are 2 and 5, we have another 4 permutations: 3,4,2,5; 3,4,5,2; 4,3,2,5; 4,3,5,2. So total 8 permutations where the sum is equal. Therefore, remaining permutations: 24 - 8 = 16. These 16 must be split equally between left sum > right sum and left sum < right sum. Therefore, 8 permutations where left sum > right sum. So, for case 1 (excluding 1), there are 8 valid permutations.Case 2: Excluding 2. The remaining numbers are 1,3,4,5. Let's perform a similar analysis. First, check if there are any pairs that sum to the same value.Possible pairs:1+3=4, 1+4=5, 1+5=6, 3+4=7, 3+5=8, 4+5=9. So, all sums are unique here. Therefore, there are no pairs that sum to the same value. Therefore, in this case, when splitting the four numbers into two pairs, the sums will always be different. Therefore, for each permutation, either left sum > right sum or left sum < right sum. Since there are 24 permutations, and each split is either left or right, the number of permutations where left sum > right sum should be half of 24, which is 12. Wait, but this is only true if each combination is equally likely to be left or right. But since we are dealing with permutations, each split is unique. However, for each combination of left pair and right pair, there are 2! permutations for the left pair and 2! for the right pair, so 4 permutations per combination. Since the sums are all different, for each combination, either left sum > right sum or vice versa, and since there are C(4,2)=6 combinations, each leading to 4 permutations, so 6*4=24. Since each combination is either left > right or left < right, and there are 6 combinations. Let's check how many of the combinations have left sum > right sum.For numbers 1,3,4,5:Possible left pairs (sum):1+3=4; right pair sum would be 4+5=9. 4 < 9, so left sum < right sum.1+4=5; right pair sum: 3+5=8. 5 < 8, left < right.1+5=6; right pair sum: 3+4=7. 6 < 7, left < right.3+4=7; right pair sum:1+5=6. 7 > 6, left > right.3+5=8; right pair sum:1+4=5. 8 > 5, left > right.4+5=9; right pair sum:1+3=4. 9 > 4, left > right.So out of the 6 combinations, 3 result in left sum > right sum (the last three combinations). Each of these 3 combinations can be arranged in 4 permutations (since order matters). Therefore, the number of permutations where left sum > right sum is 3 * 4 = 12. Therefore, in case 2 (excluding 2), there are 12 valid permutations.Case 3: Excluding 3. The remaining numbers are 1,2,4,5. Let's check for equal sums.Possible pairs:1+2=3, 1+4=5, 1+5=6, 2+4=6, 2+5=7, 4+5=9. Here, pairs (1,5) and (2,4) both sum to 6. So similar to case 1, we have two pairs that sum to 6. Therefore, the combinations where left pair is (1,5) and right pair is (2,4) or vice versa will have equal sums.So, how many permutations does that correspond to? For each combination, left pair ordered in 2 ways, right pair ordered in 2 ways, so 4 permutations each. Since there are two such combinations (left pair (1,5) vs (2,4) and left pair (2,4) vs (1,5)), total 8 permutations with equal sums. Therefore, remaining permutations: 24 - 8 = 16. These split equally into 8 permutations where left sum > right sum and 8 where left sum < right sum. Therefore, case 3 (excluding 3) gives 8 valid permutations.Case 4: Excluding 4. Remaining numbers: 1,2,3,5. Let's check for equal sums.Possible pairs:1+2=3, 1+3=4, 1+5=6, 2+3=5, 2+5=7, 3+5=8. All sums here are unique. So no pairs that sum to the same value. Therefore, similar to case 2, we can calculate the number of valid permutations.Total permutations: 24. Let's check the combinations:Possible left pairs (sum):1+2=3; right sum:3+5=8. Left < right.1+3=4; right sum:2+5=7. Left < right.1+5=6; right sum:2+3=5. Left > right.2+3=5; right sum:1+5=6. Left < right.2+5=7; right sum:1+3=4. Left > right.3+5=8; right sum:1+2=3. Left > right.So out of 6 combinations, 3 result in left sum > right sum (last three: (1,5), (2,5), (3,5)). Each of these combinations can be arranged in 4 permutations, so 3*4=12. Therefore, case 4 (excluding 4) gives 12 valid permutations.Case 5: Excluding 5. Remaining numbers: 1,2,3,4. Let's check for equal sums. As mentioned earlier, pairs (1,4) and (2,3) both sum to 5. So similar to case 1 and 3.Possible pairs:1+2=3, 1+3=4, 1+4=5, 2+3=5, 2+4=6, 3+4=7. So pairs (1,4) and (2,3) sum to 5. Therefore, permutations where the left pair is (1,4) and right pair is (2,3) or vice versa will have equal sums. Each combination can be arranged in 4 permutations, so total 8 permutations with equal sums. Therefore, remaining permutations: 24 - 8 = 16. Split equally into 8 left > right and 8 left < right. Therefore, case 5 (excluding 5) gives 8 valid permutations.Now, summing up all cases:Case 1 (exclude 1): 8Case 2 (exclude 2): 12Case 3 (exclude 3): 8Case 4 (exclude 4): 12Case 5 (exclude 5): 8Total: 8 + 12 + 8 + 12 + 8 = 48Therefore, there are 48 different ways to make the equation true.Wait, let me check if that makes sense. Total permutations of 4 numbers from 5 is 5P4 = 120. So, if the answer is 48, then 48 out of 120 permutations satisfy the condition. That's 40%. Alternatively, when we broke down each case, for each exclusion, we got either 8 or 12. Since there are two exclusions leading to 12 (excluding 2 and 4) and three exclusions leading to 8 (excluding 1,3,5). Wait, no, actually, when excluding 2 and 4, which are middle numbers, maybe there's more possible permutations where left sum exceeds right sum because the numbers are spread out more. While when excluding extremes (1,3,5), you have overlapping pairs that sum the same. So, the counts make sense.But let's verify with another approach. Let's think of all possible 4-number combinations. Each time we exclude one number, the remaining numbers have certain properties, but the key is that in some cases there are pairs that sum equally, and in others not. So, when excluding 2 or 4, which are middle numbers, the remaining numbers don't have such equal pairs, leading to more valid permutations. When excluding 1,3, or 5, which are the extremes, there are pairs that sum equally, reducing the number of valid permutations.Alternatively, perhaps there's a mathematical formula or symmetry here. But given that we have 5 cases, each with different counts, adding up to 48. Let me check again.Case 1: 8Case2: 12Case3:8Case4:12Case5:8Total: 8+12+8+12+8 = 48. Yes, that's correct.But just to be thorough, let's check with a smaller example. For example, take numbers 1,2,3,4. Let's say we want to compute the number of permutations where the first two numbers sum > the last two. Let's compute this manually.Numbers 1,2,3,4. Total permutations:24.Possible pairs:Possible left pairs:1+2=3, 1+3=4, 1+4=5, 2+1=3, 2+3=5, 2+4=6, 3+1=4, 3+2=5, 3+4=7, 4+1=5, 4+2=6, 4+3=7.But actually, it's easier to split into combinations. The pairs (1,2) sum to 3, (1,3)=4, (1,4)=5, (2,3)=5, (2,4)=6, (3,4)=7.So, comparing each left pair sum to the corresponding right pair sum:Left pair (1,2): sum 3 vs right pair (3,4): sum 7. Left < right. So all permutations with left pair (1,2) in any order and right pair (3,4) in any order: 4 permutations. These are invalid.Left pair (1,3): sum 4 vs right pair (2,4): sum 6. Left < right. 4 permutations, invalid.Left pair (1,4): sum 5 vs right pair (2,3): sum 5. Equal, so invalid. 4 permutations.Left pair (2,3): sum 5 vs right pair (1,4): sum 5. Equal, invalid. 4 permutations.Left pair (2,4): sum 6 vs right pair (1,3): sum 4. Left > right. 4 permutations, valid.Left pair (3,4): sum 7 vs right pair (1,2): sum 3. Left > right. 4 permutations, valid.So total valid permutations: 4 (from 2,4) + 4 (from 3,4) = 8. Which matches our earlier calculation when excluding 5 (since numbers 1,2,3,4 exclude 5, and we found 8 permutations). So that's correct. Similarly, if we take numbers 1,3,4,5 (excluding 2), the valid permutations are 12. Let me check one combination:Left pair (3,4): sum 7 vs right pair (1,5): sum 6. Left > right. So permutations like 3,4,1,5; 4,3,1,5; 3,4,5,1; 4,3,5,1; etc. Each combination contributes 4 permutations, and we had 3 such combinations (3,4;3,5;4,5). Each gives 4 permutations, so 3*4=12. That seems correct.Therefore, the total of 48 seems plausible.But let me check another way. The total number of permutations is 120. Each permutation either has left sum > right, left sum < right, or left sum = right. But when can left sum = right sum?In the entire 120 permutations, how many have left sum = right sum? For each exclusion case, we had some permutations with equal sums. For example, when excluding 1,3,5, there were 8 permutations each with equal sums, and when excluding 2,4, there were none. So total equal permutations: 3 cases * 8 = 24. Therefore, total permutations where left sum = right sum:24. Therefore, remaining permutations:120 -24=96. These should be equally split between left sum > right and left sum < right. So 48 each. Therefore, answer is 48. Which matches our previous calculation.Therefore, the correct answer is 48.Final Answerboxed{48}</think>To solve the problem of determining how many different ways we can make the equation square + square > square + square true by selecting 4 different numbers from the set {1, 2, 3, 4, 5}, we need to consider each possible excluded number and calculate the valid permutations for each case.1. Excluding 1: The remaining numbers are {2, 3, 4, 5}. There are 24 permutations. The pairs (2, 5) and (3, 4) sum to 7, leading to 8 permutations with equal sums. The remaining 16 permutations split evenly, resulting in 8 valid permutations.2. Excluding 2: The remaining numbers are {1, 3, 4, 5}. All pairs have unique sums. Half of the 24 permutations are valid, resulting in 12 valid permutations.3. Excluding 3: The remaining numbers are {1, 2, 4, 5}. The pairs (1, 5) and (2, 4) sum to 6, leading to 8 permutations with equal sums. The remaining 16 permutations split evenly, resulting in 8 valid permutations.4. Excluding 4: The remaining numbers are {1, 2, 3, 5}. All pairs have unique sums. Half of the 24 permutations are valid, resulting in 12 valid permutations.5. Excluding 5: The remaining numbers are {1, 2, 3, 4}. The pairs (1, 4) and (2, 3) sum to 5, leading to 8 permutations with equal sums. The remaining 16 permutations split evenly, resulting in 8 valid permutations.Summing up all valid permutations:- Case 1 (excluding 1): 8- Case 2 (excluding 2): 12- Case 3 (excluding 3): 8- Case 4 (excluding 4): 12- Case 5 (excluding 5): 8Total valid permutations: 8 + 12 + 8 + 12 + 8 = 48.Thus, the number of different ways to make the equation true is boxed{48}.,<think>Okay, let's see. The problem is asking how many different ways we can select 4 different numbers from 1,2,3,4,5 and arrange them in the equation: □ + □ > □ + □. Each arrangement where the sum of the first two numbers is greater than the sum of the last two counts as a valid way, and different orders are considered different even if they use the same numbers. For example, 1+5>2+3 and 5+1>2+3 are different. First, I need to figure out all possible ways to choose 4 numbers out of the 5 given. Since we have to use 4 different numbers, and the original set is 1,2,3,4,5, the number of combinations is C(5,4) which is 5. Each combination is missing one number. The combinations would be:1. Missing 1: {2,3,4,5}2. Missing 2: {1,3,4,5}3. Missing 3: {1,2,4,5}4. Missing 4: {1,2,3,5}5. Missing 5: {1,2,3,4}For each of these combinations, we need to consider all permutations of the 4 numbers into the four boxes. Each permutation is a different way. However, since the problem is about the inequality a + b > c + d, the order of a and b matters, as well as the order of c and d. But wait, actually, even if we swap a and b, the sum remains the same. Wait, but the problem says that different arrangements are considered different even if the sums are the same. For example, 1+5>2+3 and 5+1>2+3 are different. So even though the left side is the same sum, the arrangement is different because the numbers are swapped. Therefore, the order matters for the individual numbers, not just the sums.Therefore, for each combination of 4 numbers, the total number of arrangements is 4! = 24. But since there are 5 combinations, the total number of possible arrangements is 5 * 24 = 120. However, many of these will satisfy the inequality a + b > c + d, and we need to count how many do.But maybe instead of brute-forcing through all 120 possibilities, we can find a smarter way. Let's think about the possible sums. The maximum possible sum of two numbers from 1-5 is 5 + 4 = 9, and the minimum is 1 + 2 = 3. The sum of the other two numbers would be the total of the four numbers minus the first sum. So, for a given combination of four numbers, the total sum is fixed. Let's denote the four numbers as w, x, y, z, sorted in ascending order. Then, the total sum S = w + x + y + z. Then, if we pick two numbers a and b, their sum is s, and the sum of the remaining two is S - s. So the inequality a + b > c + d is equivalent to s > S - s, which simplifies to 2s > S, or s > S/2.Therefore, for each combination of four numbers, we can compute S, then determine how many pairs (a,b) have a sum greater than S/2. Then, since each such pair (a,b) can be arranged in 2 ways (a,b and b,a), and the remaining pair (c,d) can be arranged in 2 ways as well (c,d and d,c), each such pair contributes 2*2=4 arrangements. Therefore, for each combination, the number of valid arrangements is 4 * number of pairs (a,b) with sum > S/2.Alternatively, for each combination, we can compute how many ordered pairs (a,b) satisfy a + b > c + d, considering all permutations.But maybe let's proceed step by step.First, let's analyze each combination of four numbers.Combination 1: Missing 1: {2,3,4,5}Total sum S = 2 + 3 + 4 + 5 = 14Therefore, S/2 = 7. So we need pairs (a,b) such that a + b > 7.Possible pairs from 2,3,4,5:2+3=5, 2+4=6, 2+5=7, 3+4=7, 3+5=8, 4+5=9.So sums: 5,6,7,7,8,9. The pairs with sum >7 are 8 and 9. So the pairs (3,5), (4,5). Each of these pairs contributes 4 arrangements (since a,b can be ordered 2 ways and c,d can be ordered 2 ways). Wait, but how many such pairs are there? Let's see:The unordered pairs that sum to >7: (3,5), (4,5). So two unordered pairs. Each unordered pair can be arranged in 2 ways (a,b and b,a). Similarly, the remaining two numbers form an unordered pair, which can be arranged in 2 ways. Therefore, for each unordered pair (a,b) that sums to >7, there are 2 * 2 =4 ordered arrangements.So total valid arrangements for this combination: 2 unordered pairs * 4 = 8.Wait, but let's check:For pair (3,5):Possible arrangements for (a,b): (3,5) and (5,3)The remaining numbers are 2 and 4, which can be arranged as (2,4) or (4,2). So for each (a,b), there are 2 possibilities for (c,d). Therefore, 2 * 2 = 4 arrangements for the pair (3,5). Similarly for pair (4,5):Arrangements (4,5) and (5,4). Remaining numbers 2 and 3. So (2,3) and (3,2). So again 4 arrangements. So total 4 + 4 =8.So 8 valid arrangements for combination 1.Combination 2: Missing 2: {1,3,4,5}Total sum S =1 +3 +4 +5=13S/2=6.5. So pairs (a,b) with sum >6.5, which means sum >=7.Possible pairs from 1,3,4,5:1+3=4, 1+4=5, 1+5=6, 3+4=7, 3+5=8, 4+5=9.Sums: 4,5,6,7,8,9. So pairs with sum >=7: (3,4), (3,5), (4,5). That's three unordered pairs.For each of these:(3,4): arrangements (3,4) and (4,3). Remaining numbers 1 and5, arranged as (1,5) and (5,1). So 2*2=4.(3,5): arrangements (3,5) and (5,3). Remaining numbers 1 and4: 2*2=4.(4,5): arrangements (4,5) and (5,4). Remaining numbers 1 and3: 2*2=4.Total: 3 unordered pairs *4=12 arrangements.Therefore, combination 2 has 12 valid arrangements.Combination3: Missing3: {1,2,4,5}Sum S=1+2+4+5=12S/2=6. So pairs (a,b) with sum >6.Possible pairs:1+2=3, 1+4=5, 1+5=6, 2+4=6, 2+5=7, 4+5=9.Sums:3,5,6,6,7,9. So pairs with sum >6: (2,5), (4,5). That's two unordered pairs.For (2,5):Arrangements (2,5) and (5,2). Remaining numbers 1 and4: 2*2=4.For (4,5):Arrangements (4,5) and (5,4). Remaining numbers 1 and2: 2*2=4.Total: 2*4=8 arrangements.Combination3: 8 valid arrangements.Combination4: Missing4: {1,2,3,5}Sum S=1+2+3+5=11S/2=5.5. So pairs (a,b) with sum >5.5, i.e., >=6.Possible pairs:1+2=3, 1+3=4, 1+5=6, 2+3=5, 2+5=7, 3+5=8.Sums:3,4,6,5,7,8. So sums >=6: (1,5), (2,5), (3,5). Three unordered pairs.For (1,5):Arrangements (1,5) and (5,1). Remaining numbers 2 and3: 2*2=4.For (2,5):Arrangements (2,5) and (5,2). Remaining numbers 1 and3: 2*2=4.For (3,5):Arrangements (3,5) and (5,3). Remaining numbers 1 and2: 2*2=4.Total:3*4=12 arrangements.Combination4:12 valid arrangements.Combination5: Missing5: {1,2,3,4}Sum S=1+2+3+4=10S/2=5. So pairs (a,b) with sum >5.Possible pairs:1+2=3, 1+3=4, 1+4=5, 2+3=5, 2+4=6, 3+4=7.Sums:3,4,5,5,6,7. So pairs with sum >5: (2,4), (3,4). Two unordered pairs.For (2,4):Arrangements (2,4) and (4,2). Remaining numbers 1 and3: 2*2=4.For (3,4):Arrangements (3,4) and (4,3). Remaining numbers 1 and2: 2*2=4.Total:2*4=8 arrangements.Combination5:8 valid arrangements.Now, adding up all the valid arrangements for each combination:Combination1:8Combination2:12Combination3:8Combination4:12Combination5:8Total:8+12=20, 20+8=28, 28+12=40, 40+8=48.Wait, 8+12 is 20, then +8 is 28, +12 is 40, +8 is 48. So total 48 different ways.But wait, let me check again. Maybe there's a mistake in the calculation.Combination1:8Combination2:12 → 8+12=20Combination3:8 → 20+8=28Combination4:12 →28+12=40Combination5:8 →40+8=48.Yes, 48.But wait, let me verify with another approach. Because each combination of 4 numbers, when considering all permutations, has 4! =24 permutations. For each permutation, the first two numbers are a and b, the last two are c and d. The inequality a + b > c + d. How many of the 24 permutations satisfy this?Alternatively, for each combination, the number of ordered pairs (a,b) where a + b > c + d. Note that for each permutation, (a,b,c,d), the inequality is a + b > c + d. But the same four numbers can be arranged in different ways. However, the sum a + b can range over different values. For each combination, how many ordered pairs (a,b) have sum greater than the sum of the remaining two numbers.But since the total sum S is fixed, the sum a + b must be greater than (S - (a + b)), so as before, 2(a + b) > S, so a + b > S/2. Therefore, for each combination, the number of ordered pairs (a,b) where a + b > S/2. The number of ordered pairs is equal to the number of ordered pairs (a,b) where a and b are distinct elements of the combination. The number of ordered pairs is 4*3=12 for each combination (since we choose two distinct numbers, order matters). So for each combination, out of 12 ordered pairs (a,b), how many satisfy a + b > S/2. Then, once (a,b) is chosen, the remaining two numbers can be ordered in 2 ways (c,d and d,c). Therefore, for each valid (a,b), there are 2 permutations. So total valid permutations for each combination is 2 * number of valid (a,b) ordered pairs.Wait, but let me check.Suppose in a combination, we have numbers w, x, y, z. The number of ordered pairs (a,b) is 4*3=12. For each of these, if a + b > S/2, then we can arrange the remaining two numbers as (c,d) in 2 ways, so total 2 permutations per valid (a,b). Therefore, total valid permutations for the combination would be 2 * (number of ordered pairs (a,b) where a + b > S/2).Alternatively, since (a,b,c,d) is a permutation, the total number is 4! =24. But if we fix (a,b), then (c,d) is determined as the remaining two numbers, and can be ordered in 2 ways. So the number of permutations where a + b > c + d is equal to the number of ordered pairs (a,b) with a + b > S/2 multiplied by 2.Therefore, to compute for each combination, compute the number of ordered pairs (a,b) where a + b > S/2, multiply by 2, and sum over all combinations.Let's see if this aligns with our previous calculations.Take combination1: {2,3,4,5}, S=14, S/2=7.Number of ordered pairs (a,b) where a + b >7:Possible ordered pairs:(3,5)=8, (5,3)=8, (4,5)=9, (5,4)=9, (4,3)=7, (3,4)=7, (5,2)=7, (2,5)=7, etc.Wait, actually, to compute the number of ordered pairs (a,b) where a + b >7.List all ordered pairs (a,b) where a and b are distinct elements of {2,3,4,5}:(2,3)=5, (2,4)=6, (2,5)=7,(3,2)=5, (3,4)=7, (3,5)=8,(4,2)=6, (4,3)=7, (4,5)=9,(5,2)=7, (5,3)=8, (5,4)=9.Now, sums greater than 7: sums of 8,9.So:(3,5)=8, (4,5)=9, (5,3)=8, (5,4)=9, (4,5)=9, (5,4)=9. Wait, no:Wait, let's list all 12 ordered pairs and their sums:1. (2,3)=52. (2,4)=63. (2,5)=74. (3,2)=55. (3,4)=76. (3,5)=87. (4,2)=68. (4,3)=79. (4,5)=910. (5,2)=711. (5,3)=812. (5,4)=9Now, sums >7 are:6. (3,5)=89. (4,5)=911. (5,3)=812. (5,4)=9So that's 4 ordered pairs. Therefore, number of valid ordered pairs (a,b) is 4. Then, the number of permutations is 4 * 2 =8, which matches our previous calculation. So for combination1, 8 permutations.Similarly, combination2: {1,3,4,5}, S=13, S/2=6.5.Compute ordered pairs (a,b) where a + b >6.5:List all 12 ordered pairs:1. (1,3)=42. (1,4)=53. (1,5)=64. (3,1)=45. (3,4)=76. (3,5)=87. (4,1)=58. (4,3)=79. (4,5)=910. (5,1)=611. (5,3)=812. (5,4)=9Sums >6.5 are sums >=7:5. (3,4)=76. (3,5)=88. (4,3)=79. (4,5)=911. (5,3)=812. (5,4)=9That's 6 ordered pairs. Therefore, number of permutations is 6 *2=12, which matches combination2's 12.Similarly, combination3: {1,2,4,5}, S=12, S/2=6.Ordered pairs (a,b) where a + b >6:List:1. (1,2)=32. (1,4)=53. (1,5)=64. (2,1)=35. (2,4)=66. (2,5)=77. (4,1)=58. (4,2)=69. (4,5)=910. (5,1)=611. (5,2)=712. (5,4)=9Sums >6: 7,9,7,9. The ordered pairs are:6. (2,5)=79. (4,5)=911. (5,2)=712. (5,4)=9Total 4 ordered pairs. So permutations:4*2=8, matches combination3.Combination4: {1,2,3,5}, S=11, S/2=5.5.Ordered pairs (a,b) with a + b >5.5 (i.e., >=6):List:1. (1,2)=32. (1,3)=43. (1,5)=64. (2,1)=35. (2,3)=56. (2,5)=77. (3,1)=48. (3,2)=59. (3,5)=810. (5,1)=611. (5,2)=712. (5,3)=8Sums >=6:3. (1,5)=66. (2,5)=79. (3,5)=810. (5,1)=611. (5,2)=712. (5,3)=8That's 6 ordered pairs. So permutations:6*2=12, matches combination4.Combination5: {1,2,3,4}, S=10, S/2=5.Ordered pairs (a,b) where a + b >5:List:1. (1,2)=32. (1,3)=43. (1,4)=54. (2,1)=35. (2,3)=56. (2,4)=67. (3,1)=48. (3,2)=59. (3,4)=710. (4,1)=511. (4,2)=612. (4,3)=7Sums >5:6,5,6,7,5,6,7. Wait:Specifically:6. (2,4)=69. (3,4)=711. (4,2)=612. (4,3)=7That's 4 ordered pairs. Therefore, permutations:4*2=8, matches combination5.So all combinations check out with both methods: counting unordered pairs and multiplying by 4, and counting ordered pairs and multiplying by 2. Therefore, total is indeed 8+12+8+12+8=48.Therefore, the answer should be 48 different ways.But wait, let me just verify with another approach. Since each combination of 4 numbers contributes some number of valid permutations, and there are 5 combinations, each with 24 permutations (total 120), and we found 48 of them satisfy the inequality. Is 48/120=0.4, which is 40%. That seems plausible. But let's think if there's another way to compute it.Alternatively, for any four distinct numbers, when arranging them as a + b > c + d, how many such arrangements exist.But since the numbers are distinct, the sum a + b can be compared to the sum c + d. For four distinct numbers, the total number of ways to split them into two pairs is 3: since the number of ways to partition 4 elements into two pairs is 3. However, since order matters here (both in the pairs and the order within the pairs), it's different.Alternatively, the total number of permutations of four numbers is 24. For each permutation, the first two positions are a and b, and the last two are c and d. The number of permutations where a + b > c + d.But for any four distinct numbers, the number of permutations where the first two sum to more than the last two should be equal to the number of permutations where the last two sum to more than the first two, plus the cases where they are equal. But since all numbers are distinct, the sums a + b and c + d can be equal only if the pairs are different but sum to the same. However, since the total sum S is fixed, if a + b = c + d, then 2(a + b) = S, so a + b = S/2. Therefore, the number of permutations where a + b = c + d is equal to the number of ordered pairs (a,b) where a + b = S/2 multiplied by 2 (for the two orderings of c and d). But since S must be even for there to be such pairs.In our problem, each combination has a certain S. For combinations where S is even, there might be permutations where a + b = c + d, but in our previous counts, we only considered a + b > c + d. Therefore, the total number of permutations for a combination is equal to the number where a + b > c + d plus the number where a + b < c + d plus the number where a + b = c + d. But since we are only counting the cases where a + b > c + d, the number should be (total permutations - number of equal cases)/2, assuming that when a + b ≠ c + d, they split evenly between > and <.But let's check for combination1: S=14 (even). Number of permutations where a + b =7. The ordered pairs (a,b) where a + b=7 are (2,5) and (5,2), (3,4) and (4,3). So there are 4 ordered pairs where sum=7. Each such ordered pair corresponds to permutations where a + b =7 and c + d=7. For each of these 4 ordered pairs, the remaining two numbers can be arranged in 2 ways, so total permutations with a + b = c + d is 4*2=8. Total permutations for combination1 is24. Therefore, permutations where a + b ≠ c + d is24 -8=16. Therefore, permutations where a + b >c +d is16/2=8, which matches our previous count. So this method works.Similarly, for combination2: S=13 (odd). So there are no permutations where a + b =c + d, because S is odd. Therefore, total permutations where a + b >c +d is24/2=12, which matches.Combination3: S=12 (even). The permutations where a + b =6. The ordered pairs (a,b) that sum to6 are (1,5),(5,1),(2,4),(4,2). Each of these 4 ordered pairs, and remaining numbers arranged in2 ways. So total permutations with a + b =c +d is4*2=8. Therefore, permutations where a + b ≠c +d is24-8=16, so a + b >c +d is8, which matches.Combination4:S=11 (odd). Therefore, all permutations are either a + b >c +d or a + b <c +d, equally, so 12 each. Which matches our count of12.Combination5:S=10 (even). The permutations where a + b =5. The ordered pairs (a,b) that sum to5 are (1,4),(4,1),(2,3),(3,2). So4 ordered pairs, each with remaining numbers arranged in2 ways. So total permutations with a + b =c +d is4*2=8. Therefore, permutations where a + b ≠c +d is16, so a + b >c +d is8, which matches.Therefore, using this method, we can calculate for each combination:For even S: permutations with a + b >c +d is (24 - E)/2, where E is the number of permutations with a + b =c +d.For odd S: permutations with a + b >c +d is12.Where E= number of ordered pairs (a,b) with a + b =S/2 multiplied by 2.But we already saw that this gives the same counts as before.Therefore, the total number of valid permutations is:For combinations with even S:(24 - E)/2For combinations with odd S:12Now, let's compute E for each combination:Combination1: S=14 (even). S/2=7. Number of ordered pairs (a,b) with sum7: (2,5),(5,2),(3,4),(4,3). So 4 ordered pairs. Therefore, E=4*2=8. Therefore, valid permutations=(24-8)/2=8.Combination2:S=13 (odd). E=0. Valid permutations=12.Combination3:S=12 (even). S/2=6. Ordered pairs (a,b) with sum6: (1,5),(5,1),(2,4),(4,2). 4 ordered pairs. E=8. Valid permutations= (24-8)/2=8.Combination4:S=11 (odd). Valid permutations=12.Combination5:S=10 (even). S/2=5. Ordered pairs (a,b) with sum5: (1,4),(4,1),(2,3),(3,2). 4 ordered pairs. E=8. Valid permutations=(24-8)/2=8.So same counts. Therefore, total permutations=8+12+8+12+8=48.Therefore, the answer is48 different ways.But wait, the problem says "select 4 different numbers to fill in the 4 squares". So the numbers have to be selected from the 5, and then arranged. So our method is correct: considering all combinations of 4 numbers and all permutations. And total valid ways is48.Therefore, the answer is boxed{48}.</think>To determine the number of different ways to arrange 4 numbers from the set {1, 2, 3, 4, 5} such that the equation (square + square > square + square) holds true, we need to consider each combination of 4 numbers and count the valid permutations for each combination.1. Combination 1 (missing 1): {2, 3, 4, 5} - Total sum (S = 14), (S/2 = 7) - Valid pairs (sum > 7): (3, 5), (4, 5) - Valid permutations: (2 text{ pairs} times 4 text{ arrangements} = 8)2. Combination 2 (missing 2): {1, 3, 4, 5} - Total sum (S = 13), (S/2 = 6.5) - Valid pairs (sum ≥ 7): (3, 4), (3, 5), (4, 5) - Valid permutations: (3 text{ pairs} times 4 text{ arrangements} = 12)3. Combination 3 (missing 3): {1, 2, 4, 5} - Total sum (S = 12), (S/2 = 6) - Valid pairs (sum > 6): (2, 5), (4, 5) - Valid permutations: (2 text{ pairs} times 4 text{ arrangements} = 8)4. Combination 4 (missing 4): {1, 2, 3, 5} - Total sum (S = 11), (S/2 = 5.5) - Valid pairs (sum ≥ 6): (1, 5), (2, 5), (3, 5) - Valid permutations: (3 text{ pairs} times 4 text{ arrangements} = 12)5. Combination 5 (missing 5): {1, 2, 3, 4} - Total sum (S = 10), (S/2 = 5) - Valid pairs (sum > 5): (2, 4), (3, 4) - Valid permutations: (2 text{ pairs} times 4 text{ arrangements} = 8)Summing up all valid permutations for each combination:- Combination 1: 8- Combination 2: 12- Combination 3: 8- Combination 4: 12- Combination 5: 8Total valid permutations: (8 + 12 + 8 + 12 + 8 = 48)The final answer is (boxed{48}).]

🔑:【Analysis】We can start by considering the first and last digits:For combinations greater than 1+5 (1 type), there is 3+4>1+5For combinations greater than 1+4 (2 types), there are 2+5>1+4,3+5>1+4For combinations greater than 1+3 (3 types), there are 2+4>1+3,2+5>1+3,4+5>1+3And so on, for combinations greater than 1+2 there are 3 typesFor combinations greater than 2+3 there are 2 typesFor combinations greater than 2+4 there is 1 typeEach combination has 4 different ways to fill in, and the solution can be found accordingly.【Solution】Solution: For combinations greater than 1+5 (1 type), there is 3+4>1+5For combinations greater than 1+4 (2 types), there are 2+5>1+4,3+5>1+4For combinations greater than 1+3 (3 types), there are 2+4>1+3,2+5>1+3,4+5>1+3And so on, for combinations greater than 1+2 there are 3 typesFor combinations greater than 2+3 there are 2 typesFor combinations greater than 2+4 there is 1 typebegin{aligned}& (1+2+3) times 2 times 4 = & 12 times 4 = & 48 text { (types) }end{aligned}Answer: There are 48 different ways to fill in the blanks to make the equation true.Therefore, the answer is: 48.

🧠:[<think>Okay, so I have this problem here: If 3^m = 4 and 3^n = 5, then what is the value of 3^{m-2n}? Hmm, let me see. I need to figure out 3 raised to the power of (m minus 2n). Alright, let's start by recalling some exponent rules because I think that will help here. First, I remember that when you have exponents with the same base and you're subtracting in the exponent, like a^{b - c}, that can be rewritten as a^b divided by a^c. So maybe 3^{m - 2n} is the same as 3^m divided by 3^{2n}? Let me check that. Yeah, the rule is a^{b - c} = a^b / a^c. So that part seems right.So, applying that here, 3^{m - 2n} = 3^m / 3^{2n}. Now, I already know that 3^m is 4, from the problem statement. So substituting that in, that gives me 4 divided by 3^{2n}. But what is 3^{2n}?Hmm, 3^{2n} can be written as (3^n)^2, right? Because when you have a power raised to another power, you multiply the exponents. So (3^n)^2 is 3^{n*2} = 3^{2n}. That makes sense. And from the problem, we know that 3^n = 5. So substituting that in, (3^n)^2 becomes 5^2, which is 25. So 3^{2n} = 25.So going back to the previous expression, 3^{m - 2n} = 3^m / 3^{2n} = 4 / 25. Wait, that seems straightforward. Is there anything I'm missing here?Let me just verify each step again. Starting with the exponent rule: a^{b - c} = a^b / a^c. Applied here, b is m and c is 2n, so 3^{m - 2n} = 3^m / 3^{2n}. Then, 3^m is given as 4. Then, 3^{2n} is (3^n)^2, which is 5^2 = 25. So 4 divided by 25. That gives 4/25. Hmm, 4 divided by 25 is 0.16 in decimal, but the problem probably expects a fraction. So 4/25 is the answer. Let me just make sure there's no alternative way to approach this.Alternatively, maybe I can express m and n in terms of logarithms? Let's see. If 3^m = 4, then m = log base 3 of 4. Similarly, n = log base 3 of 5. Then, m - 2n would be log3(4) - 2*log3(5). Then, using logarithm rules, 2*log3(5) is log3(5^2) = log3(25). So, log3(4) - log3(25) is log3(4/25). Therefore, 3^{m - 2n} = 3^{log3(4/25)}. Since 3 raised to the log base 3 of something is just that something, so that simplifies to 4/25. So that confirms the same answer.Okay, so both methods lead me to 4/25. So I think that's correct. Let me check if the problem is expecting maybe a different form, like mixed number or decimal? But since 4/25 is already a simple fraction, and there's no indication otherwise, I think that's the answer. Wait, another way: Maybe express 3^{m - 2n} as 3^m * 3^{-2n} which is 3^m * (3^n)^{-2}. Then substituting the given values: 4 * (5)^{-2} = 4 * (1/25) = 4/25. Yep, same result. So regardless of the approach, it's 4/25. Alright, so I think that's the answer. Seems straightforward once I break it down using exponent rules. Maybe I was overcomplicating it at first, but verifying through different methods gives me confidence.Final AnswerThe value of (3^{m-2n}) is boxed{dfrac{4}{25}}.</think>Given the problem: If (3^m = 4) and (3^n = 5), we need to find the value of (3^{m-2n}).First, we use the exponent rule (a^{b - c} = frac{a^b}{a^c}). Applying this rule, we can rewrite (3^{m - 2n}) as:[3^{m - 2n} = frac{3^m}{3^{2n}}]We know from the problem statement that (3^m = 4). Next, we need to find (3^{2n}). Using the exponent rule ((a^b)^c = a^{bc}), we can express (3^{2n}) as:[3^{2n} = (3^n)^2]Given (3^n = 5), substituting this value gives:[(3^n)^2 = 5^2 = 25]Now substituting (3^m = 4) and (3^{2n} = 25) into the expression for (3^{m - 2n}), we get:[3^{m - 2n} = frac{3^m}{3^{2n}} = frac{4}{25}]Thus, the value of (3^{m-2n}) is (boxed{dfrac{4}{25}}).,<think>Okay, let's see. I need to find the value of (3^{m - 2n}) given that (3^m = 4) and (3^n = 5). Hmm, exponents. Alright, let me start by recalling some exponent rules. First, I know that when you subtract exponents with the same base, you can express that as division. Specifically, (3^{m - 2n}) should be the same as (3^m) divided by (3^{2n}). So, (3^{m - 2n} = frac{3^m}{3^{2n}}). Wait, but (3^{2n}) can be written as ((3^n)^2), right? Because when you have an exponent raised to another exponent, you multiply them. So, (3^{2n} = (3^n)^2). That seems correct. Let me check that: ( (3^n)^2 = 3^n times 3^n = 3^{n + n} = 3^{2n}). Yep, that works out.So substituting the known values here. They gave me (3^m = 4) and (3^n = 5). So replacing those in the expression, I should have:(3^{m - 2n} = frac{3^m}{(3^n)^2} = frac{4}{(5)^2} = frac{4}{25}). Wait, so is that all? Let me make sure I didn't skip any steps or make a mistake. Let me go through it again.Starting with (3^{m - 2n}), which is a single exponent with base 3. The exponent here is (m - 2n). According to the laws of exponents, specifically the property (a^{b - c} = frac{a^b}{a^c}), this can be rewritten as (frac{3^m}{3^{2n}}). Then, (3^{2n}) can be written as ((3^n)^2), which is 5 squared since (3^n = 5). So (5^2 = 25). Therefore, substituting back, we have (frac{4}{25}). Seems straightforward, but let me verify using logarithms to check if there's another way. If I take the logarithm base 3 of both sides of the original equations, then for (3^m = 4), we get (m = log_3 4), and for (3^n = 5), (n = log_3 5). Then (m - 2n = log_3 4 - 2 log_3 5). Using logarithm properties, this is (log_3 4 - log_3 5^2 = log_3 left(frac{4}{25}right)). Therefore, (3^{m - 2n} = 3^{log_3 (4/25)} = frac{4}{25}). Okay, so that's the same answer. So both methods confirm that the value is (frac{4}{25}). I think that's solid. Let me check once more. If (3^m = 4) and (3^n = 5), then substituting directly into (3^{m - 2n}) gives (4 / 5^2 = 4/25). Yep, that's right. No mistakes here. So I feel confident that the answer is 4/25.Final AnswerThe value of (3^{m - 2n}) is boxed{dfrac{4}{25}}.</think>Given (3^m = 4) and (3^n = 5), we need to find the value of (3^{m - 2n}).First, we use the properties of exponents to rewrite (3^{m - 2n}):[3^{m - 2n} = frac{3^m}{3^{2n}}]Next, we recognize that (3^{2n}) can be written as ((3^n)^2):[3^{2n} = (3^n)^2]Substituting the given values (3^m = 4) and (3^n = 5) into the expression, we get:[3^{m - 2n} = frac{3^m}{(3^n)^2} = frac{4}{5^2} = frac{4}{25}]Thus, the value of (3^{m - 2n}) is (boxed{dfrac{4}{25}}).]

🔑:Given that 3^{m}=4 and 3^{n}=5, we want to find the value of 3^{m-2n}. First, we recognize that the properties of exponents allow us to manipulate the given equations in useful ways. Specifically, we can use the fact that a^{b-c} = frac{a^b}{a^c} and (a^b)^c = a^{bc} for any real numbers a, b, and c. Starting with the expression 3^{m-2n}, we apply these properties as follows:[3^{m-2n} = 3^{m} div 3^{2n} = 3^{m} div (3^{n})^{2}]Substituting the given values for 3^{m} and 3^{n}, we get:[3^{m-2n} = 4 div 5^{2}]Since 5^2 = 25, this simplifies to:[3^{m-2n} = frac{4}{25}]Therefore, the value of 3^{m-2n} is boxed{frac{4}{25}}.

🧠:[<think>Okay, let's see. I need to find all bijective and monotonic functions f from the real numbers to the real numbers such that for every real number t, f(t) + f⁻¹(t) equals 2t. Hmm. Alright, let's break this down step by step.First, since f is bijective, it has an inverse function f⁻¹. The equation given is f(t) + f⁻¹(t) = 2t for all t. And we need to find all such f that are monotonic. So, monotonic could mean either strictly increasing or strictly decreasing. But since f is bijective, it must be either strictly increasing or strictly decreasing anyway, right? Because a bijection on ℝ has to be strictly monotonic. So maybe that part is redundant? Or maybe the problem wants to ensure we consider both possibilities? Not sure, but I'll keep that in mind.Let me start by trying to express f⁻¹ in terms of f. If f(t) + f⁻¹(t) = 2t, then rearranging gives f⁻¹(t) = 2t - f(t). Wait, but f⁻¹(t) is the inverse function, so that means if I let y = f(t), then t = f⁻¹(y). So substituting into the equation, we get y + f⁻¹(y) = 2t. But since t = f⁻¹(y), this becomes y + f⁻¹(y) = 2f⁻¹(y). Wait, that seems confusing. Let me write that again.Suppose y = f(t). Then t = f⁻¹(y). The original equation is f(t) + f⁻¹(t) = 2t. Let's substitute t with f⁻¹(y). Then the equation becomes f(f⁻¹(y)) + f⁻¹(f⁻¹(y)) = 2f⁻¹(y). Simplifying f(f⁻¹(y)) gives y. So, y + f⁻¹(f⁻¹(y)) = 2f⁻¹(y). Therefore, f⁻¹(f⁻¹(y)) = 2f⁻¹(y) - y.Hmm, that's an equation involving the inverse function composed with itself. Let me denote g(y) = f⁻¹(y). Then the equation becomes g(g(y)) = 2g(y) - y. So, we have a functional equation: g(g(y)) + y = 2g(y). So, this is the same as g²(y) - 2g(y) + y = 0, where g² is the composition of g with itself. Interesting.So, if I can solve this functional equation for g, then f would be the inverse of g. Since f and g are inverses, and f is monotonic, so is g. So, maybe solving this functional equation for g will help me find f.So, let's think about how to solve g(g(y)) + y = 2g(y). Let me rearrange it as g(g(y)) = 2g(y) - y. This looks similar to a linear difference equation or something. Maybe we can look for linear functions g(y) that satisfy this. Let me try assuming that g is linear, say g(y) = a y + b. Then let's see what the equation becomes.If g(y) = a y + b, then g(g(y)) = a(a y + b) + b = a² y + a b + b. Then, according to the equation, we have:a² y + a b + b = 2(a y + b) - ySimplify the right-hand side: 2a y + 2b - y = (2a - 1) y + 2bTherefore, equate coefficients:For the y term: a² = 2a - 1For the constant term: a b + b = 2b => a b = bSo, solving these equations:First equation: a² - 2a + 1 = 0 => (a - 1)^2 = 0 => a = 1Second equation: If a = 1, then 1 * b = b => 0 = 0, which is always true. So b can be any real number? Wait, but then if a = 1 and b is arbitrary, then g(y) = y + b. Let's check if this satisfies the original functional equation.If g(y) = y + b, then g(g(y)) = g(y + b) = y + b + b = y + 2b. Then according to the equation, this should equal 2g(y) - y = 2(y + b) - y = 2y + 2b - y = y + 2b. So yes, it works. So g(y) = y + b is a solution for any real number b.But wait, but we need g to be invertible because g is f⁻¹, which is the inverse of a bijection. So, if g(y) = y + b, then it's invertible (its inverse is y - b). So, f would be the inverse of g, which is f(t) = t - b.Let me check if f(t) = t - b satisfies the original equation. Let's compute f(t) + f⁻¹(t). f(t) = t - b, so f⁻¹(t) is t + b. Therefore, f(t) + f⁻¹(t) = (t - b) + (t + b) = 2t. Perfect, that works. So for any real number b, f(t) = t - b is a solution.But wait, the problem says "Find all elements of E that are monotonic mappings." So, these linear functions are certainly monotonic (they are increasing if the slope is positive, which in this case the slope is 1, so they are increasing). So, these are all linear solutions. But are there any other solutions?Wait, but in the functional equation for g, we assumed g is linear, but maybe there are non-linear solutions? The problem asks for all monotonic bijections, so we need to check if only linear functions satisfy this or if there are non-linear ones.Let me see. Suppose g is a non-linear function satisfying g(g(y)) = 2g(y) - y. Let's see if such a function can exist. Maybe not. Let's test some simple non-linear functions.For example, suppose g(y) is quadratic: g(y) = a y² + b y + c. Then g(g(y)) would be a(a y² + b y + c)² + b(a y² + b y + c) + c. That's going to be a quartic function. On the other hand, 2g(y) - y = 2a y² + 2b y + 2c - y. So, a quartic equals a quadratic? Only possible if all higher degree coefficients are zero. So unless a = 0, which would make g linear. So quadratic functions won't work. Similarly, cubic functions would lead to higher degrees. So, perhaps only linear functions satisfy this equation?Alternatively, maybe piecewise functions? Let's consider a piecewise linear function. Suppose g(y) is piecewise linear with different slopes in different intervals. But then composing g with itself would probably not result in a function that's 2g(y) - y everywhere. It might get complicated. Let me try a simple case.Suppose g(y) is defined as y + 1 for y ≤ 0 and y - 1 for y > 0. Then let's compute g(g(y)). For y ≤ 0: g(y) = y + 1. If y ≤ 0, then y + 1 ≤ 1. So if y + 1 ≤ 0, i.e., y ≤ -1, then g(g(y)) = g(y + 1) = (y + 1) + 1 = y + 2. If y > -1 but y ≤ 0, then y + 1 is between 0 and 1, so g(g(y)) = g(y + 1) = (y + 1) - 1 = y. For y > 0: g(y) = y - 1. Then y - 1 could be ≤ 0 or > 0. If y > 1, then y - 1 > 0, so g(g(y)) = g(y - 1) = (y - 1) - 1 = y - 2. If 0 < y ≤ 1, then y - 1 ≤ 0, so g(g(y)) = g(y - 1) = (y - 1) + 1 = y. So putting it all together:g(g(y)) =- y + 2, if y ≤ -1- y, if -1 < y ≤ 1- y - 2, if y > 1But according to the equation g(g(y)) = 2g(y) - y. Let's compute 2g(y) - y:For y ≤ 0: 2(y + 1) - y = 2y + 2 - y = y + 2For y > 0: 2(y - 1) - y = 2y - 2 - y = y - 2So, comparing:- For y ≤ -1: g(g(y)) = y + 2, and 2g(y) - y = y + 2. So matches.- For -1 < y ≤ 0: g(g(y)) = y, but 2g(y) - y = y + 2. Wait, this doesn't match. Here, when -1 < y ≤ 0, g(g(y)) = y, but 2g(y) - y = y + 2. So unless y = y + 2, which is not possible, this doesn't hold. Similarly, for 0 < y ≤ 1: g(g(y)) = y, but 2g(y) - y = y - 2. Again, y ≠ y - 2 unless y is infinity. So, this piecewise function only satisfies the equation for y ≤ -1 and y > 1, but not in the middle intervals. Therefore, this piecewise function is not a solution.So, maybe trying to make a non-linear or piecewise linear function doesn't work. Therefore, perhaps the only solutions are linear functions. Let's verify this.Suppose that g is a differentiable function (since it's monotonic, it's differentiable almost everywhere, but maybe assuming smoothness for simplicity). Let me try differentiating the equation g(g(y)) = 2g(y) - y. Taking derivative of both sides:Using the chain rule, the left-hand side derivative is g’(g(y)) * g’(y). The right-hand side derivative is 2g’(y) - 1. So:g’(g(y)) * g’(y) = 2g’(y) - 1If we suppose that g is linear, then g’(y) is constant. Let’s check. Let’s let g(y) = a y + b, so g’(y) = a. Then the left-hand side becomes a * a = a². The right-hand side is 2a - 1. So a² = 2a - 1, which is the same equation as before, leading to a = 1. Then, with a = 1, the equation is 1 = 2*1 - 1 = 1, which holds. So that works. But if g is non-linear, then g’(y) is not constant. Suppose that g is non-linear, can the equation still hold?Let me suppose that g is a strictly increasing function (since f is bijective and monotonic, so g = f⁻¹ is also strictly monotonic; if f is increasing, then g is increasing; if f is decreasing, then g is decreasing). Wait, the problem says "monotonic mappings", so f could be increasing or decreasing. But if f is decreasing, then so is g = f⁻¹. Hmm. Let me consider both cases.First, suppose that g is strictly increasing. Then, since it's differentiable (assuming smoothness), the derivative is positive. Let’s write the equation again:g’(g(y)) * g’(y) = 2g’(y) - 1Let’s divide both sides by g’(y) (since it's positive, we can do that):g’(g(y)) = 2 - 1/g’(y)Hmm, this is a functional equation for g’(y). Let’s denote h(y) = g’(y). Then the equation becomes:h(g(y)) = 2 - 1/h(y)But since g is strictly increasing and h(y) is its derivative, h(y) > 0 for all y. So, we have:h(g(y)) = 2 - 1/h(y)This seems complicated. Let me see if there's a constant solution. Suppose h(y) = k, constant. Then:k = 2 - 1/k => k² - 2k + 1 = 0 => (k - 1)^2 = 0 => k = 1. So h(y) = 1, which gives g’(y) = 1, so g(y) = y + b. Which brings us back to the linear solution. So the only constant solution is the linear one. But maybe there are non-constant solutions?Suppose we let h(y) = 1 for all y. Then g(y) = y + b, which works. What if h(y) is not constant? Let's try to find a function h(y) such that h(g(y)) = 2 - 1/h(y). Let me think recursively. Suppose we have some function h(y), and g(y) is its integral. But this seems too vague. Alternatively, maybe look for a function h satisfying this relation.Alternatively, assume that h(y) is linear. Suppose h(y) = m y + c. Then, we have:h(g(y)) = m g(y) + c = 2 - 1/(m y + c)But g(y) is the integral of h(y), so g(y) = ∫ h(y) dy + d = (m/2)y² + c y + d. Then:h(g(y)) = m*( (m/2)y² + c y + d ) + c = (m²/2)y² + m c y + m d + cBut according to the equation, this should equal 2 - 1/(m y + c). So:(m²/2)y² + m c y + m d + c = 2 - 1/(m y + c)This is a quadratic function on the left and a function with a reciprocal term on the right. These can only be equal if all coefficients of the quadratic terms are zero and the reciprocal term somehow cancels out. But the reciprocal term 1/(m y + c) can't be expressed as a quadratic function. Therefore, this suggests that h(y) cannot be linear. Therefore, perhaps there are no non-linear solutions if we assume h(y) is linear? But this is getting too convoluted.Alternatively, let's consider the case where f is a decreasing function. If f is decreasing, then its inverse f⁻¹ is also decreasing. Because the inverse of a decreasing function is decreasing. So, if f is decreasing, then g = f⁻¹ is also decreasing. Let's see if such functions can satisfy the original equation.Suppose f is decreasing. Then, since both f and f⁻¹ are decreasing, the left-hand side f(t) + f⁻¹(t) would be the sum of two decreasing functions, which might be a bit tricky. But the right-hand side 2t is increasing. So, the sum of two decreasing functions equals an increasing function. That seems possible, but maybe restrictive.But let's try to see if there are any decreasing solutions. Suppose f is decreasing and linear. Let’s assume f(t) = -a t + b, where a > 0. Then f⁻¹(t) can be computed. To find the inverse, set y = -a t + b, solve for t: t = (b - y)/a. So f⁻¹(t) = (b - t)/a. Then, according to the equation f(t) + f⁻¹(t) = 2t:(-a t + b) + (b - t)/a = 2tLet’s simplify:Multiply through by a to eliminate the denominator:-a² t + a b + b - t = 2a tCombine like terms:(-a² - 1 - 2a) t + (a b + b) = 0For this to hold for all t, the coefficients must be zero:Coefficient of t: -a² - 1 - 2a = 0 => a² + 2a + 1 = 0 => (a + 1)^2 = 0 => a = -1But a was supposed to be positive (since f is decreasing, slope is negative). So a = -1 gives a = -1. Wait, but a = -1 would mean the slope is -(-1) = 1? Wait, hold on. If f(t) = -a t + b, then the slope is -a. If a = -1, then the slope is -(-1) = 1, which is positive. But we wanted a decreasing function, so slope must be negative. Therefore, a must be positive. So this leads to a contradiction. Therefore, there are no linear decreasing functions satisfying the equation. So perhaps all solutions are increasing linear functions?But let's confirm. If a = -1, then f(t) = -(-1)t + b = t + b. But that's an increasing function, which we already considered earlier. So in the decreasing case, we end up with a = -1, which gives an increasing function. Therefore, there are no decreasing linear solutions. So perhaps all solutions are increasing linear functions, i.e., f(t) = t + b. Wait, but earlier we had f(t) = t - b. Wait, let me check again.Wait, when we set g(y) = y + b, then f(t) is the inverse of g, so f(t) = t - b. So, the function f(t) = t + c (where c is a constant) would have an inverse f⁻¹(t) = t - c. Then f(t) + f⁻¹(t) = (t + c) + (t - c) = 2t. So yes, any function of the form f(t) = t + c satisfies the equation. But also, if I set f(t) = t - c, then f⁻¹(t) = t + c, and again f(t) + f⁻¹(t) = (t - c) + (t + c) = 2t. So, whether we write f(t) = t + c or f(t) = t - c, it's the same set of functions, just with c being any real constant. So the general linear solution is f(t) = t + c where c is a real constant.But wait, in the earlier calculation when we assumed g(y) is linear, we found that a = 1 and b is arbitrary, leading to f(t) = t - b. So yes, c can be any real number, positive or negative. Therefore, the set of linear solutions is all functions of the form f(t) = t + c for some constant c ∈ ℝ.But the problem says "Find all elements of E that are monotonic mappings." Since we've considered linear functions, which are monotonic (specifically, strictly increasing because the slope is 1), and we saw that decreasing functions can't be linear solutions. But could there be non-linear monotonic solutions?Suppose there exists a non-linear monotonic function f satisfying the equation. Let's assume f is differentiable for simplicity. Then, as we saw earlier, the derivative condition is:g’(g(y)) * g’(y) = 2g’(y) - 1, where g = f⁻¹But if f is non-linear, then g is also non-linear, and their derivatives are not constant. Let me see if there's a way to construct such a function or prove that no non-linear function can satisfy this.Suppose we try an exponential function. Let’s say g(y) = e^{k y} + c. But this is probably not going to work. Alternatively, maybe something else.Alternatively, let's consider the functional equation g(g(y)) = 2g(y) - y. Let me try to iterate this equation. Suppose we apply g three times:g(g(g(y))) = g(2g(y) - y). But also, from the original equation, replacing y with g(y), we have g(g(g(y))) = 2g(g(y)) - g(y) = 2(2g(y) - y) - g(y) = 4g(y) - 2y - g(y) = 3g(y) - 2y.On the other hand, g(2g(y) - y) should equal 3g(y) - 2y. So:g(2g(y) - y) = 3g(y) - 2yThis is another functional equation. Maybe this can help us. Let me see if this can lead us somewhere.If we let z = 2g(y) - y, then the equation becomes g(z) = 3g(y) - 2y. But z = 2g(y) - y, so perhaps expressing y in terms of z? Maybe not straightforward.Alternatively, let's assume that g is linear, which we already know works. If g(y) = y + b, then let's check the equation g(2g(y) - y) = 3g(y) - 2y.Left-hand side: g(2(y + b) - y) = g(y + 2b) = (y + 2b) + b = y + 3bRight-hand side: 3(y + b) - 2y = 3y + 3b - 2y = y + 3bSo equality holds. So linear functions satisfy this. What if we suppose a different function?Alternatively, suppose g is affine linear, which we already considered. So perhaps the only solutions are affine linear functions. But how to prove that?Alternatively, consider writing the functional equation as g(g(y)) - 2g(y) + y = 0. If we think of this as a second-order linear recurrence, but in functions. Wait, not sure.Alternatively, treat y as a variable and consider the equation g(g(y)) + y = 2g(y). Let’s rearrange it as g(g(y)) - 2g(y) + y = 0. If we consider the left-hand side as a sort of "operator" applied to g, maybe we can factor it.Suppose we think of this as (g ∘ g - 2g + id)(y) = 0, where id is the identity function. Maybe factor this operator. If we think in terms of linear operators, which might be a stretch, but suppose that (g ∘ g - 2g + id) = 0. If we can factor this as (g - id)^2 = 0, then that would imply g(y) = id(y) + c, but I'm not sure.Alternatively, if we let h(y) = g(y) - y, then substituting into the equation:g(g(y)) = 2g(y) - yExpressing g(y) as h(y) + y:g(h(y) + y) = 2(h(y) + y) - y => h(h(y) + y) + (h(y) + y) = 2h(y) + 2y - y => h(h(y) + y) + h(y) + y = 2h(y) + ySimplify:h(h(y) + y) = 2h(y) + y - h(y) - y = h(y)So h(h(y) + y) = h(y)This is another functional equation. Let me see if this helps. Let me denote k(y) = h(y) + y. Then the equation becomes h(k(y)) = h(y). So h(k(y)) = h(y). If h is injective, then k(y) = y. But h(y) = g(y) - y. If g is injective, which it is because it's bijective, then h(y) = g(y) - y is also injective? Not necessarily. For example, if g(y) = y + b, then h(y) = b, which is constant, hence not injective. So in the linear case, h(y) is constant, so h(k(y)) = h(y) trivially holds because h is constant.But in the non-linear case, if h is injective, then k(y) = y. But k(y) = h(y) + y = g(y). So if h is injective, then h(g(y)) = h(y) implies g(y) = y. But g(y) = y implies h(y) = 0, which is the trivial solution. So if h is injective, only the trivial solution exists, which is h(y) = 0, leading to g(y) = y. But we already have solutions where h(y) = b, a constant. So this approach might not be helpful.Alternatively, maybe considering fixed points. A fixed point of g is a y such that g(y) = y. Suppose y is a fixed point. Then, from the equation g(g(y)) = 2g(y) - y, substituting g(y) = y, we get y = 2y - y => y = y. So every fixed point satisfies the equation, but gives no new information. So fixed points can be any real numbers. However, if g(y) = y + b, then fixed points would satisfy y + b = y => b = 0. So if b ≠ 0, there are no fixed points. If b = 0, then every y is a fixed point. So that's consistent.Alternatively, suppose that g has no fixed points. For example, if g(y) = y + b with b ≠ 0, then there are no fixed points. If g(y) = y, then all points are fixed. But how does this help?Alternatively, consider the function k(y) = g(y) - y. Then, as before, k(y) + y = g(y). Then, from the equation g(g(y)) = 2g(y) - y, substitute g(y) = k(y) + y:g(k(y) + y) = 2(k(y) + y) - y => k(k(y) + y) + (k(y) + y) = 2k(y) + 2y - y => k(k(y) + y) + k(y) + y = 2k(y) + ySimplify:k(k(y) + y) = 2k(y) + y - k(y) - y = k(y)Thus, k(k(y) + y) = k(y)So, k is a function such that applying it to (k(y) + y) gives k(y). If k is a constant function, say k(y) = c, then:k(k(y) + y) = c = k(y) = cWhich holds. So constant functions k(y) = c satisfy this. This corresponds to g(y) = y + c, which are the linear solutions we found earlier.If k is not constant, is there a non-constant k that satisfies k(k(y) + y) = k(y)? For example, suppose k(y) = 0 for some y and non-zero otherwise. But since g is bijective, k(y) = g(y) - y must be such that g(y) is bijective. If k(y) is zero except at some points, g(y) would be y except at those points, which might not be bijective. For example, if k(y) is zero except at y = 0 where k(0) = 1, then g(0) = 1, but g(1) = 1 + k(1) = 1 + 0 = 1. So g(0) = g(1) = 1, which means g is not injective. Therefore, k(y) must be such that g(y) = y + k(y) is bijective. This seems to restrict k(y) to being a constant function. Because if k(y) is not constant, then the slope of g(y) would be 1 + k’(y). To be bijective, the derivative must not change sign. If k’(y) is always zero, then k is constant. If k’(y) is non-zero somewhere, then the slope would vary, but still needs to maintain monotonicity. However, if k(y) is non-constant and smooth, then the derivative 1 + k’(y) must be always positive or always negative. Suppose 1 + k’(y) > 0 for all y. Then g(y) is strictly increasing. But even so, unless k(y) is linear, the function k(k(y) + y) = k(y) would be difficult to satisfy.Alternatively, suppose k(y) is linear, k(y) = m y + c. Then:k(k(y) + y) = m(k(y) + y) + c = m(m y + c + y) + c = m((m + 1)y + c) + c = m(m + 1)y + m c + cThis should equal k(y) = m y + c. So:m(m + 1)y + m c + c = m y + cEquate coefficients:For y term: m(m + 1) = m => m² + m = m => m² = 0 => m = 0For constant term: m c + c = c => (m + 1)c = c => (0 + 1)c = c => c = c, which holds.Therefore, the only linear solution for k(y) is m = 0, which gives k(y) = c, a constant. Hence, this again leads us back to the linear solution g(y) = y + c. Therefore, non-linear k(y) don't work even if they are linear in y. Wait, that's a bit confusing. If k(y) is linear, we end up with m = 0, hence constant k(y). Therefore, the only solutions for k(y) are constants. Hence, the only solutions for g(y) are linear functions of the form y + c. Therefore, the only solutions for f are inverses of these, which are f(t) = t - c.Hence, combining all this, the only monotonic bijections satisfying f(t) + f⁻¹(t) = 2t for all t are the linear functions f(t) = t + c where c is a real constant. Wait, but earlier we had f(t) = t - c as the inverse of g(y) = y + c. So f(t) = g⁻¹(t) = t - c. So yes, both forms are equivalent, with c being any real constant.Therefore, the set E consists of all functions f(t) = t + c for some constant c ∈ ℝ. These are all strictly increasing functions, as their derivative is 1, which is positive. There are no decreasing functions in E because, as we saw earlier, assuming a decreasing function leads to a contradiction unless it's actually increasing.So, after all this, the conclusion is that the only monotonic bijections in E are the linear functions f(t) = t + c for some constant c. Therefore, the answer should be all functions of the form f(t) = t + c where c is a real number.Final AnswerThe monotonic mappings in ( E ) are all linear functions of the form (boxed{f(t) = t + c}) where ( c ) is a real constant.</think>Given the problem of finding all bijective and monotonic mappings ( f ) from (mathbb{R}) to (mathbb{R}) that satisfy the equation ( f(t) + f^{-1}(t) = 2t ) for all ( t in mathbb{R} ), we start by analyzing the functional equation and its implications.1. Functional Equation Analysis: - The given equation ( f(t) + f^{-1}(t) = 2t ) can be rearranged to express the inverse function ( f^{-1} ) in terms of ( f ). Let ( g = f^{-1} ), then the equation becomes ( g(g(t)) = 2g(t) - t ).2. Linear Function Assumption: - Assuming ( g ) is a linear function, we let ( g(t) = at + b ). Substituting this into the functional equation, we find that ( a = 1 ) and ( b ) is an arbitrary constant. This gives ( g(t) = t + b ).3. Inverse Function Calculation: - The inverse of ( g(t) = t + b ) is ( f(t) = t - b ). Verifying this in the original equation, we get ( f(t) + f^{-1}(t) = (t - b) + (t + b) = 2t ), which satisfies the given condition.4. Monotonicity and Bijectivity: - Since ( f(t) = t + c ) (where ( c ) is a real constant) is a linear function with a slope of 1, it is strictly increasing and hence bijective. Any constant ( c ) will maintain the bijectivity and monotonicity of ( f ).5. Non-linear Solutions: - Considering the possibility of non-linear solutions, we analyze the functional equation and its derivatives. However, non-linear functions lead to contradictions or do not satisfy the necessary conditions for bijectivity and monotonicity.6. Conclusion: - The only solutions that satisfy the given functional equation, bijectivity, and monotonicity are linear functions of the form ( f(t) = t + c ) where ( c ) is a real constant.Thus, the monotonic mappings in ( E ) are all linear functions of the form (boxed{f(t) = t + c}) where ( c ) is a real constant.,<think>Okay, let's see. I need to find all bijective and monotonic functions f from ℝ to ℝ such that f(t) + f⁻¹(t) = 2t for all real t. Hmm, interesting problem. Let me start by understanding what this condition implies.First, since f is bijective, it has an inverse function f⁻¹. The equation given is f(t) + f⁻¹(t) = 2t. Wait, but here the argument is t, not f(t) or something else. So for every real number t, the sum of f(t) and its inverse function evaluated at t is twice t. That seems like a functional equation. Let's try to manipulate it.Maybe I can express f⁻¹(t) in terms of f. Let me denote y = f(t). Then, by definition of inverse function, f⁻¹(y) = t. But the equation given is f(t) + f⁻¹(t) = 2t. So substituting y = f(t), we get y + f⁻¹(t) = 2t. But since f⁻¹(y) = t, maybe we can write f⁻¹(t) in terms of y?Wait, maybe I should express f⁻¹(t) in terms of f. Let me think. If I let t be any real number, then f⁻¹(t) is the value s such that f(s) = t. So s = f⁻¹(t). Then, substituting into the equation, we have f(s) + f⁻¹(f(s)) = 2s. But f⁻¹(f(s)) is s, right? Because applying f and then f inverse gets you back to s. So this becomes t + s = 2s, which simplifies to t = s. But s = f⁻¹(t), so t = f⁻¹(t). Wait, that would mean f⁻¹(t) = t for all t, which would imply f(t) = t. But that's only one function. However, the problem says "find all elements of E that are monotonic mappings." So maybe f(t) = t is the only solution? But that seems too straightforward. Let me check my reasoning again.Wait, perhaps I made a mistake here. Let me go back step by step. The original equation is f(t) + f⁻¹(t) = 2t for all t. If I take t as a variable, then for each t, f(t) is some value, and f⁻¹(t) is the s such that f(s) = t. So let's denote s = f⁻¹(t). Then, by definition, f(s) = t. So substituting into the original equation, we have f(t) + s = 2t. But s = f⁻¹(t). So f(t) + f⁻¹(t) = 2t. But also, since s = f⁻¹(t), then t = f(s). Therefore, we can write:f(t) + s = 2t,t = f(s).So substituting s = f⁻¹(t) into the first equation, we get:f(t) + f⁻¹(t) = 2t.But that's just the original equation. Maybe I need to combine these equations in some way. Let's try to express s in terms of t. Since s = f⁻¹(t), then t = f(s). Let's replace t with f(s) in the first equation. Then we have:f(f(s)) + s = 2f(s).So f(f(s)) + s = 2f(s).Rewriting, f(f(s)) = 2f(s) - s.Hmm, this gives a functional equation for f composed with f. Maybe we can write this as f(f(s)) = 2f(s) - s. Let me denote this as equation (1).Alternatively, if I let u = f(s), then s = f⁻¹(u), so the equation becomes f(u) = 2u - f⁻¹(u). But from the original equation, f⁻¹(u) = 2u - f(u). Wait, substituting that into the previous equation, we get f(u) = 2u - (2u - f(u)) = 2u - 2u + f(u) = f(u). Hmm, that just gives an identity, so no new information.Maybe we can iterate equation (1). Since f(f(s)) = 2f(s) - s, then applying f again, we get f(f(f(s))) = 2f(f(s)) - f(s). Substituting the previous expression for f(f(s)), this becomes f(f(f(s))) = 2(2f(s) - s) - f(s) = 4f(s) - 2s - f(s) = 3f(s) - 2s.But perhaps this isn't helpful yet. Let me see if I can find a form for f. Since f is bijective and monotonic, it's either strictly increasing or strictly decreasing. But let's first assume it's linear. Suppose f(t) = at + b. Then, f⁻¹(t) would be (t - b)/a, provided that a ≠ 0. Let's check if such a function satisfies the equation.So, substituting into f(t) + f⁻¹(t) = 2t:at + b + (t - b)/a = 2t.Simplify the left-hand side:at + b + t/a - b/a = (a + 1/a)t + b - b/a.Set this equal to 2t. So, coefficients must match:(a + 1/a) = 2,and the constant term:b - b/a = 0.First, solve for a:a + 1/a = 2 => a² - 2a + 1 = 0 => (a - 1)^2 = 0 => a = 1.Then, substituting a = 1 into the constant equation:b - b/1 = 0 => 0 = 0, which is always true. So for a linear function, the only solution is f(t) = t + b, but since a=1, it's f(t) = t + b, but wait, the inverse of f(t) = t + b is f⁻¹(t) = t - b. Then, check the original equation:f(t) + f⁻¹(t) = (t + b) + (t - b) = 2t, which works. So all linear functions of the form f(t) = t + b satisfy the equation. Wait, but hold on, but if b ≠ 0, then f(t) = t + b is a bijection and monotonic. But according to the problem, we need to find all elements of E that are monotonic. So does this mean there are multiple solutions? Wait, but earlier when I tried substituting, I thought f(t) = t is the only solution, but now with linear functions, f(t) = t + b also works? But wait, let's check with b ≠ 0.Take f(t) = t + b. Then f⁻¹(t) = t - b. Then f(t) + f⁻¹(t) = (t + b) + (t - b) = 2t. So yes, it works. But the problem states "bijective mappings from ℝ to ℝ", so these are bijective. Also, they are monotonic since they are linear with slope 1, which is positive. So these are all strictly increasing functions. But wait, earlier when I tried substituting, I thought f⁻¹(t) = t implies f(t) = t, but here f(t) = t + b also works. So that suggests that my initial reasoning was flawed.Wait, but why did I think f⁻¹(t) = t? Because when I substituted t = f(s), I had f(t) + s = 2t, but s = f⁻¹(t). But then, if f(t) = t + b, then s = f⁻¹(t) = t - b. Then f(t) + s = (t + b) + (t - b) = 2t. So yes, that's correct. Therefore, these linear functions are solutions. So there are infinitely many solutions of the form f(t) = t + b. But wait, but in the problem statement, the condition is f(t) + f⁻¹(t) = 2t for all t. So if f(t) = t + b, then f⁻¹(t) = t - b, and indeed f(t) + f⁻¹(t) = 2t. So those are valid solutions.But then, why did my initial substitution lead me to think f⁻¹(t) = t? Maybe I made a mistake there.Let me revisit that. Suppose I take the original equation f(t) + f⁻¹(t) = 2t. Let me denote s = f⁻¹(t). Then, since f(s) = t, we can write f(t) + s = 2t. But s = f⁻¹(t) = t - b if f(t) = t + b. Then, substituting into f(t) + s = (t + b) + (t - b) = 2t. So that works. So perhaps my initial mistake was thinking that substituting t = f(s) leads to a contradiction, but actually, when f is linear, it's consistent.Therefore, maybe there are infinitely many linear solutions. But wait, when I set up the equation for a linear function f(t) = at + b, we found that a must be 1, and b can be any real number. So indeed, all functions f(t) = t + b are solutions. But wait, but if f is linear and invertible, then it must have a non-zero slope. Here, slope is 1, so invertible. Therefore, linear solutions are f(t) = t + b for any real b.But the problem says "Find all elements of E that are monotonic mappings." So, if all linear functions of the form f(t) = t + b are solutions, and they are monotonic (since slope 1 is positive), then these would all be valid. But perhaps there are non-linear solutions as well? The problem doesn't restrict to linear functions, so maybe there are non-linear monotonic bijections that satisfy the equation.Wait, let's test a non-linear function. Suppose f(t) is some non-linear function. Let's assume f is strictly increasing, differentiable maybe. Let me try f(t) = t. Then f⁻¹(t) = t, so f(t) + f⁻¹(t) = 2t, which works. If I take f(t) = t + b, as before, works. What if I try f(t) = -t + c? Wait, but then f is decreasing, but bijective. Let's check if it satisfies the equation. Suppose f(t) = -t + c. Then f⁻¹(t) = c - t. Then f(t) + f⁻¹(t) = (-t + c) + (c - t) = 2c - 2t. For this to equal 2t for all t, we need 2c - 2t = 2t => 2c = 4t => c = 2t, which is not possible unless c is variable, but c is a constant. So no solution here. Therefore, decreasing functions don't work. So maybe only increasing functions can satisfy the equation.But the problem allows for monotonic mappings, which can be increasing or decreasing. But our test shows that decreasing linear functions don't work. So perhaps only increasing functions are possible.So maybe all solutions are linear functions of the form f(t) = t + b. But let's check another non-linear function. Suppose f(t) = t + sin(t). Wait, is this bijective? The derivative is 1 + cos(t), which is non-negative since cos(t) ≥ -1, so 1 + cos(t) ≥ 0. But 1 + cos(t) = 0 when cos(t) = -1, i.e., at t = π + 2πk. So the derivative is zero at those points, which means the function is not strictly increasing, hence not invertible. So f(t) = t + sin(t) is not bijective. So maybe hard to find a non-linear bijective function.Alternatively, suppose f(t) is a piecewise function. For example, f(t) = t for t ≤ 0 and f(t) = t + 1 for t > 0. But this is not continuous, though bijective. Let's check if it satisfies f(t) + f⁻¹(t) = 2t.First, find f⁻¹(t). For t ≤ 0, f⁻¹(t) = t. For t > 0, since f(t) = t + 1, then f⁻¹(t) = t - 1. So f(t) + f⁻¹(t) would be:For t ≤ 0: f(t) = t, f⁻¹(t) = t, so sum is 2t. Good.For t > 0: f(t) = t + 1, but wait, no. Wait, f(t) is defined as t for t ≤ 0 and t + 1 for t > 0. So when t > 0, f(t) = t + 1, so the inverse function for y > 1 is f⁻¹(y) = y - 1. For y ≤ 1, since f(t) = t for t ≤ 0, but if t > 0, f(t) = t + 1 > 1. So the inverse function f⁻¹(t) is:If t ≤ 0, then f⁻¹(t) = t.If t > 1, then f⁻¹(t) = t - 1.But what about t between 0 and 1? There's no t such that f(t) = t between 0 and 1, because for t ≤ 0, f(t) = t ≤ 0, and for t > 0, f(t) = t + 1 > 1. Therefore, f is not surjective onto ℝ, since the interval (0,1] is not covered. Therefore, this function is not bijective. So that attempt fails.Alternatively, maybe a more complex piecewise function. But this seems complicated. Alternatively, maybe the only solutions are linear functions f(t) = t + b. But let's try to see if that's the case.Suppose f is a monotonic bijection satisfying f(t) + f⁻¹(t) = 2t for all t. Let's consider that equation. For any t, f(t) + f⁻¹(t) = 2t. Let me denote y = f(t). Then, f⁻¹(y) = t. So substituting into the original equation, we have y + f⁻¹(y) = 2t. But t = f⁻¹(y), so substituting that in, we get y + f⁻¹(y) = 2f⁻¹(y). Therefore, y = f⁻¹(y). Wait, that would imply f⁻¹(y) = y for all y, which would mean f(y) = y. But this contradicts the earlier result that f(t) = t + b is a solution.Wait, so where is the mistake here?Let me go through this again. Start with f(t) + f⁻¹(t) = 2t. Let y = f(t), so t = f⁻¹(y). Substitute into the equation:y + f⁻¹(y) = 2f⁻¹(y) => y = f⁻¹(y).Therefore, f⁻¹(y) = y => f(y) = y. But this suggests that the only solution is f(t) = t. But earlier, we saw that f(t) = t + b works. So which one is correct?Wait, if f(t) = t + b, then f⁻¹(t) = t - b. Then, according to the equation f(t) + f⁻¹(t) = (t + b) + (t - b) = 2t. That works, but according to the substitution above, y + f⁻¹(y) = 2f⁻¹(y) => y = f⁻¹(y). But in this case, y = t + b, and f⁻¹(y) = y - b. So y - b = t. But t = f⁻¹(y). So substituting back, y - b = f⁻¹(y). Therefore, y = f⁻¹(y) + b. But this contradicts y = f⁻¹(y). Wait, unless b = 0. So here's the problem. If we do the substitution:Starting with f(t) + f⁻¹(t) = 2t.Let y = f(t), so t = f⁻¹(y). Substitute into the equation:y + f⁻¹(f(t)) = 2t. But f⁻¹(f(t)) is t, so y + t = 2t => y = t. Therefore, y = t, which would mean f(t) = t. But this contradicts the example where f(t) = t + b.Wait, this suggests that f(t) = t is the only solution. But that contradicts the earlier calculation where f(t) = t + b satisfies the equation. So something is wrong here. Let me check with f(t) = t + b.Take f(t) = t + 1. Then f⁻¹(t) = t - 1. Let's pick a specific t, say t = 0. Then f(0) = 1, and f⁻¹(0) = -1. So f(0) + f⁻¹(0) = 1 + (-1) = 0, which is equal to 2*0 = 0. That works. For t = 1, f(1) = 2, f⁻¹(1) = 0. So 2 + 0 = 2*1 = 2. That works. Wait, so why does the substitution suggest that f(t) must be t?Ah! Wait, perhaps the substitution is not considering that when you set y = f(t), then the equation becomes y + f⁻¹(y) = 2t, but t is f⁻¹(y). Therefore, substituting t = f⁻¹(y) gives y + f⁻¹(y) = 2f⁻¹(y) => y = f⁻¹(y). Which would require f⁻¹(y) = y, hence f(y) = y. But in our example, f⁻¹(y) = y - b, so substituting into y = f⁻¹(y) gives y = y - b => b = 0. So this suggests that only f(t) = t satisfies the equation. But we have an example where f(t) = t + b does satisfy the equation. So where is the mistake here?Wait, maybe the substitution is correct, but my example is wrong? Wait, let's check again. If f(t) = t + b, then f⁻¹(t) = t - b. Then for any t, f(t) + f⁻¹(t) = (t + b) + (t - b) = 2t. That's correct. But according to the substitution:Start with y = f(t), so t = f⁻¹(y). Then, original equation f(t) + f⁻¹(t) = 2t becomes y + f⁻¹(t) = 2t. But here, t is still the original t. Wait, maybe I confused variables here. Let me clarify.Original equation: For all t in ℝ, f(t) + f⁻¹(t) = 2t.Let y = f(t). Then f⁻¹(y) = t. So the equation can be rewritten for y: f(f⁻¹(y)) + f⁻¹(f⁻¹(y)) = 2f⁻¹(y). But f(f⁻¹(y)) = y. So this gives y + f⁻¹(f⁻¹(y)) = 2f⁻¹(y). Let me denote z = f⁻¹(y). Then, this equation becomes y + f⁻¹(z) = 2z. But since z = f⁻¹(y), and y = f(z), so substituting, we get f(z) + f⁻¹(z) = 2z. Wait, but this is just the original equation applied to z. So this substitution leads us back to the original equation, which doesn't give new information.Alternatively, maybe I need to iterate the function. Let me suppose that f is a linear function f(t) = t + b. Then, as we saw, it satisfies the equation. But according to the substitution approach, if we set y = f(t), then we get y + f⁻¹(y) = 2t. But t = f⁻¹(y) = y - b. So substituting into the equation: y + (y - b) = 2(y - b). Which simplifies to 2y - b = 2y - 2b => -b = -2b => b = 0. Therefore, only when b = 0 does this hold. But wait, this contradicts our earlier example where b ≠ 0 worked. Wait, but in reality, when f(t) = t + b, f⁻¹(t) = t - b, so for the substitution:Original equation: f(t) + f⁻¹(t) = 2t.Set y = f(t) => t = f⁻¹(y). Then, equation becomes y + f⁻¹(t) = 2t. But t = f⁻¹(y), so f⁻¹(t) = f⁻¹(f⁻¹(y)) = f⁻²(y). Therefore, the equation becomes y + f⁻²(y) = 2f⁻¹(y).But for f(t) = t + b, f⁻¹(y) = y - b, so f⁻²(y) = f⁻¹(y - b) = (y - b) - b = y - 2b.Therefore, substituting into the equation: y + (y - 2b) = 2(y - b) => 2y - 2b = 2y - 2b. Which holds for all y. So actually, the substitution does not require b = 0. Therefore, my previous mistake was in miscalculating the substitution. Therefore, the substitution does not lead to a contradiction, and f(t) = t + b is indeed a valid solution.Therefore, the problem arises when we incorrectly applied substitution. So, in fact, functions of the form f(t) = t + b do satisfy the equation, and the substitution approach when done correctly does not lead to a contradiction. Therefore, the previous conclusion that only f(t) = t works was wrong because of incorrect substitution steps.Therefore, the set E includes all functions f such that f(t) + f⁻¹(t) = 2t, and we need to find all monotonic ones. So far, we have linear functions f(t) = t + b. Are there any others?Suppose f is non-linear, but monotonic. Let's assume f is differentiable for simplicity. Let's differentiate both sides of the equation f(t) + f⁻¹(t) = 2t with respect to t.The derivative of f(t) is f’(t). The derivative of f⁻¹(t) is 1/f’(f⁻¹(t)) by the inverse function theorem. Therefore, differentiating both sides:f’(t) + [1/f’(f⁻¹(t))] = 2.But since f⁻¹(t) = s where f(s) = t, so s = f⁻¹(t). Then, f’(s) = f’(f⁻¹(t)). Therefore, the derivative equation becomes:f’(t) + [1/f’(s)] = 2, where s = f⁻¹(t).But s = f⁻¹(t) => t = f(s). So we can write this as:f’(t) + [1/f’(s)] = 2, with t = f(s).Let me denote t = f(s), so s = f⁻¹(t). Then, the equation becomes:f’(f(s)) + [1/f’(s)] = 2.This is a differential equation in terms of s. Let me write it as:f’(f(s)) + 1/f’(s) = 2.This seems a bit complicated, but perhaps we can assume f is linear. If f is linear, f(s) = s + b, then f’(s) = 1, so f’(f(s)) = 1. Then, 1 + 1/1 = 2, which satisfies the equation. So linear functions satisfy the differential equation.But maybe there are non-linear solutions. Let's suppose f is non-linear but satisfies the differential equation. Let me set g(s) = f(s) - s. Then f(s) = s + g(s), and we want g(s) to be such that f is bijective and monotonic.But this might not lead anywhere. Alternatively, let's consider the differential equation:f’(f(s)) + 1/f’(s) = 2.Let’s denote u(s) = f’(s). Then, f’(f(s)) = u(f(s)). So the equation becomes:u(f(s)) + 1/u(s) = 2.This is a functional-differential equation. It relates u(s) and u(f(s)). Let’s assume that u is a constant function. If u(s) = c for all s, then u(f(s)) = c. Then, the equation becomes c + 1/c = 2. Solving this, c^2 - 2c + 1 = 0 => (c - 1)^2 = 0 => c = 1. Therefore, constant derivative 1 implies f(s) = s + b, which is consistent with our previous result.What if u is not constant? Suppose u(s) = 1 for some s and different elsewhere, but this would contradict the equation unless u(s) is always 1. Alternatively, perhaps there's a non-constant solution.Suppose we set u(f(s)) = 2 - 1/u(s). Let me think recursively. If I know u(s), then u(f(s)) is determined. But since f(s) = integral of u(s) ds + constant, it's a bit intertwined.Alternatively, let me suppose that u(s) satisfies u(f(s)) = 2 - 1/u(s). If I iterate this, then u(f(f(s))) = 2 - 1/u(f(s)) = 2 - 1/(2 - 1/u(s)).But this seems complicated. Let me check if a linear function is the only solution. Suppose f is non-linear but satisfies the equation. For example, suppose f(s) = ks + b. Wait, we already saw that k must be 1. So even affine functions must have slope 1. So maybe only translations are allowed. Therefore, perhaps the only solutions are f(t) = t + b.Alternatively, consider another example. Suppose f(t) = t for t ≤ 0 and f(t) = t + 1 for t > 0. As before, this is piecewise linear with slope 1 everywhere except at t = 0, where there's a jump. However, as discussed earlier, this function is not bijective over ℝ because it's not surjective (misses the interval (0,1]). Therefore, such a function is invalid.Alternatively, maybe a function with varying slopes but still maintaining the condition f(t) + f⁻¹(t) = 2t. Suppose f(t) = t + b(t), where b(t) is some function. Then, f⁻¹(t) = t - b(f⁻¹(t)). Substituting into the original equation:f(t) + f⁻¹(t) = t + b(t) + t - b(f⁻¹(t)) = 2t + b(t) - b(f⁻¹(t)) = 2t.Therefore, we must have b(t) = b(f⁻¹(t)) for all t. Let’s denote s = f⁻¹(t). Then, t = f(s) = s + b(s). So the equation becomes b(t) = b(s). But s = f⁻¹(t) = t - b(t). Therefore, b(t) = b(t - b(t)).This is a functional equation for b. Let’s suppose b is a constant function, then b(t) = b(t - b(t)) implies b = b, which is always true. Hence, constant functions b(t) = b are solutions, leading to f(t) = t + b.If b is not constant, then we need b(t) = b(t - b(t)) for all t. Let me check if there are non-constant solutions. Suppose b(t) = c(t), then c(t) = c(t - c(t)). Let's try simple functions. Let’s try c(t) = kt, linear function. Then, c(t) = kt, so c(t - c(t)) = k(t - kt) = kt - k^2 t = kt(1 - k). Setting this equal to c(t) = kt:kt(1 - k) = kt => (1 - k) = 1 => k = 0. So only solution is k = 0, which gives b(t) = 0, the constant function. So no non-trivial linear solutions.What about quadratic? Let’s suppose c(t) = at^2 + bt + c. Then c(t - c(t)) = a(t - c(t))^2 + b(t - c(t)) + c. This seems too complicated, and setting equal to c(t) would require solving a functional equation which likely only has constant solutions.Alternatively, suppose c(t) is periodic. For example, c(t + T) = c(t). Then, the equation c(t) = c(t - c(t)) would require that t - c(t) is also a period shift. But this seems difficult to satisfy unless c(t) is constant.Alternatively, perhaps only constant functions satisfy c(t) = c(t - c(t)). Therefore, the only solutions are f(t) = t + b. Therefore, combining all this, it seems that the only monotonic bijections satisfying the given condition are the linear functions f(t) = t + b for some constant b.But let's check if there could be any non-linear solutions. Suppose f is a non-linear monotonic bijection. For example, take f(t) = t + e^t. This function is strictly increasing since its derivative is 1 + e^t > 0 for all t. It's bijective because as t approaches -∞, f(t) approaches -∞, and as t approaches +∞, f(t) approaches +∞. So it's invertible. Let’s compute f⁻¹(t). Let’s set y = t + e^t. To find f⁻¹(y) = t, we need to solve y = t + e^t for t. This equation is transcendental and cannot be solved analytically, but let's check if f(t) + f⁻¹(t) = 2t.Suppose for a specific t, we have y = t + e^t. Then f⁻¹(y) = t. Then, f(t) + f⁻¹(t) = y + f⁻¹(t). But f⁻¹(t) is the value s such that s + e^s = t. So unless s + e^s = t and y = t + e^t, this would need to satisfy y + s = 2t. Substituting y = t + e^t and s = f⁻�¹(t), which is s such that s + e^s = t. So the equation becomes (t + e^t) + s = 2t. Therefore, e^t + s = t. But s is defined by s + e^s = t. Therefore, substituting s = t - e^t into s + e^s = t gives (t - e^t) + e^{t - e^t} = t. Therefore, e^{t - e^t} = e^t. Taking natural logarithm: t - e^t = t => -e^t = 0, which is impossible. Therefore, f(t) = t + e^t does not satisfy the equation. Therefore, such a function is not a solution.Therefore, it seems that non-linear functions do not satisfy the equation. Hence, the only solutions are linear functions of the form f(t) = t + b.But let me check another example. Suppose f(t) = t for t < 0 and f(t) = t + 1 for t ≥ 0. This function is strictly increasing, bijective (since every real number is covered: for t < 0, f(t) = t; for t ≥ 0, f(t) = t + 1, so the inverse is f⁻¹(t) = t for t < 0 and f⁻¹(t) = t - 1 for t ≥ 1. The interval [0,1) is not covered in the range, hence f is not surjective. Therefore, this function is not bijective. Therefore, invalid.Alternatively, make a piecewise function that is surjective. For example, f(t) = t for t ≤ 0 and f(t) = t + 1 for t > 0. Then the inverse would be f⁻¹(t) = t for t ≤ 0 and f⁻¹(t) = t - 1 for t > 0. Wait, but then for t > 0, f(t) = t + 1, so the inverse function for y > 1 is f⁻¹(y) = y - 1. For y ≤ 0, f⁻¹(y) = y. But for 0 < y ≤ 1, there is no t such that f(t) = y because for t ≤ 0, f(t) = t ≤ 0, and for t > 0, f(t) = t + 1 > 1. Therefore, this function is not surjective on ℝ, hence not bijective. Therefore, such piecewise functions won't work unless they cover all real numbers.Therefore, constructing a non-linear bijective function that satisfies f(t) + f⁻¹(t) = 2t is difficult. All the examples I can think of either fail to be bijective or don't satisfy the equation.Therefore, based on the differential equation approach, substitution tests, and trying examples, it seems the only solutions are the linear functions f(t) = t + b where b is a real constant. These functions are bijective, monotonic (since they're strictly increasing with derivative 1), and satisfy the given functional equation.Therefore, the elements of E that are monotonic are precisely all the linear functions of the form f(t) = t + b for some constant b ∈ ℝ.Final AnswerAll such functions are linear; the solutions are precisely the functions ( f(t) = t + c ) for some constant ( c ). Thus, the answer is (boxed{f(t) = t + c}) where ( c ) is a real constant.</think>Given the problem of finding all bijective and monotonic mappings ( f ) from (mathbb{R}) to (mathbb{R}) that satisfy the equation ( f(t) + f^{-1}(t) = 2t ) for all ( t in mathbb{R} ), we start by analyzing the functional equation and its implications.1. Functional Equation Analysis: - For any ( t in mathbb{R} ), the equation ( f(t) + f^{-1}(t) = 2t ) must hold. - By substituting ( y = f(t) ), we get ( t = f^{-1}(y) ), leading to the equation ( y + f^{-1}(y) = 2f^{-1}(y) ). This simplifies to ( y = f^{-1}(y) ), suggesting ( f(y) = y ). However, further analysis shows this is not the only solution.2. Linear Function Check: - Assume ( f(t) = at + b ). The inverse function is ( f^{-1(t)} = frac{t - b}{a} ). - Substituting into the equation, we get ( at + b + frac{t - b}{a} = 2t ). Solving for ( a ) and ( b ), we find ( a = 1 ) and ( b ) can be any real number. Thus, ( f(t) = t + b ) satisfies the equation.3. Verification of Linear Solutions: - For ( f(t) = t + b ), the inverse is ( f^{-1}(t) = t - b ). Substituting back, we get ( (t + b) + (t - b) = 2t ), which holds true. - Differentiating the equation confirms that the derivative ( f'(t) = 1 ), ensuring monotonicity.4. Non-linear Function Consideration: - Attempts to construct non-linear solutions (e.g., exponential, piecewise) either fail to be bijective or do not satisfy the functional equation. - Differential equation analysis shows that non-linear solutions would require complex conditions not satisfied by simple non-linear functions.5. Conclusion: - The only solutions that are bijective, monotonic, and satisfy the functional equation are linear functions of the form ( f(t) = t + b ).Thus, the elements of ( E ) that are monotonic mappings are precisely the functions ( f(t) = t + c ) for some constant ( c in mathbb{R} ).[boxed{f(t) = t + c}] where ( c ) is a real constant.]

🔑:1. Given Condition and Initial Analysis: We are given that ( f ) is a bijective mapping from (mathbb{R}) to (mathbb{R}) satisfying: [ f(t) + f^{-1}(t) = 2t, quad forall t in mathbb{R}, ] where ( f^{-1} ) is the inverse of ( f ). We need to find all elements of ( E ) that are monotonic mappings.2. Monotonicity and Inverse Function: Since ( f ) is bijective and monotonic, ( f ) must be either strictly increasing or strictly decreasing. Similarly, ( f^{-1} ) must also be strictly monotonic in the same direction as ( f ) because the inverse of a strictly increasing function is strictly increasing, and the inverse of a strictly decreasing function is strictly decreasing.3. Sum of Functions: Given the condition ( f(t) + f^{-1}(t) = 2t ), we can analyze the behavior of ( f ) and ( f^{-1} ). If ( f ) is strictly increasing, then ( f^{-1} ) is also strictly increasing, and their sum ( f(t) + f^{-1}(t) ) must be strictly increasing.4. Assumption and Contradiction: Suppose ( f(x) - x ) is not constant. Let ( u neq v ) such that ( f(u) - u = a ) and ( f(v) - v = b ) with ( a > b ). Using the given condition, we can derive: [ f(x) + f^{-1}(x) = 2x. ] It follows that: [ f(u + na) = u + (n+1)a quad text{and} quad f(v + nb) = v + (n+1)b quad forall n in mathbb{Z}. ] Let ( n = leftlfloor frac{v - u}{a - b} rightrfloor ). Then: [ n + 1 > frac{v - u}{a - b} geq n. ] This implies: [ frac{v - u}{a - b} geq n implies v - u geq n(a - b) implies v + nb geq u + na implies f(v + nb) geq f(u + na). ] Since ( f ) is increasing: [ v + (n+1)b geq u + (n+1)a implies frac{v - u}{a - b} geq n + 1. ] This leads to a contradiction.5. Conclusion: Therefore, ( f(x) - x ) must be constant. Let ( f(x) = x + c ) for some constant ( c ). Substituting ( f(x) = x + c ) into the given condition: [ f(t) + f^{-1}(t) = 2t implies (t + c) + (t - c) = 2t, ] which holds true for any real ( c ).The final answer is (boxed{f(x) = x + c}) for any real constant ( c ).

🧠:[<think>Okay, let's see. The problem is about a rectangle where two opposite sides are increased by 50%, and the other two sides are decreased by 20%. Then we need to find the percentage increase in the area, which is given as 'a%'. Hmm, so I need to figure out how changing the length and width by those percentages affects the area. Let me break this down step by step.First, let's recall that the area of a rectangle is calculated by multiplying its length and width. So if the original length is, say, 'l' and the original width is 'w', the original area would be A = l * w. Now, the problem says two opposite sides are increased by 50%, and the other two are decreased by 20%. I need to make sure I interpret this correctly. Since a rectangle has two pairs of equal sides, increasing two opposite sides (which are equal) by 50% and decreasing the other two opposite sides (which are the other pair) by 20%. So essentially, one pair of sides is increased by 50%, and the other pair is decreased by 20%. So this would mean that if the original length is 'l' and the original width is 'w', then after the change, the new length would be... hmm, actually, wait. It's not specified whether the length or the width is being changed. Maybe I should assign variables to the original sides and then apply the percentage changes accordingly.Let me define the original sides as length 'l' and width 'w'. The problem doesn't specify which sides are being increased or decreased, just that two opposite sides are increased by 50% and the other two are decreased by 20%. So perhaps it doesn't matter which ones we choose as length or width, since multiplication is commutative. So let's say we increase the length by 50% and decrease the width by 20%. Wait, but actually, if two opposite sides are increased, that would mean both the length and the width? Wait, no. A rectangle has two sides of each. So if two opposite sides (which are lengths) are increased by 50%, and the other two opposite sides (which are widths) are decreased by 20%. So, actually, the length becomes longer by 50%, and the width becomes shorter by 20%. Is that the correct interpretation? Because in a rectangle, opposite sides are equal. So if you have two sides (lengths) increased by 50%, then both of those sides are increased, and the other two sides (widths) are decreased by 20%. So the new length would be original length plus 50% of it, and the new width would be original width minus 20% of it. Then the new area would be new length times new width. Then the percentage increase in area would be (new area - original area)/original area * 100%. Then that's 'a%'. So the answer is a = [(new area/original area - 1) * 100]. Let's compute that.Let me write this out step by step.Original length = lOriginal width = wOriginal area = l * wAfter the changes:New length = l + 50% of l = l * (1 + 0.5) = 1.5lNew width = w - 20% of w = w * (1 - 0.2) = 0.8wTherefore, new area = new length * new width = 1.5l * 0.8w = (1.5 * 0.8) * l * w = 1.2 * l * wSo the new area is 1.2 times the original area.Therefore, the increase in area is 1.2 - 1 = 0.2, which is 20%.Therefore, a = 20.Wait, that seems straightforward. But let me verify if I interpreted the problem correctly. The problem says two opposite sides are increased by 50% while the other two are decreased by 20%. So depending on which sides you choose to increase or decrease, does it make a difference? Let's see. Suppose instead of changing the length, you changed the width. But in a rectangle, length and width are just labels; switching them doesn't affect the area. So increasing one pair by 50% and decreasing the other pair by 20% would result in the same area regardless of which pair you choose. So the calculation remains the same.Alternatively, maybe I can test with numbers. Let me take an example. Suppose the original rectangle is 10 units by 10 units, so it's a square. Original area is 100.If I increase two opposite sides (lengths) by 50%, so new length is 10 + 5 = 15. Decrease the other two sides (widths) by 20%, so new width is 10 - 2 = 8. Then new area is 15 * 8 = 120. The original area was 100, so the increase is 20. Therefore, 20% increase. So a = 20. That's the same result as before. So that seems to check out.Alternatively, if the original rectangle is 20 by 5. Original area is 100. Increase length by 50%: 20 * 1.5 = 30. Decrease width by 20%: 5 * 0.8 = 4. New area is 30 * 4 = 120. Again, 20% increase. So the percentage increase is consistent regardless of original dimensions. So seems like a is 20.Therefore, the answer is 20. Therefore, a = 20. So the area is increased by 20%.But wait, let me check another example to be thorough. Let's say original rectangle is 2 by 3. Area is 6. Increase the length by 50%: 2 * 1.5 = 3. Decrease the width by 20%: 3 * 0.8 = 2.4. New area is 3 * 2.4 = 7.2. The increase is 7.2 - 6 = 1.2. 1.2 / 6 = 0.2, so 20% increase. Same result. So regardless of the original dimensions, the percentage increase in area is 20%. Therefore, a = 20. So the answer is 20.But let me think again. Is there another way someone might misinterpret this problem? For example, if someone thought that all four sides are adjusted, but no, the problem says two opposite sides are increased by 50%, while the other two are decreased by 20%. In a rectangle, two opposite sides are length and the other two are width. So you can't have all four sides adjusted unless it's a square. So in general, for a rectangle, you have two pairs of equal sides. So changing two opposite sides (a pair) by 50% and the other pair by 20%. So the calculation is straightforward as we did before. So I think the answer is correct.Another way: Suppose the original sides are x and y. Then the new sides are 1.5x and 0.8y. So area becomes 1.5x * 0.8y = 1.2xy. So 1.2 times original area, which is 20% increase. So same result. Yeah, this seems solid.Therefore, the answer is 20. So a = 20.Final AnswerThe value of ( a ) is boxed{20}.</think>Let the original length and width of the rectangle be ( l ) and ( w ) respectively. The original area of the rectangle is ( A = l times w ).When two opposite sides (length) are increased by 50%, the new length becomes:[1.5l]When the other two opposite sides (width) are decreased by 20%, the new width becomes:[0.8w]The new area of the rectangle is:[text{New Area} = 1.5l times 0.8w = 1.2lw]The increase in area is:[1.2lw - lw = 0.2lw]The percentage increase in area is:[left( frac{0.2lw}{lw} right) times 100% = 20%]Thus, the value of ( a ) is (boxed{20}).,<think>Okay, so here's this problem: we have a rectangle, and two opposite sides are increased by 50%, while the other two are decreased by 20%. Then, we need to find the percentage increase in the area, which is denoted by 'a'. Hmm, let me try to break this down step by step.First off, let's recall that the area of a rectangle is calculated by multiplying its length and width. So, if we change the length and the width, the area will change accordingly. The problem says two opposite sides are increased by 50%, and the other two are decreased by 20%. Since opposite sides of a rectangle are equal, this means that one pair of sides (let's say the length) is being increased by 50%, and the other pair (the width) is being decreased by 20%. Or it could be the other way around, but since multiplication is commutative, it doesn't really matter which one is which. The result should be the same.Let me assign some variables to make this clearer. Let's denote the original length as L and the original width as W. Therefore, the original area is A = L * W.Now, according to the problem, two opposite sides are increased by 50%. So, if we take the length as being increased by 50%, the new length becomes L + 50% of L, which is L * (1 + 0.50) = 1.5L. Similarly, the other two sides (the width) are decreased by 20%, so the new width is W - 20% of W = W * (1 - 0.20) = 0.8W.Therefore, the new area after the changes will be A' = new length * new width = 1.5L * 0.8W. Let me compute that: 1.5 * 0.8 = 1.2, so A' = 1.2 * L * W = 1.2A. Wait, so the new area is 1.2 times the original area. That means the area has increased by 0.2 times the original area, right? So, 0.2 is equivalent to 20%, which would make 'a' equal to 20. But hold on, let me make sure I did that correctly.Wait, 1.5 multiplied by 0.8... Let me check that arithmetic again. 1.5 times 0.8. Hmm, 1 times 0.8 is 0.8, and 0.5 times 0.8 is 0.4, so adding them together gives 0.8 + 0.4 = 1.2. Yep, that's correct. So the new area is indeed 1.2 times the original area, which is a 20% increase. So, a = 20. Therefore, the answer should be 20. But let me think again to ensure there isn't a trick here.Wait, the problem says "two opposite sides are increased by 50% while the other two are decreased by 20%". So, if the rectangle has length L and width W, increasing two opposite sides (say, the lengths) by 50% gives 1.5L, and decreasing the other two (widths) by 20% gives 0.8W. Then, the area becomes 1.5L * 0.8W = 1.2LW, which is indeed a 20% increase. So, the percentage increase 'a' is 20. That seems straightforward.But maybe I should verify this with actual numbers to be sure. Let's pick specific values for L and W. Suppose the original length is 10 units and the original width is 5 units. Then, the original area is 10 * 5 = 50 square units.After increasing the length by 50%, the new length is 10 + 5 = 15 units. After decreasing the width by 20%, the new width is 5 - 1 = 4 units. The new area is 15 * 4 = 60 square units. Comparing the new area to the original area: 60 - 50 = 10, which is an increase of 10 units. The percentage increase is (10/50)*100 = 20%. So, that's a 20% increase. Hmm, so the answer is indeed 20.Wait, but maybe I need to present this in terms of variables, not specific numbers. Let me do that again. Let the original length be L and original width be W. Then, original area is LW. The new dimensions are 1.5L and 0.8W. The new area is 1.5L * 0.8W = (1.5 * 0.8)LW = 1.2LW. So, the new area is 1.2 times the original area, which is a 20% increase. So, the value of 'a' is 20.But wait, the problem says "the area of the rectangle is increased by a%", so they want to know by what percentage the area has increased. So, since 1.2 times the original area is a 20% increase, then a = 20. Therefore, the answer should be 20.Is there any chance that the problem is referring to something else? For example, maybe the sides are increased and decreased in a different way? Wait, the problem says "two opposite sides are increased by 50% while the other two are decreased by 20%". In a rectangle, opposite sides are equal, so increasing two opposite sides by 50% would mean both of those sides become 1.5 times their original length, and the other two sides (the other pair of opposite sides) are decreased by 20%, so they become 0.8 times their original length. Therefore, the area is 1.5 * 0.8 = 1.2 times the original area, so 20% increase. Yes, that's correct.Alternatively, if the sides were adjacent sides, but no, the problem says opposite sides. So, in a rectangle, each pair of opposite sides are of equal length, so increasing two opposite sides (length) and decreasing the other two opposite sides (width). Therefore, the calculations hold.Alternatively, if they had increased one length and one width, but no, the problem states "two opposite sides", so both of the same dimension. So, for example, both lengths are increased, and both widths are decreased. So, no ambiguity there.Hence, I think my answer is correct. The area increases by 20%, so a = 20.Final AnswerThe value of ( a ) is boxed{20}.</think>Let the original length of the rectangle be ( L ) and the original width be ( W ). The original area of the rectangle is ( A = L times W ).When two opposite sides (length) are increased by 50%, the new length becomes ( 1.5L ). When the other two opposite sides (width) are decreased by 20%, the new width becomes ( 0.8W ).The new area of the rectangle is calculated as:[text{New Area} = 1.5L times 0.8W = (1.5 times 0.8) times L times W = 1.2 times L times W = 1.2A]This shows that the new area is 1.2 times the original area, which is a 20% increase. Therefore, the percentage increase ( a ) is 20.[boxed{20}]]

🔑:Let the length and width be x and y respectively.begin{array}{l}1.5 x times 0.8 y=1.2 x y Rightarrow a=20end{array}