🧠:Okay, so I need to solve this problem where real numbers x, y, z satisfy two equations, and I have to find the value of this expression (y² + z² - x²)/(2yz). Let me start by writing down the given equations and see how I can manipulate them to find the required value.First equation is (y)/(x - y) = (x)/(y + z). Let's denote that as Equation (1). The second equation is z² = x(y + z) - y(x - y), which we'll call Equation (2). The expression we need to find is (y² + z² - x²)/(2yz). Hmm, that looks a bit like the formula for the cosine of an angle in a triangle using the Law of Cosines. Maybe that's a hint, but let's not jump to conclusions yet. Let's first work with the equations given.Starting with Equation (1): (y)/(x - y) = (x)/(y + z). Let me cross-multiply to eliminate the fractions. So, multiplying both sides by (x - y)(y + z), we get:y(y + z) = x(x - y)Expanding both sides:Left side: y² + yzRight side: x² - xySo, Equation (1) simplifies to:y² + yz = x² - xyLet me rearrange terms to bring everything to one side:y² + yz + xy - x² = 0Hmm, perhaps factor some terms. Let's see:Group terms: (y² - x²) + yz + xy = 0Note that y² - x² factors into (y - x)(y + x). So,(y - x)(y + x) + y(z + x) = 0Wait, maybe that's not the most helpful. Let me check my algebra again. The original equation after cross-multiplying was:y² + yz = x² - xySo, moving all terms to the left:y² + yz + xy - x² = 0Alternatively, we can write:y² + xy + yz - x² = 0Factor y from the first three terms? Let's see:y(y + x + z) - x² = 0Hmm, not sure if that helps. Maybe another approach. Let's keep this in mind and move to Equation (2):z² = x(y + z) - y(x - y)Let me expand the right-hand side:x(y + z) = xy + xzand -y(x - y) = -xy + y²So, combining these:xy + xz - xy + y² = xz + y²Therefore, Equation (2) simplifies to:z² = xz + y²So, z² - xz - y² = 0Hmm, so from Equation (2), we have z² = xz + y².That's a quadratic in z. Maybe we can solve for z in terms of x and y? Let's see:z² - xz - y² = 0Using quadratic formula, z = [x ± sqrt(x² + 4y²)] / 2But not sure if that's immediately helpful. Let's see if we can combine this with Equation (1). From Equation (1), we had:y² + yz + xy - x² = 0Wait, maybe express x from Equation (1) in terms of y and z, then substitute into Equation (2)?Starting with Equation (1):y/(x - y) = x/(y + z)Cross-multiplied gives:y(y + z) = x(x - y)Which is y² + yz = x² - xyLet me rearrange this equation:x² - xy - y² - yz = 0x² - xy = y² + yzAlternatively, x² = xy + y² + yzHmm. Let's see if we can express x in terms of y and z, maybe. Let's consider x as a variable and solve for x.From Equation (1):y² + yz = x² - xyBring all terms to one side:x² - xy - y² - yz = 0This is a quadratic in x: x² - xy - (y² + yz) = 0So, solving for x:x = [y ± sqrt(y² + 4(y² + yz))]/2Wait, discriminant D = y² + 4*(y² + yz) = y² + 4y² + 4yz = 5y² + 4yzSo, x = [y ± sqrt(5y² + 4yz)] / 2Hmm, this seems complicated. Maybe there's a better way. Let's check Equation (2):z² = xz + y²From here, z² - xz - y² = 0If we can express x from here, maybe. Let's solve for x:z² - y² = xzSo, x = (z² - y²)/zAssuming z ≠ 0. Wait, if z = 0, then from Equation (2): 0 = x*0 + y² => y² = 0 => y = 0. Then from Equation (1), 0/(x - 0) = x/(0 + 0), which would be 0 = undefined unless x is also 0. But if x, y, z are all zero, then the expression (y² + z² - x²)/(2yz) would be 0/0, which is undefined. So, maybe z cannot be zero. Therefore, we can assume z ≠ 0, so x = (z² - y²)/z = z - y²/zHmm, okay. So, from Equation (2), x is expressed in terms of y and z. Let's substitute this into Equation (1) and see if we can find a relation between y and z.So, substituting x = (z² - y²)/z into Equation (1):Original Equation (1): y/(x - y) = x/(y + z)Substitute x:Left side: y / [( (z² - y²)/z ) - y] = y / [ (z² - y² - yz)/z ] = y * [ z / (z² - y² - yz) ) ] = (yz)/(z² - y² - yz )Right side: x/(y + z) = ( (z² - y²)/z ) / (y + z) = (z² - y²)/( z(y + z) )So, setting left side equal to right side:(yz)/(z² - y² - yz) = (z² - y²)/( z(y + z) )Cross-multiplying:(yz) * z(y + z) = (z² - y² - yz)(z² - y²)Let's compute each side:Left side: yz * z(y + z) = y z² (y + z)Right side: (z² - y² - yz)(z² - y²) = Let me expand this.First, note that z² - y² - yz = (z² - y²) - yz. Let me denote A = z² - y², so expression is A - yz. Then, right side is (A - yz) * A = A² - yz A.So, expanding:(z² - y²)^2 - yz(z² - y²)Therefore, left side is y z² (y + z), right side is (z² - y²)^2 - y z (z² - y²)So, equation:y z² (y + z) = (z² - y²)^2 - y z (z² - y²)Let me factor (z² - y²) on the right side:(z² - y²)[(z² - y²) - y z]= (z² - y²)(z² - y² - y z)So, the equation becomes:y z² (y + z) = (z² - y²)(z² - y² - y z)Hmm, this seems complicated. Maybe expand both sides and see if terms cancel.First, left side:y z² (y + z) = y² z² + y z³Right side:(z² - y²)(z² - y² - y z)Let me expand this:First, let me write (z² - y² - y z) as (z² - y z - y²)Multiply by (z² - y²):Multiply term by term:z²*(z² - y z - y²) - y²*(z² - y z - y²)= z^4 - y z³ - y² z² - y² z² + y^3 z + y^4Combine like terms:z^4 - y z³ - y² z² - y² z² + y^3 z + y^4= z^4 - y z³ - 2 y² z² + y^3 z + y^4So, right side is z^4 - y z³ - 2 y² z² + y^3 z + y^4Left side is y² z² + y z³Setting equal:y² z² + y z³ = z^4 - y z³ - 2 y² z² + y^3 z + y^4Bring all terms to the left:y² z² + y z³ - z^4 + y z³ + 2 y² z² - y^3 z - y^4 = 0Combine like terms:(y² z² + 2 y² z²) + (y z³ + y z³) + (- z^4) + (- y^3 z) + (- y^4)= 3 y² z² + 2 y z³ - z^4 - y^3 z - y^4 = 0So, the equation simplifies to:- z^4 + 2 y z³ + 3 y² z² - y^3 z - y^4 = 0Let me write this as:- z^4 + 2 y z³ + 3 y² z² - y^3 z - y^4 = 0Multiply both sides by -1:z^4 - 2 y z³ - 3 y² z² + y^3 z + y^4 = 0Hmm, this is a quartic equation in z. Maybe factor it?Let me try to factor this polynomial. Let's consider it as a polynomial in z with coefficients in terms of y. Let's denote w = z/y, assuming y ≠ 0. If y = 0, then from Equation (2), z² = x* z + 0 => z² = x z. Then either z = 0, which would lead to x from Equation (1): 0/(x - 0) = x/0, which is undefined, so y can't be zero. Therefore, y ≠ 0, so we can set w = z/y.Then, z = w y. Substitute into the equation:(z)^4 - 2 y z³ - 3 y² z² + y^3 z + y^4 = 0Substituting z = w y:(w y)^4 - 2 y (w y)^3 - 3 y² (w y)^2 + y^3 (w y) + y^4 = 0Compute each term:= w^4 y^4 - 2 y * w³ y³ - 3 y² * w² y² + y^3 * w y + y^4= w^4 y^4 - 2 w³ y^4 - 3 w² y^4 + w y^4 + y^4Factor out y^4:y^4 (w^4 - 2 w³ - 3 w² + w + 1) = 0Since y ≠ 0, then y^4 ≠ 0, so the polynomial in w must be zero:w^4 - 2 w³ - 3 w² + w + 1 = 0Let me try to factor this quartic polynomial. Let's look for rational roots using Rational Root Theorem. Possible roots are ±1.Test w = 1:1 - 2 - 3 + 1 + 1 = (1 - 2) + (-3 + 1 + 1) = (-1) + (-1) = -2 ≠ 0Test w = -1:1 + 2 - 3 - 1 + 1 = (1 + 2) + (-3 -1 +1) = 3 + (-3) = 0Yes, w = -1 is a root. Therefore, (w + 1) is a factor.Let's perform polynomial division or use synthetic division.Divide w^4 - 2w³ - 3w² + w + 1 by (w + 1):Using synthetic division:- Coefficients: 1 (w^4), -2 (w³), -3 (w²), 1 (w), 1 (constant)Root at w = -1:Bring down 1.Multiply by -1: 1*(-1) = -1. Add to next coefficient: -2 + (-1) = -3.Multiply by -1: -3*(-1) = 3. Add to next coefficient: -3 + 3 = 0.Multiply by -1: 0*(-1) = 0. Add to next coefficient: 1 + 0 = 1.Multiply by -1: 1*(-1) = -1. Add to last coefficient: 1 + (-1) = 0.So, the quotient is w³ - 3w² + 0w + 1, i.e., w³ - 3w² + 1Therefore, the quartic factors as (w + 1)(w³ - 3w² + 1)Now, factor the cubic w³ - 3w² + 1. Let's check for rational roots again. Possible roots ±1.Test w=1: 1 - 3 + 1 = -1 ≠ 0Test w=-1: -1 - 3 + 1 = -3 ≠ 0No rational roots. Maybe factor it as a product of quadratics and linear terms, but since it's a cubic without rational roots, maybe use the rational root theorem or Cardano's method. Alternatively, see if it can be factored.Alternatively, perhaps use substitution. Let me set u = w - a to eliminate the quadratic term. But since the cubic is w³ - 3w² + 1, maybe let u = w - 1:u = w - 1 => w = u + 1Substitute into cubic:(u + 1)^3 - 3(u + 1)^2 + 1Expand:= u³ + 3u² + 3u + 1 - 3(u² + 2u + 1) + 1= u³ + 3u² + 3u + 1 - 3u² - 6u - 3 + 1= u³ - 3u -1So, the cubic becomes u³ - 3u - 1 = 0. Not sure if this helps. Maybe it's easier to consider that this cubic may have real roots. Let me check discriminant. For cubic equation u³ + pu + q = 0, discriminant is -4p³ -27q².Here, equation is u³ -3u -1 = 0, so p = -3, q = -1Discriminant: -4*(-3)^3 -27*(-1)^2 = -4*(-27) -27*1 = 108 - 27 = 81 > 0Since discriminant is positive, there are three real roots. But since exact roots are messy, maybe we can just note that the quartic equation in w has one rational root w = -1 and three irrational roots. Therefore, possible solutions for w are w = -1 and roots of the cubic equation.But since we need real solutions, and given that z and y are real numbers, this could be possible. But maybe we can use the original equations to find relations between variables without getting into the quartic.Wait, perhaps there is another approach. Let's recall that the expression we need to compute is (y² + z² - x²)/(2yz). Let's denote this expression as C. So,C = (y² + z² - x²)/(2yz)If we can express x in terms of y and z, then substitute into C and simplify. From Equation (2), we had z² = x z + y² => x = (z² - y²)/z, as before.So, substitute x = (z² - y²)/z into C:C = [ y² + z² - ( (z² - y²)/z )² ] / (2 y z )Let me compute numerator:Numerator = y² + z² - [ (z² - y²)^2 / z² ]So,= [ y² z² + z^4 - (z² - y²)^2 ] / z²Expand (z² - y²)^2:= z^4 - 2 y² z² + y^4So, numerator becomes:[ y² z² + z^4 - z^4 + 2 y² z² - y^4 ] / z²Simplify numerator:= ( y² z² + z^4 - z^4 + 2 y² z² - y^4 ) / z²= (3 y² z² - y^4 ) / z²Factor y²:= y² (3 z² - y² ) / z²Therefore, numerator is y² (3 z² - y² ) / z²Denominator of C is 2 y zTherefore, C = [ y² (3 z² - y² ) / z² ] / (2 y z )Simplify:= [ y² / z² * (3 z² - y²) ] / (2 y z )= [ y² (3 z² - y²) ] / (2 y z * z² )= [ y (3 z² - y²) ] / (2 z³ )Hmm, that seems complicated. Let me check my steps again.Wait, starting again:C = (y² + z² - x²)/(2 y z )From Equation (2): z² = x z + y² => x = (z² - y²)/zSo, x² = (z² - y²)^2 / z²Thus, y² + z² - x² = y² + z² - (z^4 - 2 y² z² + y^4)/z²= (y² z² + z^4 - z^4 + 2 y² z² - y^4)/z²= (3 y² z² - y^4)/z²= 3 y² - y^4/z²Therefore, C = (3 y² - y^4/z²)/(2 y z ) = [ y² (3 - y²/z²) ] / (2 y z ) = [ y (3 - y²/z²) ] / (2 z )= (3 y - y³/z² ) / (2 z )Hmm, not sure if this is helpful yet. Maybe we can relate y and z through Equation (1). Recall that from Equation (1), we had x expressed in terms of y and z, so perhaps we can substitute x into Equation (1) and find a relation between y and z, then use that here.Wait, earlier, we tried substituting x into Equation (1) and ended up with a quartic equation. That might not be the best path. Let's think differently.Let me recall that in Equation (1):From y/(x - y) = x/(y + z), cross-multiplying gives y(y + z) = x(x - y)But from Equation (2), z² = x z + y² => x z = z² - y² => x = (z² - y²)/zTherefore, substituting x into the first equation:y(y + z) = [(z² - y²)/z ] * ( (z² - y²)/z - y )Simplify the right-hand side:First, compute x - y:[(z² - y²)/z - y] = (z² - y² - y z)/zTherefore, RHS:[ (z² - y²)/z ] * [ (z² - y² - y z)/z ] = (z² - y²)(z² - y² - y z)/z²So, equation becomes:y(y + z) = (z² - y²)(z² - y² - y z)/z²But this is exactly the same equation we derived before, leading to the quartic. So perhaps there's another way to relate variables.Wait, maybe use the expression for C we had earlier, which is:C = (y² + z² - x²)/(2 y z )If we can express x in terms of y and z, then substitute into C. We have x = (z² - y²)/z from Equation (2). Let's compute x²:x² = (z² - y²)^2 / z²Therefore, numerator of C:y² + z² - x² = y² + z² - (z^4 - 2 y² z² + y^4)/z²= y² + z² - (z^4/z² - 2 y² z²/z² + y^4/z²)= y² + z² - (z² - 2 y² + y^4/z² )= y² + z² - z² + 2 y² - y^4/z²= 3 y² - y^4/z²Thus, numerator is 3 y² - y^4/z², and denominator is 2 y z.So, C = (3 y² - y^4/z²)/(2 y z ) = [ y² (3 - y²/z²) ]/(2 y z ) = [ y (3 - y²/z²) ]/(2 z )Let me denote t = y/z. Then, y = t z. Substitute into C:C = [ t z (3 - (t z )² / z² ) ] / (2 z ) = [ t z (3 - t² ) ] / (2 z ) = [ t (3 - t² ) ] / 2Therefore, C = (3 t - t³)/2, where t = y/z.So, if we can find the value of t, then we can compute C. So, the problem reduces to finding t = y/z.Let's try to find t. From Equation (1):From Equation (1): y/(x - y) = x/(y + z)Express x in terms of y and z: x = (z² - y²)/zSo, x - y = (z² - y²)/z - y = (z² - y² - y z)/zTherefore, Equation (1) becomes:y / [ (z² - y² - y z)/z ] = [ (z² - y²)/z ] / (y + z )Simplify left side: y * [ z / (z² - y² - y z ) ] = y z / (z² - y² - y z )Right side: (z² - y²)/z / (y + z ) = (z² - y²)/( z(y + z ) )Set left side equal to right side:y z / (z² - y² - y z ) = (z² - y²)/( z(y + z ) )Cross-multiplying:y z * z(y + z ) = (z² - y²)(z² - y² - y z )Which is the same as before, leading to the quartic. So, perhaps express this equation in terms of t = y/z.Let me let t = y/z, so y = t z. Substitute into the equation.First, z ≠ 0 (as established before). So, substituting y = t z:Left side: y z / (z² - y² - y z ) = (t z ) z / ( z² - (t z )² - (t z ) z )= t z² / ( z² - t² z² - t z² )Factor z² in denominator:= t z² / [ z² (1 - t² - t ) ]Cancel z²:= t / (1 - t² - t )Right side: (z² - y²)/( z(y + z ) ) = ( z² - t² z² ) / ( z ( t z + z ) )= z²(1 - t² ) / ( z * z ( t + 1 ) )= z²(1 - t² ) / ( z² ( t + 1 ) )Cancel z²:= (1 - t² ) / ( t + 1 )Simplify numerator:(1 - t)(1 + t ) / ( t + 1 ) ) = 1 - t, provided that t + 1 ≠ 0. Since t + 1 = 0 would mean y/z = -1, so y = -z. Let's check if that's possible.If t = -1, then y = -z. Then from Equation (2): z² = x z + y² = x z + z². So, 0 = x z. If z ≠ 0, then x = 0. Then from Equation (1): y/(x - y ) = (-z)/(0 - (-z)) = (-z)/z = -1. On the right side: x/(y + z ) = 0/( -z + z ) = 0/0, which is undefined. Therefore, t = -1 is invalid. So, t + 1 ≠ 0, so we can safely simplify (1 - t²)/(t + 1) to 1 - t.Thus, the equation becomes:t / (1 - t - t² ) = 1 - tSo, t / (1 - t - t² ) = 1 - tMultiply both sides by (1 - t - t² ):t = (1 - t)(1 - t - t² )Expand the right side:(1 - t)(1 - t - t² ) = (1)(1 - t - t² ) - t(1 - t - t² )= 1 - t - t² - t + t² + t³Simplify:= 1 - 2t + 0 t² + t³So, equation becomes:t = t³ - 2t + 1Bring all terms to left side:t³ - 2t + 1 - t = 0 => t³ - 3t + 1 = 0So, the equation reduces to t³ - 3t + 1 = 0Now, we need to solve this cubic equation: t³ - 3t + 1 = 0Let me check for rational roots using Rational Root Theorem. Possible roots are ±1.Test t=1: 1 - 3 + 1 = -1 ≠ 0Test t=-1: -1 + 3 +1 = 3 ≠ 0So, no rational roots. Let's try to find real roots. Maybe use the method for solving cubics.The general cubic equation is t³ + pt² + qt + r = 0. In this case, equation is t³ - 3t + 1 = 0, so p = 0, q = -3, r = 1.The discriminant Δ = ( (27 r² + 4 p³ - 18 p q r + 4 q³ + p² q² ) ) Wait, for depressed cubic t³ + pt + q = 0, the discriminant is Δ = -4p³ - 27q². Wait, our equation is t³ + 0 t² -3 t +1 =0. So, as a depressed cubic, it's t³ + mt + n = 0, where m = -3, n = 1.So discriminant Δ = -4 m³ - 27 n² = -4*(-3)^3 - 27*(1)^2 = -4*(-27) -27 = 108 - 27 = 81 >0Since Δ >0, there are three distinct real roots. Let's use trigonometric substitution to find them.For a depressed cubic t³ + mt + n = 0 with Δ >0, the roots can be expressed using cosine:t = 2 sqrt( -m/3 ) cos( θ/3 + 2πk/3 ), k=0,1,2where θ = arccos( ( -n/2 ) / sqrt( -m³/27 ) )First, compute m = -3, n =1Compute sqrt( -m /3 ) = sqrt( 3/3 ) = sqrt(1) =1Compute the argument inside arccos:( -n / 2 ) / sqrt( -m³ / 27 ) = ( -1/2 ) / sqrt( 27 / 27 ) = (-1/2)/1 = -1/2Thus, θ = arccos(-1/2 ) = 2π/3Therefore, the roots are:t = 2*1*cos( (2π/3)/3 + 2πk/3 ) = 2 cos( 2π/9 + 2πk/3 ), k=0,1,2Compute for k=0: 2 cos(2π/9) ≈ 2 cos(40°) ≈ 2*0.7660 ≈ 1.532k=1: 2 cos(2π/9 + 2π/3) = 2 cos(8π/9) ≈ 2 cos(160°) ≈ 2*(-0.9397) ≈ -1.8794k=2: 2 cos(2π/9 + 4π/3) = 2 cos(14π/9) ≈ 2 cos(280°) ≈ 2*0.1736 ≈ 0.347So, the three real roots are approximately 1.532, -1.8794, and 0.347.But since t = y/z, and we need real numbers y and z, these are valid. However, we need to determine which of these roots are valid in the context of the original equations.Recall from Equation (2): z² = x z + y². If z and y are real numbers, then z² - y² = x z. Since z ≠0, x = (z² - y²)/z. Now, if t = y/z is positive or negative, depending on the values.But perhaps we can check each root:1. t ≈ 1.532: y = 1.532 zFrom Equation (2): z² = x z + y² = x z + (1.532 z )² = x z + 2.347 z²Thus, z² = x z + 2.347 z² => -1.347 z² = x z => x = -1.347 zFrom Equation (1): y/(x - y ) = x/(y + z )Substitute:1.532 z / ( -1.347 z - 1.532 z ) = (-1.347 z ) / (1.532 z + z )Simplify:1.532 / ( -2.879 ) ≈ -1.347 / 2.532Compute left side: ≈ -0.532Right side: ≈ -0.532So, approximately equal. So this is a valid solution.2. t ≈ -1.8794: y = -1.8794 zFrom Equation (2): z² = x z + y² = x z + ( -1.8794 z )² = x z + 3.532 z²Thus, z² = x z + 3.532 z² => -2.532 z² = x z => x = -2.532 zFrom Equation (1):y/(x - y ) = (-1.8794 z ) / ( -2.532 z - (-1.8794 z ) ) = (-1.8794) / ( -0.6526 ) ≈ 2.879x/(y + z ) = ( -2.532 z ) / ( -1.8794 z + z ) = (-2.532 ) / ( -0.8794 ) ≈ 2.879So, this checks out as well.3. t ≈ 0.347: y = 0.347 zFrom Equation (2): z² = x z + y² = x z + (0.347 z )² = x z + 0.1204 z²Thus, z² = x z + 0.1204 z² => 0.8796 z² = x z => x = 0.8796 zFrom Equation (1):y/(x - y ) = 0.347 z / (0.8796 z - 0.347 z ) = 0.347 / 0.5326 ≈ 0.651x/(y + z ) = 0.8796 z / (0.347 z + z ) = 0.8796 / 1.347 ≈ 0.652Approximately equal. So, all three roots are valid. Therefore, t can take three possible values. However, the problem states "Given real numbers x, y, z" satisfying the equations, so there might be multiple solutions, but the expression (y² + z² - x²)/(2yz) might evaluate to the same value for all possible t.Recall that C = (3 t - t³)/2. Let's substitute each t into this expression.1. For t ≈ 1.532:C ≈ (3*1.532 - (1.532)^3)/2 ≈ (4.596 - 3.597)/2 ≈ (0.999)/2 ≈ 0.52. For t ≈ -1.8794:C ≈ (3*(-1.8794) - (-1.8794)^3)/2 ≈ (-5.638 - (-6.638))/2 ≈ (1.0)/2 ≈ 0.53. For t ≈ 0.347:C ≈ (3*0.347 - (0.347)^3)/2 ≈ (1.041 - 0.041)/2 ≈ 1.0/2 ≈ 0.5Wow, so in all cases, C ≈ 0.5. Therefore, regardless of the value of t, the expression equals 0.5, which is 1/2.Therefore, the value of (y² + z² - x²)/(2 y z ) is 1/2.Final Answerboxed{dfrac{1}{2}}

✅:Given real numbers ( x, y, z ) satisfying the equations:[frac{y}{x - y} = frac{x}{y + z} quad text{(1)}][z^2 = x(y + z) - y(x - y) quad text{(2)}]we need to find the value of (frac{y^2 + z^2 - x^2}{2yz}).First, we simplify Equation (1) by cross-multiplying:[y(y + z) = x(x - y)]Expanding both sides, we get:[y^2 + yz = x^2 - xy]Rearranging terms, we have:[x^2 - xy - y^2 - yz = 0]Next, we simplify Equation (2):[z^2 = x(y + z) - y(x - y)]Expanding the right-hand side, we get:[z^2 = xy + xz - xy + y^2]Simplifying, we find:[z^2 = xz + y^2]Rearranging terms, we have:[z^2 - xz - y^2 = 0]Solving for ( x ), we get:[x = frac{z^2 - y^2}{z}]Substituting ( x = frac{z^2 - y^2}{z} ) into the expression (frac{y^2 + z^2 - x^2}{2yz}), we need to compute the numerator and denominator. The numerator is:[y^2 + z^2 - left( frac{z^2 - y^2}{z} right)^2]Expanding and simplifying the numerator:[y^2 + z^2 - frac{(z^2 - y^2)^2}{z^2} = frac{y^2 z^2 + z^4 - (z^4 - 2y^2 z^2 + y^4)}{z^2}][= frac{y^2 z^2 + z^4 - z^4 + 2y^2 z^2 - y^4}{z^2} = frac{3y^2 z^2 - y^4}{z^2}][= 3y^2 - frac{y^4}{z^2}]The denominator is:[2yz]Thus, the expression becomes:[frac{3y^2 - frac{y^4}{z^2}}{2yz} = frac{y^2 left( 3 - frac{y^2}{z^2} right)}{2yz} = frac{y left( 3 - left( frac{y}{z} right)^2 right)}{2z}]Let ( t = frac{y}{z} ). Then the expression simplifies to:[frac{3t - t^3}{2}]We solve the cubic equation ( t^3 - 3t + 1 = 0 ) derived from substituting ( t = frac{y}{z} ) into the equations. The roots of this cubic equation are found using trigonometric methods, leading to three real roots. However, substituting these roots into the expression (frac{3t - t^3}{2}) results in the value (frac{1}{2}) for all valid roots.Thus, the value of (frac{y^2 + z^2 - x^2}{2yz}) is:[boxed{frac{1}{2}}]

🔑:1. Given the equations involving real numbers ( x ), ( y ), and ( z ):[frac{y}{x-y} = frac{x}{y+z}, quad z^2 = x(y+z) - y(x-y)]2. From (frac{y}{x-y} = frac{x}{y+z}), we can cross-multiply to obtain:[y(y + z) = x(x - y)]3. Expanding and rearranging terms:[y^2 + yz = x^2 - xy ]4. Using the second given equation ( z^2 = x(y+z) - y(x-y) ):[z^2 = xy + xz - yx + y^2 = xz + y^2 ]5. Substitute ( y^2 + yz = x^2 - xy ) into the problem:[frac{y^2 + z^2 - x^2}{2yz}]6. First calculate ( y^2 + z^2 - x^2 ):[y^2 + z^2 - x^2 = y^2 + (xz + y^2) - x^2 = 2y^2 + xz - x^2]7. Rewrite the numerator:[frac{2y^2 + xz - x^2}{2yz}]8. Next, we know from the rearranged form ( y^2 + yz = x^2 - xy ):[y^2 + yz = x^2 - xy ]9. Substitute the rearranged term:[frac{2(y^2 + yz)}{2yz} = frac{2(x^2 - xy)}{2yz} = frac{x^2 - xy}{yz}]10. Simplify by dividing:[frac{2(x^2 - xy)}{2yz} = frac{x^2 - xy}{yz} = frac{1}{2}]Conclusion:[boxed{frac{1}{2}}]

🧠:Okay, so I have this problem here where I need to find the minimum value of the expression ( x_{1} + frac{x_{2}^{2}}{2} + frac{x_{3}^{3}}{3} + ldots + frac{x_{n}^{n}}{n} ), given that ( x_1, x_2, ldots, x_n ) are strictly positive real numbers and the sum of their reciprocals is exactly ( n ). Hmm, let's try to break this down step by step.First, the objective is to minimize the expression which is a sum of terms each involving a different power of ( x_i ) divided by their respective indices. The constraint is that the sum of the reciprocals of these variables equals ( n ). So, the problem is a constrained optimization problem. That makes me think of using methods like Lagrange multipliers. But since all variables are positive, maybe there's a way to apply inequalities like the AM-GM inequality or others. Let me consider both approaches.Let's start by writing the problem formally. We need to minimize:[f(x_1, x_2, ldots, x_n) = x_1 + frac{x_2^2}{2} + frac{x_3^3}{3} + ldots + frac{x_n^n}{n}]subject to the constraint:[g(x_1, x_2, ldots, x_n) = frac{1}{x_1} + frac{1}{x_2} + ldots + frac{1}{x_n} = n]All ( x_i > 0 ).Since we have a constraint and an objective function, Lagrange multipliers might be the way to go. Let's recall that in the method of Lagrange multipliers, we set up the gradient of the function equal to a scalar multiple of the gradient of the constraint. That is, we introduce a multiplier ( lambda ) such that:[nabla f = lambda nabla g]So, let's compute the partial derivatives.First, the partial derivative of ( f ) with respect to ( x_i ):For each ( i ), ( frac{partial f}{partial x_i} = frac{d}{dx_i} left( frac{x_i^i}{i} right) = x_i^{i - 1} ).For ( x_1 ), it's just 1, since ( frac{d}{dx_1} (x_1) = 1 ).Now, the partial derivative of the constraint ( g ) with respect to ( x_i ):For each ( i ), ( frac{partial g}{partial x_i} = -frac{1}{x_i^2} ).Therefore, according to the Lagrange multiplier condition, for each ( i ), we have:[x_i^{i - 1} = lambda left( -frac{1}{x_i^2} right)]Wait, but hold on. The Lagrange multiplier equation is:[nabla f = lambda nabla g]Which component-wise gives:For each ( i ), ( frac{partial f}{partial x_i} = lambda frac{partial g}{partial x_i} ).So, substituting the derivatives we computed:For ( i = 1 ):[1 = lambda left( -frac{1}{x_1^2} right)]For ( i geq 2 ):[x_i^{i - 1} = lambda left( -frac{1}{x_i^2} right)]But here, the right-hand sides are negative because the derivative of ( 1/x_i ) is ( -1/x_i^2 ). However, since all ( x_i ) are positive, and the left-hand sides (for ( i geq 2 )) are positive (since ( x_i > 0 )), this implies that ( lambda ) must be negative. Let's denote ( lambda = -mu ), where ( mu > 0 ). Then, the equations become:For ( i = 1 ):[1 = mu cdot frac{1}{x_1^2} implies x_1^2 = mu implies x_1 = sqrt{mu}]For ( i geq 2 ):[x_i^{i - 1} = mu cdot frac{1}{x_i^2} implies x_i^{i - 1 + 2} = mu implies x_i^{i + 1} = mu implies x_i = mu^{1/(i + 1)}]So, each ( x_i ) is expressed in terms of ( mu ). Therefore, all variables can be written as ( x_i = mu^{1/(i + 1)} ) for ( i geq 1 ). Wait, but for ( i = 1 ), according to the above, ( x_1 = sqrt{mu} ), which is ( mu^{1/2} ), which is consistent with ( 1/(i + 1) ) when ( i = 1 ): ( 1/(1 + 1) = 1/2 ). Similarly, for ( i = 2 ), ( x_2 = mu^{1/3} ), and so on. So, yes, in general, ( x_i = mu^{1/(i + 1)} ) for each ( i ).Now, since all ( x_i ) are expressed in terms of ( mu ), we can substitute these into the constraint equation to solve for ( mu ).The constraint is:[sum_{i=1}^n frac{1}{x_i} = n]Substituting ( x_i = mu^{1/(i + 1)} ), we get:[sum_{i=1}^n frac{1}{mu^{1/(i + 1)}} = n implies sum_{i=1}^n mu^{-1/(i + 1)} = n]So, this simplifies to:[sum_{i=1}^n mu^{-1/(i + 1)} = n]Let me write each term in the sum:For ( i = 1 ): ( mu^{-1/2} )For ( i = 2 ): ( mu^{-1/3} )...For ( i = n ): ( mu^{-1/(n + 1)} )Therefore, the sum is:[mu^{-1/2} + mu^{-1/3} + ldots + mu^{-1/(n + 1)} = n]Hmm, this equation in ( mu ) seems a bit complicated. Let me see if there's a pattern or if all terms can be made equal. That is, perhaps ( mu^{-1/(i + 1)} ) is the same for each ( i ), which would be the case if each term in the sum is 1. Because if each term is 1, then the sum is ( n ), which matches the right-hand side. Therefore, if we set ( mu^{-1/(i + 1)} = 1 ) for all ( i ), then the sum would be ( n times 1 = n ), which satisfies the constraint.But wait, ( mu^{-1/(i + 1)} = 1 ) implies ( mu = 1 ) for all ( i ). But if ( mu = 1 ), then all ( x_i = 1^{1/(i + 1)} = 1 ). Therefore, all ( x_i = 1 ).But let's check if this is indeed the case. If we set all ( x_i = 1 ), then the sum of reciprocals is ( 1 + 1 + ldots + 1 = n ), which satisfies the constraint. And the value of the expression to minimize would be ( 1 + frac{1^2}{2} + frac{1^3}{3} + ldots + frac{1^n}{n} = 1 + frac{1}{2} + frac{1}{3} + ldots + frac{1}{n} ), which is the harmonic series up to ( n ). However, the problem states to find the minimum value of the expression. If this is indeed the case, then the minimal value would be the harmonic number ( H_n ). But wait, this contradicts the fact that when we use Lagrange multipliers, we found that ( x_i = mu^{1/(i + 1)} ), but if all ( x_i = 1 ), then ( mu = 1 ), but in that case, all the equations must be satisfied. Let's check that.If ( x_i = 1 ) for all ( i ), then for ( i = 1 ), the equation is ( 1 = mu cdot frac{1}{1^2} implies mu = 1 ). For ( i geq 2 ), the equation is ( 1^{i - 1} = mu cdot frac{1}{1^2} implies 1 = mu ). So yes, all equations are satisfied when ( mu = 1 ). Therefore, the critical point occurs at all ( x_i = 1 ). So, is this the minimum?Wait, but let's verify if this critical point is indeed a minimum. To confirm, we might need to check the second derivatives or use another method, but perhaps there's another way.Alternatively, maybe using the method of inequalities. Let's consider if we can apply the Holder's inequality or the AM-GM inequality.The objective function is a sum of terms ( frac{x_i^i}{i} ), and the constraint is the sum of reciprocals ( sum frac{1}{x_i} = n ). Let's see if we can use the method of weighted AM-GM.For each term ( frac{x_i^i}{i} ), perhaps we can pair it with the reciprocal term ( frac{1}{x_i} ).Wait, Holder's inequality relates the product of sums, each raised to a certain power. Alternatively, the Cauchy-Schwarz inequality is a specific case. Alternatively, since the variables are connected reciprocally, maybe dual variables in some sense.Alternatively, let's think of each term ( frac{x_i^i}{i} ) and its corresponding reciprocal ( frac{1}{x_i} ). Let's consider for each ( i geq 1 ), the pair ( frac{x_i^i}{i} ) and ( frac{1}{x_i} ).Maybe we can use the inequality ( a + b geq 2sqrt{ab} ), which is the AM-GM inequality for two variables. But here, we have different terms. Alternatively, perhaps use a more general form of AM-GM.Suppose for each ( i ), we have:( frac{x_i^i}{i} + (i) cdot frac{1}{x_i} geq something ).Wait, perhaps if we use weighted AM-GM for each pair.Wait, but the constraint is the sum of reciprocals, so each reciprocal term is part of the constraint. The objective is the sum of the terms ( frac{x_i^i}{i} ). Maybe we can use Lagrange multipliers as earlier.But in the Lagrange multiplier method, we found that the critical point is when all ( x_i = 1 ). Let's check what the value of the objective function is at that point. As I computed earlier, it's ( 1 + frac{1}{2} + frac{1}{3} + ldots + frac{1}{n} ), which is the harmonic series ( H_n ). But is this the minimum?Wait, let's test for a small ( n ). Let's take ( n = 1 ). Then, the problem reduces to minimizing ( x_1 ), given that ( 1/x_1 = 1 implies x_1 = 1 ). So the minimum is 1, which is indeed the harmonic number ( H_1 = 1 ). So that works.For ( n = 2 ), the problem is to minimize ( x_1 + frac{x_2^2}{2} ), subject to ( 1/x_1 + 1/x_2 = 2 ). Let's see if the minimum is ( 1 + 1/2 = 1.5 ). Let's check.If ( x_1 = 1 ), then ( 1/x_1 = 1 ), so ( 1/x_2 = 1 implies x_2 = 1 ). So the value is ( 1 + 1/2 = 1.5 ).Alternatively, suppose we take ( x_1 ) slightly larger than 1, say ( x_1 = 1 + epsilon ). Then ( 1/x_1 approx 1 - epsilon ), so ( 1/x_2 approx 1 + epsilon implies x_2 approx 1/(1 + epsilon) approx 1 - epsilon ). Then, compute the objective function:( x_1 + x_2^2 / 2 approx (1 + epsilon) + (1 - 2epsilon + epsilon^2)/2 approx 1 + epsilon + 0.5 - epsilon + 0.5 epsilon^2 approx 1.5 + 0.5 epsilon^2 ). So, for small ( epsilon ), the value is larger than 1.5. Similarly, if we take ( x_1 < 1 ), then ( x_2 > 1 ), and compute the objective function:( x_1 + x_2^2 / 2 ). Let's say ( x_1 = 1 - epsilon ), so ( 1/x_1 approx 1 + epsilon implies 1/x_2 = 2 - (1 + epsilon) = 1 - epsilon implies x_2 = 1/(1 - epsilon) approx 1 + epsilon ). Then, ( x_2^2 approx 1 + 2epsilon + epsilon^2 ), so ( x_2^2 / 2 approx 0.5 + epsilon + 0.5 epsilon^2 ). Adding ( x_1 approx 1 - epsilon ), total is ( 1 - epsilon + 0.5 + epsilon + 0.5 epsilon^2 approx 1.5 + 0.5 epsilon^2 ). Again, higher than 1.5. Therefore, in the case ( n = 2 ), the minimal value is indeed 1.5.Similarly, for ( n = 3 ), the minimal value would be ( 1 + 1/2 + 1/3 approx 1.833 ). Let's check if perturbing variables around 1 increases the sum.Take ( x_1 = 1 + epsilon ), then ( 1/x_1 approx 1 - epsilon ), so sum of reciprocals would require ( 1/x_2 + 1/x_3 = 3 - (1 - epsilon) = 2 + epsilon ). If we set ( x_2 = 1 ), then ( 1/x_3 = 1 + epsilon implies x_3 = 1/(1 + epsilon) approx 1 - epsilon ). Then compute the objective function:( x_1 + x_2^2 /2 + x_3^3 /3 approx (1 + epsilon) + 1/2 + (1 - 3epsilon + 3epsilon^2 - epsilon^3)/3 approx 1 + epsilon + 0.5 + 1/3 - epsilon + epsilon^2 - epsilon^3/3 approx 1 + 0.5 + 1/3 + epsilon^2 - epsilon^3/3 ), which is approximately ( 1.833 + epsilon^2 ), which is higher. Therefore, perturbing increases the value. So again, the minimal value is at all ( x_i = 1 ).Therefore, this suggests that the minimal value is indeed the harmonic series ( H_n ). But wait, the problem says "Find the minimum value of the expression... where ( x_1, ldots, x_n ) are strictly positive real numbers whose sum of the reciprocals is exactly ( n )." So, according to our analysis, the minimal value is the sum ( sum_{k=1}^n frac{1}{k} ), which is ( H_n ).But wait a second, when we used Lagrange multipliers, we found that all ( x_i = 1 ), leading to the harmonic series. But let me check another case. Suppose ( n = 1 ). Then the expression is just ( x_1 ), with the constraint ( 1/x_1 = 1 implies x_1 = 1 ). So the minimum is 1, which is ( H_1 ). For ( n = 2 ), as we saw, it's 1.5, which is ( H_2 ). So this seems consistent.But let me test with a different approach. Suppose we use the Cauchy-Schwarz inequality. But I'm not sure how to apply it here. Alternatively, perhaps use convexity. The function to minimize is a sum of convex functions (since each term ( frac{x_i^i}{i} ) is convex in ( x_i )), and the constraint is a convex constraint (since the sum of reciprocals is a convex function for ( x_i > 0 )). Therefore, the problem is a convex optimization problem, which implies that any critical point found via Lagrange multipliers is indeed the global minimum. Therefore, since we found a critical point at ( x_i = 1 ), this must be the global minimum. Therefore, the minimal value is ( H_n ).But let me check with another perspective. Let's suppose that for each ( i ), we can pair ( frac{x_i^i}{i} ) with ( frac{1}{x_i} ). Let's consider using the inequality ( a cdot t + b cdot frac{1}{t} geq 2sqrt{ab} ), which is the AM-GM inequality. However, here the terms are not linear and reciprocal; instead, they have different exponents. So maybe for each term ( frac{x_i^i}{i} ) and the corresponding reciprocal ( frac{1}{x_i} ), we can use Hölder's inequality.Recall that Hölder's inequality states that for conjugate exponents ( p ) and ( q ) (i.e., ( 1/p + 1/q = 1 )), we have:[sum_{k=1}^n a_k b_k leq left( sum_{k=1}^n a_k^p right)^{1/p} left( sum_{k=1}^n b_k^q right)^{1/q}]But in our case, the terms are not products but separate. Alternatively, maybe think of each term in the objective and the constraint.Alternatively, consider the Lagrangian function:[mathcal{L}(x_1, ldots, x_n, lambda) = x_1 + frac{x_2^2}{2} + ldots + frac{x_n^n}{n} + lambda left( n - sum_{i=1}^n frac{1}{x_i} right)]Taking partial derivatives with respect to each ( x_i ):For ( x_1 ):[frac{partial mathcal{L}}{partial x_1} = 1 + lambda cdot frac{1}{x_1^2} = 0 implies 1 = -lambda cdot frac{1}{x_1^2}]For ( x_i ) with ( i geq 2 ):[frac{partial mathcal{L}}{partial x_i} = x_i^{i - 1} + lambda cdot frac{1}{x_i^2} = 0 implies x_i^{i - 1} = -lambda cdot frac{1}{x_i^2} implies x_i^{i + 1} = -lambda]But since ( x_i > 0 ), ( -lambda ) must be positive, so ( lambda < 0 ). Let ( lambda = -mu ), ( mu > 0 ). Then:For ( x_1 ):[1 = mu cdot frac{1}{x_1^2} implies x_1 = sqrt{mu}]For ( x_i geq 2 ):[x_i^{i + 1} = mu implies x_i = mu^{1/(i + 1)}]This is the same result as before. So all variables can be expressed in terms of ( mu ). Then, substituting into the constraint:[sum_{i=1}^n frac{1}{x_i} = sum_{i=1}^n mu^{-1/(i + 1)} = n]If we set ( mu = 1 ), then each term in the sum is ( 1 ), so the sum is ( n ), which satisfies the constraint. Hence, this gives ( x_i = 1 ) for all ( i ), and the minimal value is ( H_n ).But let's check if there could be another solution where ( mu neq 1 ). Suppose ( mu neq 1 ). Then the sum ( sum_{i=1}^n mu^{-1/(i + 1)} = n ). But is there a ( mu neq 1 ) that satisfies this equation?Consider ( n = 2 ). Then the equation becomes ( mu^{-1/2} + mu^{-1/3} = 2 ). Let's check for ( mu = 1 ): ( 1 + 1 = 2 ), which works. Suppose ( mu > 1 ): Then ( mu^{-1/2} < 1 ) and ( mu^{-1/3} < 1 ), so the sum would be less than 2, which doesn't satisfy the equation. If ( mu < 1 ), then ( mu^{-1/2} > 1 ) and ( mu^{-1/3} > 1 ), so the sum would be greater than 2, which also doesn't satisfy the equation. Therefore, for ( n = 2 ), the only solution is ( mu = 1 ).Similarly, for ( n = 3 ), the equation is ( mu^{-1/2} + mu^{-1/3} + mu^{-1/4} = 3 ). Again, ( mu = 1 ) gives 3. If ( mu > 1 ), all terms are less than 1, so sum < 3. If ( mu < 1 ), all terms are greater than 1, so sum > 3. Therefore, only ( mu = 1 ) works. The same logic applies for higher ( n ): each term ( mu^{-1/(i + 1)} ) is a decreasing function of ( mu ). Therefore, the sum ( sum_{i=1}^n mu^{-1/(i + 1)} ) is a decreasing function of ( mu ). Since at ( mu = 1 ), the sum is exactly ( n ), and the function is continuous, there's no other ( mu ) that can satisfy the equation. Therefore, the only critical point is at ( mu = 1 ), leading to all ( x_i = 1 ).Therefore, the minimal value of the expression is ( H_n = sum_{k=1}^n frac{1}{k} ).But let me verify with another approach. Suppose we consider each term ( frac{x_i^i}{i} ) and apply the inequality between the weighted arithmetic mean and the weighted geometric mean.For each term ( frac{x_i^i}{i} ), we can think of it as ( frac{x_i^i}{i} ), and the corresponding reciprocal term ( frac{1}{x_i} ). Let's consider combining these two terms with appropriate weights.Let's take for each ( i ), the two terms ( frac{x_i^i}{i} ) and ( frac{1}{x_i} ), and apply AM-GM to them with some exponents.Suppose we have positive numbers ( a ) and ( b ), and positive weights ( p ) and ( q ), then:[frac{p cdot a + q cdot b}{p + q} geq a^{p/(p + q)} b^{q/(p + q)}]In our case, for each ( i ), let's pair ( a = frac{x_i^i}{i} ) and ( b = frac{1}{x_i} ). We need to choose weights ( p_i ) and ( q_i ) such that when we apply AM-GM, we can relate these terms to a constant.But I'm not sure how to choose the weights here. Alternatively, maybe for each ( i ), find a way to express ( frac{x_i^i}{i} + lambda cdot frac{1}{x_i} geq text{something} ), but this might not directly work.Alternatively, notice that the Lagrangian conditions have already given us the optimal point where all variables are equal to 1, and given that the problem is convex, this is the unique minimum.Alternatively, maybe use induction. Suppose that for ( n = k ), the minimal value is ( H_k ). Then for ( n = k + 1 ), we can consider fixing the first ( k ) variables and optimizing the last one. But this approach might be complicated.Alternatively, think of the problem as optimizing each variable individually while keeping the others fixed. If we fix all variables except ( x_i ), then the reciprocal sum constraint allows us to express ( 1/x_i ) in terms of the others. Then, take the derivative with respect to ( x_i ). But this is essentially what the Lagrange multiplier method does.Given that all methods point to the minimal value being the harmonic number ( H_n ), and the critical point at all variables equal to 1, which satisfies the constraint and yields the harmonic series, I think this is the correct answer.But wait, let's consider an example where ( n = 1 ). Then the problem is to minimize ( x_1 ), given ( 1/x_1 = 1 implies x_1 = 1 ). The minimal value is 1, which is ( H_1 ). For ( n = 2 ), as checked earlier, it's 1.5, which is ( H_2 ). For ( n = 3 ), it's ( 1 + 1/2 + 1/3 approx 1.833 ), which is ( H_3 ). So this pattern holds.But here's a thought: if we use the Lagrange multiplier method and found that all variables must be equal to 1 at the critical point, then under the convexity of the problem, this must be the global minimum. Since the objective function is convex (each term ( frac{x_i^i}{i} ) is convex in ( x_i ), and the sum of convex functions is convex) and the constraint is linear (if we consider the reciprocal variables, but actually the constraint is convex in ( x_i )), then the problem is convex, and the critical point is indeed the unique minimum.Therefore, the minimal value is indeed the sum ( sum_{k=1}^n frac{1}{k} ), which is the nth harmonic number.Hence, the final answer should be ( H_n ), which is written as ( boxed{1 + frac{1}{2} + frac{1}{3} + ldots + frac{1}{n}} ). However, the problem might expect it written in summation notation or as ( boxed{sum_{k=1}^n frac{1}{k}} ).Alternatively, if they prefer it in terms of harmonic numbers, ( boxed{H_n} ). But since harmonic numbers are usually denoted ( H_n ), but the problem might want the explicit sum. Let me check the problem statement again.The problem says "Find the minimum value of the expression... where ( x_1, ldots, x_n ) are strictly positive real numbers whose sum of the reciprocals is exactly ( n )." The answer should be in a boxed format. Since the harmonic number ( H_n ) is standard, but in case they want the expanded form, but given that ( n ) is arbitrary, writing it as ( H_n ) is acceptable. However, in many mathematical contexts, writing it as the sum is clearer if the problem doesn't reference harmonic numbers. Since the problem didn't mention harmonic numbers, I think writing the sum explicitly is safer.But let me check for ( n = 1 ), the answer is 1, which is ( H_1 ). For ( n = 2 ), it's ( 1 + 1/2 ), etc. So yes, the sum is the harmonic series. Therefore, the answer is ( boxed{sum_{k=1}^n frac{1}{k}} ).But to confirm once more, if all variables are 1, then the sum of reciprocals is n, and the expression evaluates to the harmonic series. Since perturbing the variables increases the expression, as we saw in examples, this must indeed be the minimum.Final AnswerThe minimum value of the expression is boxed{1 + frac{1}{2} + frac{1}{3} + ldots + frac{1}{n}}.

✅:To find the minimum value of the expression[x_{1} + frac{x_{2}^{2}}{2} + frac{x_{3}^{3}}{3} + ldots + frac{x_{n}^{n}}{n}]where ( x_{1}, x_{2}, ldots, x_{n} ) are strictly positive real numbers whose sum of the reciprocals is exactly ( n ), we can use the method of Lagrange multipliers. We need to minimize the function:[f(x_1, x_2, ldots, x_n) = x_1 + frac{x_2^2}{2} + frac{x_3^3}{3} + ldots + frac{x_n^n}{n}]subject to the constraint:[g(x_1, x_2, ldots, x_n) = frac{1}{x_1} + frac{1}{x_2} + ldots + frac{1}{x_n} = n]Using Lagrange multipliers, we set up the equations:[nabla f = lambda nabla g]For each ( i ), the partial derivatives give:[x_i^{i - 1} = lambda left( -frac{1}{x_i^2} right)]Solving these equations, we find that each ( x_i = mu^{1/(i + 1)} ) where ( mu ) is a constant. Substituting these into the constraint equation, we find that ( mu = 1 ), leading to ( x_i = 1 ) for all ( i ).Substituting ( x_i = 1 ) into the objective function, we get:[1 + frac{1^2}{2} + frac{1^3}{3} + ldots + frac{1^n}{n} = 1 + frac{1}{2} + frac{1}{3} + ldots + frac{1}{n}]This is the harmonic series ( H_n ), which is the minimum value of the expression. Thus, the minimum value is:[boxed{1 + frac{1}{2} + frac{1}{3} + ldots + frac{1}{n}}]

🔑:1. Restate the Problem and the Given Conditions: Let ( n ) be a positive integer. We want to find the minimum value of: [ x_{1}+frac{x_{2}^{2}}{2}+frac{x_{3}^{3}}{3}+ldots+frac{x_{n}^{n}}{n} ] where ( x_{1}, x_{2}, ldots, x_{n} ) are positive real numbers such that the sum of their reciprocals is exactly ( n ): [ frac{1}{x_1} + frac{1}{x_2} + cdots + frac{1}{x_n} = n. ]2. Apply the Harmonic Mean Inequality: The harmonic mean of ( x_1, x_2, ldots, x_n ) is given by: [ H = frac{n}{frac{1}{x_1} + frac{1}{x_2} + cdots + frac{1}{x_n}}. ] Given the constraint (frac{1}{x_1} + frac{1}{x_2} + cdots + frac{1}{x_n} = n), we have: [ H = frac{n}{n} = 1. ]3. Apply the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality): By the AM-GM inequality, for any set of positive real numbers ( a_1, a_2, ldots, a_n ): [ frac{a_1 + a_2 + cdots + a_n}{n} geq sqrt[n]{a_1 a_2 cdots a_n}. ] With equality if and only if ( a_1 = a_2 = cdots = a_n ). Setting ( a_i = x_i ) for ( i = 1, 2, ldots, n ). Then, [ frac{x_1 + x_2 + cdots + x_n}{n} geq sqrt[n]{x_1 x_2 cdots x_n}. ] Given that ( H = 1 ), we need to find conditions when ( x_1 x_2 cdots x_n geq 1 ) which is inherently satisfied from our harmonic constraint.4. Weighted AM-GM Inequality Application: Apply the weighted AM-GM inequality for the expression ( x_{1}+frac{x_{2}^{2}}{2}+frac{x_{3}^{3}}{3}+ldots+frac{x_{n}^{n}}{n} ): [ x_{1} + frac{x_{2}^{2}}{2} + frac{x_{3}^{3}}{3} + ldots + frac{x_{n}^{n}}{n} geq sum_{k=1}^{n} frac{k}{k} = 1 + frac{1}{2} + ldots + frac{1}{n}. ]5. Achieve Equality: The minimum value is reached when the equality condition of the AM-GM inequality holds, i.e., when ( x_{1} = x_{2} = cdots = x_{n} ). Since ( frac{1}{x_1} + frac{1}{x_2} + cdots + frac{1}{x_n} = n ), it must be that each ( x_i = 1 ).6. Compute the Minimum: Therefore, substituting ( x_i = 1 ): [ 1 + frac{1^2}{2} + frac{1^3}{3} + cdots + frac{1^n}{n} = 1 + frac{1}{2} + frac{1}{3} + cdots + frac{1}{n}. ] Conclusion: The minimal value of the given expression is: [ boxed{1 + frac{1}{2} + frac{1}{3} + cdots + frac{1}{n}}. ]

🧠:Okay, so I need to find the coefficient of the term (a_{1}^{k_{1}} a_{2}^{k_{2}} cdots a_{m}^{k_{m}}) in the expansion of (left(a_{1}+a_{2}+cdots+a_{m}right)^{n}), where all the exponents (k_{1}, k_{2}, ldots, k_{m}) add up to (n). Hmm, let's start by recalling what I know about polynomial expansions and coefficients. First, when you expand a multinomial like ((a_1 + a_2 + cdots + a_m)^n), each term in the expansion is of the form (a_1^{k_1} a_2^{k_2} cdots a_m^{k_m}) where the exponents (k_i) are non-negative integers such that (k_1 + k_2 + cdots + k_m = n). The coefficient of such a term is known as a multinomial coefficient. I remember that for binomials, when there are only two terms, the coefficient is given by the binomial coefficient (binom{n}{k}), which is (frac{n!}{k!(n-k)!}). For multinomials, this generalizes.So maybe the multinomial coefficient is (frac{n!}{k_1! k_2! cdots k_m!})? Let me verify this with some examples. Take a simple case where (m = 2), so we have a binomial. Let's say (n = 3), and we want the coefficient of (a_1^2 a_2^1). According to the binomial theorem, the coefficient should be (binom{3}{2} = 3). Using the formula I mentioned, it would be (frac{3!}{2!1!} = frac{6}{2 cdot 1} = 3). That works. Another example with (m = 3). Suppose we have ((a + b + c)^2). The expansion is (a^2 + b^2 + c^2 + 2ab + 2ac + 2bc). Let's check the coefficient of (abc). Wait, but in this case, each term has exponents adding up to 2. If we take (k_1 = 1), (k_2 = 1), (k_3 = 0), the coefficient is 2 for (ab), which would be (frac{2!}{1!1!0!} = 2). But 0! is 1, right? So that works. Similarly, if we take (k_1 = 1), (k_2 = 1), (k_3 = 0), the coefficient is 2. If we wanted a term like (a^1 b^1 c^1), but in the expansion of ((a + b + c)^2), such a term doesn't exist because the exponents would have to add up to 2, not 3. So the formula works here as well.Wait, another thought. Let me try with (m=3) and (n=3). The expansion of ((a + b + c)^3) should have terms like (a^3), (a^2b), (ab^2), (abc), etc. Let's check the coefficient of (a^1 b^1 c^1). How many ways can we get one a, one b, and one c when multiplying three terms? Each term in the expansion comes from picking one of the variables from each of the three factors. So to get one a, one b, and one c, we need to choose a from one factor, b from another, and c from the third. The number of ways is the number of permutations of the three variables, which is 3! = 6. So the coefficient should be 6. Using the formula (frac{3!}{1!1!1!} = 6), which matches. That's correct.So, this seems consistent. Therefore, the coefficient is given by the multinomial coefficient, which generalizes the binomial coefficient. The formula is indeed (frac{n!}{k_1! k_2! cdots k_m!}). But let me think about why this is the case. How does this formula come about? When expanding ((a_1 + a_2 + cdots + a_m)^n), each term in the expansion is formed by selecting one term from each of the (n) factors and multiplying them together. So, for example, if we pick (a_1) from the first (k_1) factors, (a_2) from the next (k_2) factors, and so on up to (a_m) from the last (k_m) factors, then the product would be (a_1^{k_1} a_2^{k_2} cdots a_m^{k_m}). However, the order in which we pick the terms doesn't matter. The number of distinct ways to arrange these selections is given by the multinomial coefficient.Alternatively, this is equivalent to arranging (n) objects where there are (k_1) identical objects of type 1, (k_2) of type 2, ..., (k_m) of type m. The number of distinct arrangements is (frac{n!}{k_1! k_2! cdots k_m!}). So, in the context of the expansion, each distinct arrangement corresponds to a different way of choosing the variables from each factor, leading to the same monomial term. Therefore, the coefficient is the number of such arrangements, which is the multinomial coefficient.Wait, but let me make sure there isn't an alternative way to think about this. For instance, using generating functions or combinatorial arguments. Another approach: The coefficient of (a_1^{k_1} a_2^{k_2} cdots a_m^{k_m}) is the number of ways to assign each of the (n) factors ((a_1 + a_2 + cdots + a_m)) to one of the variables (a_i) such that variable (a_i) is chosen exactly (k_i) times. This is equivalent to the multinomial coefficient as well.Yes, that's another way to phrase it. So, when you expand the product, you're effectively distributing (n) choices (each corresponding to a factor) among the (m) variables, with each variable being chosen (k_i) times. The number of such distributions is the multinomial coefficient. Therefore, regardless of the approach—whether through permutation of terms, arranging selections, or distributing choices—the coefficient ends up being the multinomial coefficient. Is there a case where this formula might not hold? For example, if some (k_i) are zero? Let's see. Suppose (m = 3), (n = 2), and we want the coefficient of (a_1^2 a_2^0 a_3^0). According to the formula, it should be (frac{2!}{2!0!0!} = 1). Indeed, in the expansion of ((a_1 + a_2 + a_3)^2), the term (a_1^2) appears once, so that works. Similarly, if (k_3 = 1) and others adjust accordingly. Another edge case: if all (k_i = 1) but (n = m). Then the coefficient is (frac{m!}{1!1! cdots 1!} = m!). For example, in ((a + b + c)^3), the coefficient of (abc) is 6, which is 3! as we saw earlier. So that works.Therefore, the formula seems robust across different cases. Let me think if there's another way to derive this formula. Maybe using induction?Suppose for a binomial case ((a_1 + a_2)^n), the coefficient of (a_1^{k} a_2^{n - k}) is (binom{n}{k}). This is the binomial coefficient. Now, if we add another term, making it a trinomial ((a_1 + a_2 + a_3)^n), how does the coefficient work?We can think of the expansion as a sum over all possible triples (k_1 + k_2 + k_3 = n), and the coefficient is (frac{n!}{k_1! k_2! k_3!}). This can be shown by induction. Assume it's true for (m) terms, then adding another term would involve splitting the exponent among the new term and the existing ones. But maybe that's more complicated.Alternatively, using generating functions, the generating function for each variable (a_i) is (1 + a_i + a_i^2 + cdots), but in our case, each factor is a linear term. Wait, no. The generating function here is ((a_1 + a_2 + cdots + a_m)^n), which is a polynomial generating function. The coefficient extraction for a specific term is exactly what the multinomial coefficient gives.Alternatively, using the multinomial theorem, which states that:[(a_1 + a_2 + cdots + a_m)^n = sum_{k_1 + k_2 + cdots + k_m = n} frac{n!}{k_1! k_2! cdots k_m!} a_1^{k_1} a_2^{k_2} cdots a_m^{k_m}]Therefore, by the multinomial theorem, the coefficient is indeed the multinomial coefficient as stated.So, putting this all together, the coefficient is given by the multinomial coefficient formula. Therefore, after all this thinking and verification through examples and different reasoning approaches, I conclude that the coefficient of (a_{1}^{k_{1}} a_{2}^{k_{2}} cdots a_{m}^{k_{m}}) in the expansion is (frac{n!}{k_1! k_2! cdots k_m!}).

✅:The coefficient of the term (a_{1}^{k_{1}} a_{2}^{k_{2}} cdots a_{m}^{k_{m}}) in the expansion of (left(a_{1} + a_{2} + cdots + a_{m}right)^{n}), where (k_{1} + k_{2} + cdots + k_{m} = n), is given by the multinomial coefficient:[boxed{frac{n!}{k_{1}! , k_{2}! , cdots , k_{m}!}}] Explanation:1. Multinomial Expansion: The expansion of ((a_1 + a_2 + cdots + a_m)^n) generates terms of the form (a_1^{k_1} a_2^{k_2} cdots a_m^{k_m}) where (k_1 + k_2 + cdots + k_m = n).2. Combinatorial Interpretation: The coefficient corresponds to the number of distinct ways to distribute (n) exponents among the (m) variables. This is equivalent to counting permutations of (n) items where there are (k_i) identical items of type (i).3. Formula Derivation: The multinomial coefficient (frac{n!}{k_1! , k_2! , cdots , k_m!}) accounts for these permutations by dividing (n!) (total arrangements) by the product of factorials of the counts of each variable ((k_i!)), eliminating redundancies from identical terms.This result generalizes the binomial coefficient and is derived rigorously through combinatorial reasoning and the multinomial theorem.

🔑:We are tasked with finding the coefficient of the term (a_{1}^{k_{1}} a_{2}^{k_{2}} cdots a_{m}^{k_{m}}) in the expansion of ((a_{1} + a_{2} + cdots + a_{m})^n), subject to the constraint (k_{1} + k_{2} + cdots + k_{m} = n).1. Case when (m=2): By the binomial theorem, we know that the coefficient of the term (a_{1}^{k_{1}} a_{2}^{n-k_{1}}) in the expansion of ((a_{1} + a_{2})^n) is given by [ binom{n}{k_{1}} = frac{n!}{k_{1}!(n-k_{1})!}. ] Consider the expansion: [ (a_{1} + a_{2})^n = underbrace{(a_{1} + a_{2})(a_{1} + a_{2}) cdots (a_{1} + a_{2})}_{n text{ times}}, ] where each term is a product of some (a_{1})'s and some (a_{2})'s. Specifically, a term of the form (a_{1}^{k_{1}} a_{2}^{n-k_{1}}) corresponds to selecting (k_{1}) instances of (a_{1}) and (n-k_{1}) instances of (a_{2}). The number of ways to arrange these selections is given by the binomial coefficient (binom{n}{k_{1}}).2. Case when (m=3): When expanding ((a_{1} + a_{2} + a_{3})^n), a term of the form (a_{1}^{k_{1}} a_{2}^{k_{2}} a_{3}^{k_{3}}) (with (k_{1} + k_{2} + k_{3} = n)) corresponds to the multinomial coefficient: [ binom{n}{k_{1}, k_{2}, k_{3}} = frac{n!}{k_{1}!k_{2}!k_{3}!}. ]3. General case: For ((a_{1} + a_{2} + cdots + a_{m})^n), a term of the form (a_{1}^{k_{1}} a_{2}^{k_{2}} cdots a_{m}^{k_{m}}) (with (k_{1} + k_{2} + cdots + k_{m} = n)) appears with the coefficient given by the multinomial coefficient: [ binom{n}{k_{1}, k_{2}, ldots, k_{m}} = frac{n!}{k_{1}!k_{2}! cdots k_{m}!}. ]4. Explanation via Combinatorics: The expansion of ((a_{1} + a_{2} + cdots + a_{m})^n) can be thought of as: [ underbrace{(a_{1} + a_{2} + cdots + a_{m})(a_{1} + a_{2} + cdots + a_{m}) cdots (a_{1} + a_{2} + cdots + a_{m})}_{n text{ times}}. ] Each term in the expansion results from choosing (k_{1}) copies of (a_{1}), (k_{2}) copies of (a_{2}), ..., (k_{m}) copies of (a_{m}), where (k_{1} + k_{2} + cdots + k_{m} = n). The number of such possible arrangements (ways to order (n) items where (k_{1}) are (a_{1}), (k_{2}) are (a_{2}), ..., (k_{m}) are (a_{m})) is exactly given by the multinomial coefficient.In conclusion, the coefficient of the term (a_{1}^{k_{1}} a_{2}^{k_{2}} cdots a_{m}^{k_{m}}) in the expansion of ((a_{1} + a_{2} + cdots + a_{m})^n) is:[boxed{frac{n!}{k_{1}!k_{2}! cdots k_{m}!}}.]

🧠:Okay, so I need to solve this functional equation problem. Let me take it step by step. The problem says that there's a function f defined on integers, integer-valued, and it satisfies the equation:f(m + f(f(n))) = -f(f(m + 1)) - nfor all integers m and n. Also, there's a polynomial g(n) with integer coefficients such that g(n) = g(f(n)) for all n. We need to find f(1991) and determine the most general form for g.First, let me focus on the functional equation. Functional equations can sometimes be tricky, but often substituting specific values for variables can help simplify things. Let me see if I can find some properties of f, like injectivity, surjectivity, or maybe even figure out if it's linear.Let me start by checking if f is injective or surjective. If I can show that f is injective, that might help. Suppose f(a) = f(b). Then, perhaps I can substitute into the equation in some way to show that a = b.Alternatively, maybe I can find an expression for f in terms of itself. Let me try substituting specific values for m or n.Let me first try setting m to some specific value. Let's see, maybe set m = 0. Then the equation becomes:f(0 + f(f(n))) = -f(f(0 + 1)) - nSimplify:f(f(f(n))) = -f(f(1)) - nHmm. That's an interesting equation. It relates f(f(f(n))) to -n minus a constant term f(f(1)). Let me denote some constants here. Let's let c = f(f(1)). Then the equation becomes:f(f(f(n))) = -c - nSo f^3(n) = -c - n, where f^3(n) means f applied three times. So applying f three times gives us a linear function. That seems significant. Maybe f is invertible? If f^3(n) is linear, then perhaps f itself is linear?Alternatively, maybe f is affine, meaning f(n) = an + b for some integers a and b. Let me assume that f is linear and see if that works. Suppose f(n) = an + b.Then let's compute both sides of the original functional equation.Left-hand side (LHS): f(m + f(f(n))) = a(m + f(f(n))) + b = a(m + a(f(n)) + b) + b = a(m + a(an + b) + b) + b = a(m + a^2 n + ab + b) + b = a m + a^3 n + a^2 b + a b + b.Right-hand side (RHS): -f(f(m + 1)) - n = -[a(f(m + 1)) + b] - n = -[a(a(m + 1) + b) + b] - n = -[a^2(m + 1) + a b + b] - n = -a^2 m - a^2 - a b - b - n.So equate LHS and RHS:a m + a^3 n + a^2 b + a b + b = -a^2 m - a^2 - a b - b - nNow, since this equation must hold for all integers m and n, the coefficients of like terms must be equal. Let's collect terms for m, n, and constants.Coefficients of m:Left: aRight: -a^2Thus, a = -a^2 => a + a^2 = 0 => a(a + 1) = 0. So a = 0 or a = -1.Coefficients of n:Left: a^3Right: -1Thus, a^3 = -1 => a = -1 (since we are dealing with integers, so cube root of -1 is -1).Constant terms:Left: a^2 b + a b + bRight: -a^2 - a b - bSo set them equal:a^2 b + a b + b = -a^2 - a b - bLet me substitute a = -1 (since from coefficients of m and n, a must be -1). Let's check:If a = -1,Coefficients of m: a = -1, right side is -a^2 = -1. So -1 = -1, which holds.Coefficients of n: a^3 = (-1)^3 = -1, which matches the right side of -1.Now constants:Left: a^2 b + a b + b = (-1)^2 b + (-1)b + b = 1*b - b + b = b.Right: -a^2 - a b - b = -(-1)^2 - (-1)b - b = -1 + b - b = -1.Thus, b = -1.Therefore, if f is linear, then f(n) = a n + b = -1 * n + (-1) = -n -1.Let me check if this function satisfies the original equation.Let f(n) = -n -1.Check LHS: f(m + f(f(n))).First compute f(n) = -n -1.Then f(f(n)) = f(-n -1) = -(-n -1) -1 = n + 1 -1 = n.Then f(m + f(f(n))) = f(m + n).Since f(k) = -k -1, so f(m + n) = -(m + n) -1 = -m -n -1.RHS: -f(f(m + 1)) - n.First compute f(m + 1) = -(m + 1) -1 = -m -2.Then f(f(m + 1)) = f(-m -2) = -(-m -2) -1 = m + 2 -1 = m +1.Thus, -f(f(m +1)) - n = -(m +1) -n = -m -1 -n.Compare LHS and RHS: LHS is -m -n -1, RHS is -m -n -1. They are equal. So yes, the function f(n) = -n -1 satisfies the equation. Great!Therefore, f(n) = -n -1 is a solution. Now, the question is, is this the only solution? Since we assumed f was linear, but the problem only states that f is integer-valued. However, given that the equation is quite restrictive, and we found a solution through assuming linearity, and given that polynomials with integer coefficients are involved in g(n), it's possible that this is the unique solution.But let me check if there could be other solutions. Suppose f is not linear. However, from the equation f^3(n) = -c -n, where c = f(f(1)), if we can show that c is a constant. Wait, let's see:Earlier, we set m = 0 and found f(f(f(n))) = -f(f(1)) -n. Let me denote c = f(f(1)). Then f^3(n) = -c -n. So applying f three times gives a linear function. If f is invertible, then we might be able to find f in terms of this. But let's think.Suppose f is invertible. Then since f^3(n) = -c -n, we can apply f inverse three times to both sides. But maybe this is overcomplicating.Alternatively, maybe we can find f(0) or other values. Let me try to compute some specific values.Let me try setting n = 0 in the original equation. Wait, actually, the original equation is valid for all integers m and n. Let me try to set n = 0.Set n = 0:f(m + f(f(0))) = -f(f(m + 1)) - 0Simplify:f(m + f(f(0))) = -f(f(m +1))Hmm. Let me denote d = f(f(0)), so:f(m + d) = -f(f(m + 1)).But from earlier, when we set m = 0, we had f(f(f(n))) = -c -n. If we set n = 0 there, we get f(f(f(0))) = -c -0 = -c. But f(f(f(0))) is f(d) since d = f(f(0)), so f(d) = -c. So c = -f(d).But c = f(f(1)). So f(f(1)) = -f(d). Hmm. Not sure if that helps yet.Alternatively, maybe substitute m = -d in the equation f(m + d) = -f(f(m +1)). Then m = -d:f(-d + d) = f(0) = -f(f(-d +1)).Thus, f(0) = -f(f(-d +1)). So f(f(-d +1)) = -f(0). Hmm.Alternatively, let's think about surjectivity. Since f^3(n) = -c -n, for any integer k, we can solve for n such that f^3(n) = k. Let me see. Let k be any integer. Then set n = (-k -c). Then f^3(-k -c) = -c - (-k -c) = k. So f^3 is surjective, which implies that f itself is surjective. Because if f^3 is surjective, then f must be surjective. Because if f weren't surjective, then f^3 wouldn't hit all integers. So f is surjective.Similarly, is f injective? Suppose f(a) = f(b). Then applying f to both sides, f(f(a)) = f(f(b)). Then applying f again, f(f(f(a))) = f(f(f(b))). But f(f(f(a))) = -c -a and f(f(f(b))) = -c -b. Therefore, -c -a = -c -b => a = b. Therefore, f is injective.Since f is both injective and surjective, it's bijective. Therefore, f is a bijection on the integers.Knowing that f is bijective might help. Let's see. Let me try to find f inverse. Since f is bijective, there exists an inverse function f^{-1} such that f(f^{-1}(n)) = n and f^{-1}(f(n)) = n for all n.Given that f^3(n) = -c -n, let's see if we can express f in terms of f inverse. Let's compute f^3(n) = -c -n.If I apply f inverse three times to both sides:n = f^{-3}(-c -n)But not sure if helpful.Alternatively, let's note that since f is bijective, maybe we can find a relationship between f and its inverse.Wait, let's think again about the equation f(m + f(f(n))) = -f(f(m + 1)) -n. Let me try to substitute m with some expression. Maybe set m = k - f(f(n)), so that m + f(f(n)) = k. Then the left-hand side becomes f(k). Let's try that.Set m = k - f(f(n)):Then LHS: f(k - f(f(n)) + f(f(n))) = f(k)RHS: -f(f(k - f(f(n)) + 1)) -nSo we get:f(k) = -f(f(k - f(f(n)) + 1)) -nHmm. This seems complicated. Maybe another approach.Since we have f^3(n) = -c -n, where c = f(f(1)). Let me compute c in the linear solution we found. If f(n) = -n -1, then f(f(1)) = f(-1 -1) = f(-2) = -(-2) -1 = 2 -1 = 1. So c = 1. Then f^3(n) = -1 -n. Let's check:f(f(f(n))) = f(f(-n -1)) = f(-(-n -1) -1) = f(n +1 -1) = f(n). But f(n) = -n -1, and -c -n = -1 -n. Which matches. Wait, hold on:Wait, f^3(n) is f(f(f(n))). Let's compute f(f(f(n))) with f(n) = -n -1.First f(n) = -n -1.Then f(f(n)) = f(-n -1) = -(-n -1) -1 = n +1 -1 = n.Then f(f(f(n))) = f(n) = -n -1. But according to the earlier equation, f^3(n) = -c -n = -1 -n. So indeed, -n -1 = -1 -n. So that works.Wait, but in this case, f^3(n) = f(n). Because f(f(f(n))) = f(n). But according to our previous result, f^3(n) = -c -n. So in this case, f(n) = -c -n. Wait, that would mean f(n) = -1 -n, which matches c =1. So yes, f(n) = -c -n where c =1. So f(n) = -n -1. So this seems consistent.But how does this help us in the general case? Suppose that f^3(n) = -c -n. Then f^3 is linear, so maybe f itself is linear? Since composing linear functions gives linear functions. If f is linear, then f(n) = a n + b. Then f^3(n) = a^3 n + (a^2 + a +1) b. Wait, in our case, f^3(n) = -n -1. So:a^3 = -1,(a^2 + a +1) b = -1.From a^3 = -1, since a is integer, a = -1. Then,(a^2 + a +1) = (1 -1 +1) = 1. So 1 * b = -1 => b = -1.Hence, f(n) = -n -1. So this is the only linear solution. But could there be a non-linear solution?Given that f is bijective and f^3(n) is linear, perhaps f itself must be linear? Because if f is a bijection and its third iterate is linear, maybe f has to be linear. Let me think.Suppose f is not linear. However, f^3(n) is linear. Suppose f(n) is affine, which we already considered. If f is not affine, but perhaps quadratic or something. Let me test with a quadratic function.Suppose f(n) = a n^2 + b n + c. Then f(f(n)) would be a quadratic function composed with a quadratic function, resulting in a quartic function. Then f(f(f(n))) would be a degree 8 function. But according to the equation, f^3(n) must be linear. Therefore, f cannot be quadratic. Similarly, any higher-degree polynomial would result in higher-degree compositions, which can't be linear. Hence, f must be linear. Therefore, the only possible solutions are linear functions. Since we found a linear solution, and the constraints forced a = -1 and b = -1, this is the unique solution.Therefore, f(n) = -n -1 is the only solution. Thus, f(1991) = -1991 -1 = -1992.Now, moving on to the polynomial g(n). We need to find the most general form for g(n) with integer coefficients such that g(n) = g(f(n)) for all integers n. Since f(n) = -n -1, this means that g(n) = g(-n -1) for all integers n.So the polynomial g(n) satisfies g(n) = g(-n -1). We need to find all such polynomials with integer coefficients.Let me consider the substitution x = n. Then the equation becomes g(x) = g(-x -1). So for all x, g(x) - g(-x -1) = 0. So the polynomial g(x) - g(-x -1) is the zero polynomial.Therefore, we can write this as:g(x) - g(-x -1) = 0.We need to find all polynomials g(x) with integer coefficients satisfying this equation.Let me consider that the transformation x → -x -1 is an involution. Let me check:Let τ(x) = -x -1. Then τ(τ(x)) = -(-x -1) -1 = x +1 -1 = x. So τ is indeed an involution; applying it twice gives the identity. Therefore, the polynomial g must be invariant under this transformation. Therefore, g must be a polynomial in terms that are invariant under τ.To find such polynomials, we can consider the ring of polynomials invariant under τ. The invariant polynomials under τ are generated by the elementary symmetric functions of x and τ(x). Let's compute τ(x):τ(x) = -x -1.Let me find a polynomial h(x) such that h(x) = h(τ(x)). Let me look for an invariant under τ. Let me compute x + τ(x):x + (-x -1) = -1. So x + τ(x) = -1, which is a constant. Therefore, the sum is fixed. The product x * τ(x) = x*(-x -1) = -x^2 -x.Thus, perhaps the invariant ring is generated by some function. Alternatively, consider that any polynomial invariant under τ must satisfy g(x) = g(-x -1). Let me make a substitution to simplify this.Let y = x + 1/2. Then x = y - 1/2. Then τ(x) = -x -1 = -(y -1/2) -1 = -y +1/2 -1 = -y -1/2. So in terms of y, the transformation τ becomes y → -y. So the transformation is a reflection over the origin in terms of y.Therefore, if we set y = x + 1/2, then g(x) = g(-x -1) becomes g(y - 1/2) = g(-y -1/2). Let me denote G(y) = g(y - 1/2). Then the equation becomes G(y) = G(-y). Therefore, G(y) is an even function in y.Therefore, G(y) must be a polynomial in y^2. Since G(y) is even. Therefore, G(y) = h(y^2) for some polynomial h with coefficients in terms of y^2. Therefore, G(y) = h(y^2), so g(x) = G(y) = h((x + 1/2)^2). Therefore, g(x) is a polynomial in (x + 1/2)^2.But since g(x) has integer coefficients, we need to express this in terms of integer coefficients. Let me see. Let’s write (x + 1/2)^2 = x^2 + x + 1/4. However, if we consider polynomials in (x^2 + x), which is (x(x +1)), but this might not be sufficient. Alternatively, perhaps the substitution t = x + 1/2, but since we need integer coefficients, it's tricky.Wait, let's think differently. Since G(y) is even, and G(y) = g(y - 1/2), then G(y) must be a polynomial in y^2. Therefore, G(y) = a_0 + a_1 y^2 + a_2 y^4 + ... + a_k y^{2k}. Then, substituting back y = x + 1/2, we have g(x) = a_0 + a_1 (x + 1/2)^2 + a_2 (x + 1/2)^4 + ... + a_k (x + 1/2)^{2k}. However, since g(x) must have integer coefficients, each coefficient when expanded must be integer. Let's check (x + 1/2)^2:(x + 1/2)^2 = x^2 + x + 1/4. To have integer coefficients, the coefficients of x^2 + x + 1/4 must be integers, but 1/4 is a problem. Therefore, unless a_1 is a multiple of 4, but since a_1 is an integer coefficient, this term would introduce fractions. Similarly, higher powers would introduce fractions with denominators as powers of 2. Therefore, to have g(x) with integer coefficients, all coefficients a_i must be zero except possibly a_0. Wait, but even a_0 is problematic if there's a 1/4 term.Wait, perhaps there's a different approach. Let me consider that the relation g(n) = g(-n -1) must hold for all integers n, and g is a polynomial with integer coefficients. Let me consider the polynomial g(x) - g(-x -1). This polynomial is identically zero, so all its coefficients must be zero. Let me write g(x) as a general polynomial:g(x) = c_0 + c_1 x + c_2 x^2 + ... + c_k x^k.Then g(-x -1) = c_0 + c_1 (-x -1) + c_2 (-x -1)^2 + ... + c_k (-x -1)^k.Compute the difference g(x) - g(-x -1):= [c_0 - c_0] + [c_1 x - c_1 (-x -1)] + [c_2 x^2 - c_2 (-x -1)^2] + ... + [c_k x^k - c_k (-x -1)^k]Simplify term by term:First term: 0.Second term: c_1 x - c_1 (-x -1) = c_1 x + c_1 x + c_1 = 2 c_1 x + c_1.Third term: c_2 x^2 - c_2 (x^2 + 2x +1) = c_2 x^2 - c_2 x^2 - 2 c_2 x - c_2 = -2 c_2 x - c_2.Fourth term: For degree 3:c_3 x^3 - c_3 (-x -1)^3 = c_3 x^3 - c_3 (-x^3 -3x^2 -3x -1) = c_3 x^3 + c_3 x^3 + 3 c_3 x^2 + 3 c_3 x + c_3 = 2 c_3 x^3 + 3 c_3 x^2 + 3 c_3 x + c_3.Similarly, higher terms will involve more complex combinations. Since the entire difference must be the zero polynomial, each coefficient of x^m must be zero. Let's consider this for each degree.Let me denote the difference as:g(x) - g(-x -1) = Σ_{m=0}^k d_m x^m = 0.Therefore, each coefficient d_m = 0.Let's compute d_m for each m.For each term in the expansion:For each degree m, the coefficient d_m is obtained by subtracting the coefficients of x^m in g(-x -1) from the coefficient of x^m in g(x). So:d_m = c_m - [coefficient of x^m in g(-x -1)].Let me compute coefficients of g(-x -1):Given g(-x -1) = Σ_{i=0}^k c_i (-x -1)^i.The coefficient of x^m in g(-x -1) is Σ_{i=m}^k c_i * (-1)^i * C(i, m) * 1^{i - m} }.Wait, binomial theorem: (-x -1)^i = Σ_{j=0}^i C(i, j) (-x)^j (-1)^{i - j} } = Σ_{j=0}^i (-1)^{i - j} C(i, j) x^j.Wait, actually, (-x -1)^i = (-1)^i (x +1)^i. Then expanding (x +1)^i gives Σ_{j=0}^i C(i, j) x^j. So:(-x -1)^i = (-1)^i Σ_{j=0}^i C(i, j) x^j. Therefore, the coefficient of x^m in (-x -1)^i is (-1)^i C(i, m).Therefore, the coefficient of x^m in g(-x -1) is Σ_{i=m}^k c_i (-1)^i C(i, m).Thus, d_m = c_m - Σ_{i=m}^k c_i (-1)^i C(i, m).But since d_m = 0 for all m, we have:c_m = Σ_{i=m}^k c_i (-1)^i C(i, m) for each m from 0 to k.This gives a system of equations. Let's consider this for each m.Starting from the highest degree m = k:For m = k:d_k = c_k - Σ_{i=k}^k c_i (-1)^i C(k, k) = c_k - c_k (-1)^k * 1 = c_k (1 - (-1)^k) = 0.Thus, c_k (1 - (-1)^k) = 0. Since we are working over integers, this implies either c_k = 0 or 1 - (-1)^k = 0. 1 - (-1)^k = 0 implies (-1)^k =1, so k even. Therefore:If k is even, then for m =k, 1 - (-1)^k = 0, so equation holds for any c_k. If k is odd, then 1 - (-1)^k = 2, so c_k * 2 =0 => c_k =0.But wait, in our problem, the polynomial g has integer coefficients, but the degree isn't specified. However, since g is equal to its composition with f, which is affine, the degree might be constrained. Wait, let me think.But perhaps we can approach this differently. Let's note that the relation g(n) = g(-n -1) must hold for all integers n. Therefore, the polynomial g(x) - g(-x -1) is identically zero. Let's try to find the general form of such polynomials.Let me consider that the transformation x → -x -1 can be represented as a linear transformation. Let me make a substitution to diagonalize this transformation or find eigenfunctions.Let me set t = x + 1/2. Then as before, x = t - 1/2. Then:g(x) = g(-x -1) => g(t - 1/2) = g(- (t - 1/2) -1) = g(-t + 1/2 -1) = g(-t -1/2).So in terms of t, the equation becomes g(t - 1/2) = g(-t -1/2). Let me denote h(t) = g(t - 1/2). Then the equation is h(t) = h(-t). Therefore, h(t) is an even function. Therefore, h(t) must be a polynomial in t^2.Since h(t) is even, it can be written as h(t) = a_0 + a_1 t^2 + a_2 t^4 + ... + a_n t^{2n}.Therefore, substituting back, g(x) = h(t) = h(x + 1/2) = a_0 + a_1 (x + 1/2)^2 + a_2 (x + 1/2)^4 + ... + a_n (x + 1/2)^{2n}.However, since g(x) must have integer coefficients, we need each term when expanded to have integer coefficients. Let's check the expansion of (x + 1/2)^{2m}:The expansion is Σ_{k=0}^{2m} C(2m, k) x^k (1/2)^{2m -k}.This will introduce denominators of 2^{2m -k}. To have integer coefficients, all these denominators must be canceled by the coefficients a_m. However, since a_m are integers, unless the coefficients a_m are chosen such that when multiplied by (1/2)^{2m -k}, the result is integer. This seems restrictive.Wait, but perhaps instead of working with t = x + 1/2, we can find a different substitution. Let me note that (x + 1/2)^2 = x^2 + x + 1/4. If we let y = x(x +1). Then, note that:y = x^2 + x.Then, compute y under the transformation x → -x -1:y' = (-x -1)(-x -1 +1) = (-x -1)(-x) = x(x +1) = y. So y is invariant under x → -x -1.Therefore, x(x +1) is an invariant. Thus, any polynomial in y = x(x +1) will satisfy g(x) = g(-x -1). Therefore, such polynomials are precisely the polynomials in y = x(x +1).Moreover, since x(x +1) generates the invariant ring under the transformation x → -x -1, any polynomial invariant under this transformation must be a polynomial in x(x +1). Therefore, the most general form of g is a polynomial in x(x +1). Since x(x +1) has integer coefficients, any polynomial in x(x +1) with integer coefficients will satisfy g(x) = g(-x -1).Therefore, the most general form for g is g(n) = P(n(n +1)), where P is a polynomial with integer coefficients.Let me verify this. Suppose g(n) = P(n(n +1)) for some polynomial P with integer coefficients. Then g(-n -1) = P((-n -1)(-n -1 +1)) = P((-n -1)(-n)) = P(n(n +1)) = g(n). Therefore, this works.Conversely, suppose that g(n) is a polynomial with integer coefficients such that g(n) = g(-n -1). Then, as we saw, g must be a polynomial in n(n +1). Therefore, the most general form is as stated.Therefore, putting it all together:f(n) = -n -1, so f(1991) = -1991 -1 = -1992.The most general polynomial g is any polynomial in n(n +1) with integer coefficients, i.e., g(n) = P(n(n +1)) where P is a polynomial with integer coefficients.Final AnswerThe value of ( f(1991) ) is (boxed{-1992}), and the most general form for ( g ) is ( g(n) = P(n(n + 1)) ) where ( P ) is a polynomial with integer coefficients.

✅:To solve the problem, we start by analyzing the functional equation ( f(m + f(f(n))) = -f(f(m + 1)) - n ). By substituting specific values for ( m ) and ( n ), we find that ( f ) must be a linear function. 1. Substituting ( m = 0 ): [ f(f(f(n))) = -f(f(1)) - n ] This implies ( f^3(n) = -c - n ) where ( c = f(f(1)) ).2. Assuming ( f(n) = an + b ): - Substituting into the functional equation and equating coefficients, we find ( a = -1 ) and ( b = -1 ). - This gives ( f(n) = -n - 1 ), which satisfies the original equation.3. Verification: - Check that ( f(n) = -n - 1 ) satisfies the functional equation: - Left-hand side: ( f(m + f(f(n))) = f(m + n) = -(m + n) - 1 ) - Right-hand side: ( -f(f(m + 1)) - n = -f(-m - 2) - n = -(m + 1) - n ) - Both sides are equal, confirming ( f(n) = -n - 1 ).4. Finding ( f(1991) ): - ( f(1991) = -1991 - 1 = -1992 ).Next, we determine the polynomial ( g(n) ) such that ( g(n) = g(f(n)) ). Given ( f(n) = -n - 1 ), we need ( g(n) = g(-n - 1) ).5. Invariant Polynomial: - The transformation ( n to -n - 1 ) is an involution. The invariant polynomial under this transformation is ( n(n + 1) ). - Therefore, ( g(n) ) must be a polynomial in ( n(n + 1) ). The most general form is ( g(n) = P(n(n + 1)) ) where ( P ) is a polynomial with integer coefficients. Final AnswerThe value of ( f(1991) ) is (boxed{-1992}), and the most general form for ( g ) is ( g(n) = P(n(n + 1)) ) where ( P ) is a polynomial with integer coefficients.

🔑:We are given an integer-valued function ( f(n) ) which satisfies the functional equation [ f(m + f(f(n))) = -f(f(m + 1)) - n ]for all integers ( m ) and ( n ). Additionally, we are asked to find a polynomial ( g(n) ) with integer coefficients such that ( g(n) = g(f(n)) ) for all ( n ).1. Analyzing the functional equation: Let's first substitute ( m = 1 ) into the given functional equation: [ f(1 + f(f(n))) = -f(f(2)) - n ] Define ( k = f(f(2)) ). Then the equation becomes: [ f(f(f(n)) + 1) = -n - k tag{1} ]2. Functional equation modification: Replace ( m ) by ( f(f(m)) ) in the original equation: [ f(f(f(m)) + f(f(n))) = -f(f(f(m) + 1)) - n ] Using equation (1), we substitute ( f(f(f(m) + 1)) = -m - k ): [ f(f(f(m)) + f(f(n))) = -(-m - k) - n = m + k - n tag{2} ]3. Symmetry in the equation: Swap ( m ) and ( n ) in equation (2): [ f(f(f(n)) + f(f(m))) = n + k - m ] Equate the left-hand sides of the two equations: [ f(f(f(m)) + f(f(n))) = f(f(f(n)) + f(f(m))) ] It implies: [ m + k - n = n + k - m ] Simplifying gives: [ 2m = 2n ] Hence, ( m = n ) for all integers ( m ) and ( n ). This shows that ( f(f(n)) ) is essentially a reflection about ( f ) itself.4. Form of ( f(n) ): We find the difference: [ f(n) - f(m) = m - n ] for all ( m ) and ( n ). Set ( m = 0 ) to find a specific constant ( h = f(0) ): [ f(n) = h - n ] Substitute this into the original equation and simplify: [ h - m - f(f(n)) = -f(f(m + 1)) - n ] With ( h = f(0) ): [ h - m - (h - (f(n)) = -f(h - (m + 1)) - n ] Trying ( m = n + 1 ) simplifies this: [ h = -1 ] Hence, [ f(n) = -(n + 1) ]5. For the polynomial ( g(n) ): We need ( g(n) = g(-n-1) ) for all integers ( n ): Define ( h(x) = g(x - frac{1}{2}) ). Then: [ h(-x) = g(-x - frac{1}{2}) = g(x - frac{1}{2}) ] Hence, [ h(x) text{ has only even powers of } x ] This implies that ( g(x) ) is a polynomial in ( x^2 + x ).6. Conclusion: So, we conclude: [ f(1991) = -1992 ] and [ g(x) text{ is a polynomial in } (x^2 + x) text{ with integer coefficients.} ] [ boxed{-1992} ]

🧠:Okay, let me try to figure out this aquarium fish problem. So, there are three types of fish: goldfish, silverfish, and red fish. The problem gives two scenarios about a cat eating all the goldfish or all the red fish and how that affects the total number of fish remaining. The question is asking which type is more numerous between goldfish and silverfish, and by how much. First, I need to parse the information given. Let me denote the original number of goldfish as G, silverfish as S, and red fish as R. So, the total original number of fish is G + S + R. The first scenario: if the cat eats all the goldfish, the remaining fish are S + R. According to the problem, this number is 1 less than 2/3 of the original number. So, mathematically, that would be S + R = (2/3)(G + S + R) - 1. The second scenario: if the cat eats all the red fish, the remaining fish are G + S. The problem states this is 4 more than 2/3 of the original number. So, G + S = (2/3)(G + S + R) + 4.Hmm, okay. So we have two equations here. Let me write them out again:1. S + R = (2/3)(Total) - 12. G + S = (2/3)(Total) + 4Where Total = G + S + R.Maybe we can subtract these two equations to eliminate the (2/3)Total term? Let me try that. Subtract equation 1 from equation 2:(G + S) - (S + R) = [(2/3)(Total) + 4] - [(2/3)(Total) - 1]Simplify left side: G + S - S - R = G - RRight side: 2/3 Total + 4 - 2/3 Total + 1 = 5So, G - R = 5. That tells us that the number of goldfish is 5 more than red fish. Interesting. So G = R + 5.Okay, now let's see if we can find another relation. Let me take equation 1 and equation 2 and express them in terms of Total. From equation 1: S + R = (2/3)Total - 1. Let's rearrange that to find S + R + 1 = (2/3)Total. Similarly, equation 2: G + S = (2/3)Total + 4. Rearranged, G + S - 4 = (2/3)Total.Wait, both of these equal (2/3)Total. So we can set them equal to each other:S + R + 1 = G + S - 4Simplify: S + R +1 = G + S - 4Cancel S from both sides: R + 1 = G - 4But from earlier, we found that G = R + 5. Let's substitute that into this equation.R + 1 = (R + 5) - 4Simplify right side: R +5 -4 = R +1So left side: R +1 = R +1. Which is an identity, so that doesn't give us new information. Hmm, so perhaps I need to approach this differently.Let me denote Total as T. So T = G + S + R.Equation 1: S + R = (2/3)T - 1Equation 2: G + S = (2/3)T + 4From equation 1: S + R = (2/3)T - 1. Since T = G + S + R, substitute T into the equation:S + R = (2/3)(G + S + R) - 1Multiply both sides by 3 to eliminate fractions:3(S + R) = 2(G + S + R) - 3Expand right side: 3S + 3R = 2G + 2S + 2R - 3Subtract 2S + 2R from both sides: S + R = 2G - 3Similarly, from equation 2: G + S = (2/3)T + 4. Substitute T:G + S = (2/3)(G + S + R) + 4Multiply both sides by 3:3G + 3S = 2G + 2S + 2R + 12Subtract 2G + 2S from both sides: G + S = 2R + 12So now, from equation 1 processed: S + R = 2G -3From equation 2 processed: G + S = 2R + 12And we also have from earlier: G = R + 5So let's substitute G = R + 5 into these equations.First equation: S + R = 2(R +5) -3 => S + R = 2R +10 -3 => S + R = 2R +7 => Subtract R from both sides: S = R +7Second equation: (R +5) + S = 2R +12 => R +5 + S = 2R +12 => S = 2R +12 - R -5 => S = R +7So both equations give S = R +7. That's consistent. So now we have:G = R +5S = R +7Total T = G + S + R = (R +5) + (R +7) + R = 3R +12Now, we can check if these satisfy the original equations.First original equation: S + R = (2/3)T -1Left side: (R +7) + R = 2R +7Right side: (2/3)(3R +12) -1 = 2(R +4) -1 = 2R +8 -1 = 2R +7So yes, matches.Second equation: G + S = (2/3)T +4Left side: (R +5) + (R +7) = 2R +12Right side: (2/3)(3R +12) +4 = 2(R +4) +4 = 2R +8 +4 = 2R +12Also matches. So all equations are consistent.Now, the question is asking which is more numerous: goldfish or silverfish, and by how much.From above, we have:G = R +5S = R +7Therefore, S = (R +5) +2 = G +2So silverfish are 2 more than goldfish.Therefore, silverfish are more numerous by 2.Wait, let me check again:If S = R +7 and G = R +5, then S - G = (R +7) - (R +5) = 2. Yes, so silverfish are 2 more than goldfish. So the answer is silverfish are more numerous by 2.But let me just confirm all steps once more to make sure I didn't make a mistake.Original variables:Total T = G + S + RFirst condition: After eating goldfish, remaining fish S + R = (2/3)T -1Second condition: After eating red fish, remaining fish G + S = (2/3)T +4We derived:G = R +5S = R +7Therefore, G = R +5, S = R +7, so S = G +2Therefore, silverfish are 2 more than goldfish.Yes, seems correct.Alternatively, perhaps to see in terms of actual numbers. Let's pick a value for R and compute others.Suppose R = 0 (just for testing), then G =5, S=7, T=5+7+0=12First condition: S + R =7 +0=7. 2/3 of T is 8, 8 -1=7. Correct.Second condition: G + S=12. 2/3 of T is 8, 8 +4=12. Correct.So if R=0, then S=7, G=5. Difference is 2. So seems correct.Alternatively, R=1, then G=6, S=8, T=15.First condition: S + R=8+1=9. 2/3 of 15 is 10, 10 -1=9. Correct.Second condition: G + S=14. 2/3 of 15 is 10, 10 +4=14. Correct.So in both cases, S - G=2. So regardless of R, the difference is 2. So answer is silverfish are more numerous by 2.Hence, the answer is silverfish are more numerous than goldfish by 2.

✅:Solution:Let ( G ), ( S ), and ( R ) represent the number of goldfish, silverfish, and red fish, respectively. The total number of fish is ( T = G + S + R ).Given Conditions:1. If all goldfish are eaten, the remaining fish ( S + R ) equal ( frac{2}{3}T - 1 ): [ S + R = frac{2}{3}T - 1 quad text{(Equation 1)} ]2. If all red fish are eaten, the remaining fish ( G + S ) equal ( frac{2}{3}T + 4 ): [ G + S = frac{2}{3}T + 4 quad text{(Equation 2)} ]Deriving Relationships:- Subtract Equation 1 from Equation 2: [ (G + S) - (S + R) = left(frac{2}{3}T + 4right) - left(frac{2}{3}T - 1right) ] Simplifying: [ G - R = 5 implies G = R + 5 quad text{(Equation 3)} ]- Substitute ( T = G + S + R ) into Equation 1: [ S + R = frac{2}{3}(G + S + R) - 1 ] Multiply by 3 and simplify: [ 3S + 3R = 2G + 2S + 2R - 3 implies S + R = 2G - 3 quad text{(Equation 4)} ]- Substitute ( T = G + S + R ) into Equation 2: [ G + S = frac{2}{3}(G + S + R) + 4 ] Multiply by 3 and simplify: [ 3G + 3S = 2G + 2S + 2R + 12 implies G + S = 2R + 12 quad text{(Equation 5)} ]Solving the System:- Substitute ( G = R + 5 ) (from Equation 3) into Equations 4 and 5: - Equation 4 becomes: [ S + R = 2(R + 5) - 3 implies S = R + 7 ] - Equation 5 becomes: [ (R + 5) + S = 2R + 12 implies S = R + 7 ]Conclusion:- Goldfish: ( G = R + 5 )- Silverfish: ( S = R + 7 )The difference between silverfish and goldfish is:[S - G = (R + 7) - (R + 5) = 2]Answer: Silverfish are more numerous than goldfish by 2.

🔑:To solve the problem, let us denote:- The total number of fish in the aquarium as ( n ).- The number of goldfish as ( G ).- The number of silverfish as ( S ).- The number of redfish as ( R ).We are given two conditions:1. If the cat eats all the goldfish, the number of remaining fish is ( frac{2}{3}n - 1 ).2. If the cat eats all the redfish, the number of remaining fish is ( frac{2}{3}n + 4 ).We can express these conditions mathematically as follows:# Condition 1:[ n - G = frac{2}{3}n - 1 ]# Condition 2:[ n - R = frac{2}{3}n + 4 ]Let’s solve these equations step-by-step.1. Solving for ( G ) (Number of Goldfish) Begin with the first equation: [ n - G = frac{2}{3}n - 1 ] Rearrange to isolate ( G ): [ G = n - left(frac{2}{3}n - 1right) ] Simplify inside the parenthesis: [ G = n - frac{2}{3}n + 1 ] Combine the terms involving ( n ): [ G = frac{3}{3}n - frac{2}{3}n + 1 = frac{1}{3}n + 1 ]2. Solving for ( R ) (Number of Redfish) Now consider the second equation: [ n - R = frac{2}{3}n + 4 ] Rearrange to isolate ( R ): [ R = n - left(frac{2}{3}n + 4right) ] Simplify inside the parenthesis: [ R = n - frac{2}{3}n - 4 ] Combine the terms involving ( n ): [ R = frac{3}{3}n - frac{2}{3}n - 4 = frac{1}{3}n - 4 ]3. Finding Number of Silverfish ( S ) The total number of fish is ( n ), so: [ G + S + R = n ] Substitute the expressions for ( G ) and ( R ): [ left( frac{1}{3}n + 1 right) + S + left( frac{1}{3}n - 4 right) = n ] Combine terms involving ( n ): [ frac{1}{3}n + frac{1}{3}n + 1 - 4 + S = n ] Simplify: [ frac{2}{3}n - 3 + S = n ] Rearrange to solve for ( S ): [ S = n - frac{2}{3}n + 3 ] Combine terms involving ( n ): [ S = frac{3}{3}n - frac{2}{3}n + 3 = frac{1}{3}n + 3 ]4. Comparing the Numbers Now we compare the numbers of goldfish and silverfish. We need to determine whether ( G ) or ( S ) is greater: [ G = frac{1}{3}n + 1 quad text{and} quad S = frac{1}{3}n + 3 ] Clearly, since ( frac{1}{3}n + 3 > frac{1}{3}n + 1 ), it shows that the number of silverfish ( S ) is greater than the number of goldfish ( G ). Difference: [ S - G = left( frac{1}{3}n + 3 right) - left( frac{1}{3}n + 1 right) = 3 - 1 = 2 ]The number of silverfish is greater than the number of goldfish by 2. Conclusion:[boxed{2}]

🧠:Okay, so I need to prove that if a set M on a plane has the property that every point in M is the midpoint of some line segment connecting two other points in M, then M must contain infinitely many points. Hmm, interesting. Let me start by understanding the problem thoroughly.First, let me restate the problem in my own words. We have a set M of points in the plane. For every point m in M, there exist two points p and q in M such that m is the midpoint of the segment pq. The goal is to show that such a set M cannot be finite; it must have infinitely many points.Alright, so maybe I can approach this by contradiction. Suppose M is a finite set. Then, there must be some points in M that can't be midpoints unless they are generated by other points. But wait, if every point is a midpoint, then each point is determined by two others. However, if the set is finite, there might be some dependencies or cycles that lead to a contradiction.Let me think about a simple example first. Suppose M has only one point. Then, that point would have to be the midpoint of a segment connecting two points in M. But there's only one point, so we need two points in M to generate it as a midpoint. But there aren't two points, so this is impossible. Hence, M cannot have just one point.Similarly, if M has two points, say A and B. Then, each of these points must be the midpoint of some segment connecting two points in M. The midpoint of A and B is (A+B)/2. But if A is supposed to be the midpoint of two points in M, those two points would have to be such that their average is A. Similarly for B. Let's denote the midpoint formula. If A is the midpoint of points P and Q, then P + Q = 2A. Similarly, if B is the midpoint of points R and S, then R + S = 2B.But since M only has A and B, then P, Q, R, S must be either A or B. Let's see. If A is the midpoint, then P + Q = 2A. If both P and Q are A, then A + A = 2A, which works, but that would mean the segment from A to A, which is just a single point. But midpoints are defined for distinct points, right? Wait, the problem doesn't specify that the line segment has to be non-degenerate. So maybe a segment from a point to itself is allowed? But then the midpoint would just be the same point. But if the problem allows that, then A can be the midpoint of the "segment" from A to A. But is that acceptable?Wait, the problem states that each point in M is the midpoint of a line segment connecting two points in M. If the two points are the same, then the line segment is just a single point, and its midpoint is itself. So, in that case, if M has a single point, that point would trivially be the midpoint of the "segment" from itself to itself. But earlier, the problem said "a line segment connecting two points", which could be interpreted as two distinct points. But the problem statement doesn't specify distinctness. This is a critical point.If the problem allows a point to be the midpoint of a degenerate segment (i.e., a segment where both endpoints are the same), then any singleton set M would satisfy the condition because the single point is the midpoint of the "segment" from itself to itself. However, the problem statement probably intends for the segments to be non-degenerate, meaning the two endpoints are distinct. Otherwise, the problem is trivial and the conclusion that M is infinite would be false (since a singleton set would satisfy the condition but be finite).Therefore, I need to assume that the two points connected by the segment are distinct. So, the problem should be interpreted as: each point in M is the midpoint of a line segment connecting two distinct points in M. Otherwise, if degenerate segments are allowed, then finite sets are possible. Given that the problem asks to prove M must be infinite, we can safely assume that the segments must be non-degenerate. Therefore, each point in M is the midpoint of a segment connecting two distinct points in M.So, going back. If M is finite, then each point is the midpoint of a segment between two distinct points in M. Let's try to see what happens with small numbers.Case 1: M has one point. As before, impossible because we need two distinct points to form a midpoint, but there's only one.Case 2: M has two points, A and B. Then, the midpoints of AB is (A+B)/2. So, (A+B)/2 must be in M. But M only has A and B. So, unless (A+B)/2 is either A or B, which would require A = B, which is not allowed since they are distinct. Therefore, M cannot have two points either.Wait, but if M had two points, but each is the midpoint of a segment between two distinct points. But with two points, the only possible midpoints are the midpoint between A and B, which is a new point not in M. Therefore, M cannot have two points.Case 3: M has three points. Let's say A, B, and C. Each of these must be the midpoint of a segment between two distinct points in M. Let's see.Suppose A is the midpoint of B and C. Then, A = (B + C)/2. Similarly, B must be the midpoint of two points in M. So, B = (A + D)/2, but D must be in M. However, M only has A, B, C. So D must be one of these. Similarly for C.Wait, let's write equations. Let me assign coordinates to make this concrete. Let me place the points in a coordinate system to simplify calculations.Suppose A is at (0,0), B is at (2a, 0), so that the midpoint between them is (a, 0). If that midpoint is supposed to be another point in M, say C. Then C is at (a, 0). Then, C is the midpoint of A and B. Now, each point must be a midpoint. So A must be the midpoint of two points in M. Let's see. If A is (0,0), then there must exist points P and Q in M such that (P + Q)/2 = (0,0). Therefore, P + Q = (0,0). Similarly for B and C.So, for point A: there are points P and Q in M such that P + Q = (0,0). The points in M are A(0,0), B(2a, 0), and C(a, 0). So possible pairs:If P and Q are both A: (0,0) + (0,0) = (0,0). But they need to be distinct points, so P and Q cannot both be A. So possible pairs are A and B, A and C, B and C.If we take P = B and Q = something. Let's see: If P = B(2a,0) and Q = ?, then P + Q = (2a + Q_x, 0 + Q_y) = (0,0). Therefore, Q must be (-2a, 0). But (-2a, 0) is not in M. So that's a problem. Similarly, if P = C(a, 0), then Q must be (-a, 0), which is not in M. If P = A(0,0) and Q = something, then Q would have to be (0,0), but they need to be distinct. Therefore, there's no way to have A be the midpoint of two distinct points in M. Therefore, the three-point case also fails.Therefore, three points are impossible. Hmm. So maybe any finite set is impossible? Then how do we proceed?Perhaps induction. Suppose that for any finite set with n points, it's impossible to satisfy the condition, hence the set must be infinite.Alternatively, think in terms of vector spaces. If M is a subset of the plane (which is a vector space over ℝ), and every point in M is the midpoint of two other points in M, then M must be closed under the operation of taking midpoints. But closure under midpoints implies that M is an affine subspace. But the only affine subspaces of the plane are points, lines, or the entire plane. Since each point is a midpoint, it can't be a single point. If it's a line, then being closed under midpoints would mean it's a dense set or the whole line. But even a line would require infinitely many points. Wait, but the problem is set in the plane, which is continuous, but M could be a discrete set? Wait, but if you require that every point is the midpoint of two others, then even on a line, you can't have a finite set.Wait, for example, suppose M is a set of points on a line. If it's finite, say with points at positions x₁, x₂, ..., xₙ. Then each x_i must be the average of two other points. But similar to the earlier reasoning, this creates an infinite chain. For example, suppose you have points at 0 and 1. Then the midpoint is 0.5, which must be included. Then the midpoint of 0 and 0.5 is 0.25, which must be included, and so on. So you get an infinite set.Alternatively, if we think in terms of groups or generators. If every point is the midpoint of two others, then starting from any initial point, you can generate more points by taking midpoints, but each time you take a midpoint, you might need to include more points. This process would never terminate unless you end up in a loop, but with midpoints, loops would require some kind of symmetry or periodicity.But how do we formalize this?Alternatively, suppose M is non-empty and satisfies the condition. Take any point m₀ in M. Then, m₀ is the midpoint of two points m₁ and m₂ in M. Then, m₁ is the midpoint of two points m₃ and m₄, and so on. If this process continues indefinitely without repeating points, then M is infinite. If it does repeat, then perhaps we can find a cycle, which would impose some conditions that lead to a contradiction.Wait, suppose there is a finite set M. Then, since each point is a midpoint, we can model this as a directed graph where each node is a point, and each node has two outgoing edges pointing to the two points whose midpoint it is. In a finite graph, by the pigeonhole principle, some nodes must repeat, leading to cycles. But can such a cycle exist?Let me try to construct a cycle. Suppose we have a point A which is the midpoint of B and C. Then, B is the midpoint of D and E, and C is the midpoint of F and G, and so on. If we eventually loop back to A, such that some later point is the midpoint involving A. But this seems complex. Let's attempt a simple cycle.Suppose we have points A, B, C such that:- A is the midpoint of B and C,- B is the midpoint of C and A,- C is the midpoint of A and B.Is this possible? Let's check with coordinates.Let me assign coordinates. Let’s say A = (0,0), B = (a,b), C = (c,d). Then:Since A is the midpoint of B and C:( (a + c)/2 , (b + d)/2 ) = (0,0)Therefore:a + c = 0b + d = 0So c = -a, d = -b. Thus, C = (-a, -b).Now, B is the midpoint of C and A. Let's check:Midpoint of C and A is ( (-a + 0)/2, (-b + 0)/2 ) = (-a/2, -b/2 ). But this should be equal to B = (a, b). Therefore:-a/2 = a ⇒ -a/2 = a ⇒ -3a/2 = 0 ⇒ a = 0Similarly, -b/2 = b ⇒ -3b/2 = 0 ⇒ b = 0Therefore, B = (0,0), which is the same as A. But we assumed points are distinct. Therefore, such a cycle is impossible with distinct points.Therefore, a 3-cycle where each point is the midpoint of the other two leads to all points coinciding, which violates the distinctness. Therefore, such a cycle cannot exist with distinct points.Therefore, in a finite set, the graph of midpoints must have cycles, but such cycles collapse to a single point, which is a contradiction. Therefore, no finite set can exist where each point is the midpoint of two distinct others.Alternatively, think in terms of linear algebra. If M is a finite set, then consider the vector space generated by M over the real numbers. Since M is finite, the vector space is finite-dimensional, but the midpoint operation corresponds to averaging, which is a convex combination. However, since every point is required to be the midpoint, this would necessitate the set being closed under certain linear combinations. However, in finite-dimensional spaces, generating new points via midpoints would require the set to be infinite, as each midpoint generates a new point which in turn requires its own midpoints, ad infinitum.Alternatively, consider that if M is finite, then we can look at the convex hull of M. The convex hull is a convex polygon, and any midpoint of two points in M lies within the convex hull. However, if M is required to contain all such midpoints, then M would have to include all the points in the convex hull, which is infinite unless the convex hull is a single point, which would require all points to coincide, but that would contradict the distinctness required for midpoints.Wait, but even if the convex hull is a polygon with more than one point, the midpoints would generate new points inside, which would then require their own midpoints, leading again to an infinite set.But maybe this is too hand-wavy. Let me try a more formal approach.Assume M is finite. Then, consider the set of all x-coordinates of points in M. Since M is finite, there are finitely many x-coordinates. Let x_min be the smallest x-coordinate and x_max be the largest. Consider the point in M with x-coordinate x_max. This point must be the midpoint of two points in M. The x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints. Therefore, (p_x + q_x)/2 = x_max. Since x_max is the maximum, p_x and q_x must both be equal to x_max, because if either were less, the average would be less than x_max. Therefore, both endpoints must have x-coordinate x_max. But then their midpoint also has x-coordinate x_max. However, for the point with x-coordinate x_max to be the midpoint of two distinct points, those two points must both have x-coordinate x_max and distinct y-coordinates. But then the average of their y-coordinates must be the y-coordinate of the midpoint. So, suppose we have two points (x_max, y1) and (x_max, y2). Their midpoint is (x_max, (y1 + y2)/2). Therefore, for the point (x_max, (y1 + y2)/2) to be in M, it must itself be the midpoint of two points in M. However, applying the same logic to the y-coordinate, the y-coordinate of this midpoint is (y1 + y2)/2. If this is to be the midpoint of two points with x-coordinate x_max, then their y-coordinates must average to (y1 + y2)/2, meaning the two points would have y-coordinates [(y1 + y2)/2 + a] and [(y1 + y2)/2 - a] for some a. However, unless a = 0, these points would not be in M because M has a finite number of points. Therefore, unless y1 = y2, which would make the midpoint coincide with them, but they need to be distinct. This leads to a contradiction because we cannot have infinitely many points with x-coordinate x_max in a finite set M. Therefore, our assumption that M is finite must be wrong.Similarly, this argument applies to the minimal x-coordinate, minimal and maximal y-coordinates, etc. Therefore, in any finite set, the extremal points (those with maximal or minimal coordinates) cannot exist because they would require other points with the same extremal coordinates, leading to an infinite regress.Therefore, this suggests that M cannot have extremal points, which is impossible in a finite set, as every finite set in the plane has well-defined extremal points (like the convex hull vertices). Hence, M must be infinite.Another angle: think of M as a vector space over the field of rational numbers. If every point is the midpoint of two others, then M is closed under the operation of taking midpoints, which is equivalent to being closed under the operation (p + q)/2. If we start with any point m₀ in M, then we can generate an infinite sequence of points by repeatedly taking midpoints with existing points, leading to an infinite set. For example, starting with m₀, we need m₁ and m₂ such that m₀ is their midpoint. Then, m₁ needs to be the midpoint of two other points, say m₃ and m₄, and so on. Unless this process cycles, which would require some linear dependence, but in the plane, such dependencies would force all points to coincide, which isn't allowed. Therefore, the set must be infinite.Alternatively, use induction. Suppose that for any set M with n points, it's impossible for all n points to satisfy the midpoint condition. Base cases n=1,2,3 as discussed are impossible. Assume for some k >= 3, any set with k points cannot satisfy the condition. Then, a set with k+1 points would also have to violate the condition. However, induction might not be straightforward here because adding a point might not directly relate to the previous cases. Maybe another approach.Consider the set M in the plane. Define a function f: M × M → M such that f(p, q) is the midpoint of p and q. Since every point in M is a midpoint, the function f is surjective. However, if M is finite, the number of pairs (p, q) is finite (specifically, |M|²), but the number of midpoints is also finite. However, since each midpoint is determined by at least one pair, and some pairs may produce the same midpoint. The question is whether it's possible for a finite set to have such a surjective function.But in a finite set, the number of distinct midpoints is at most C(|M|, 2) + |M| (if allowing degenerate segments). But if we disallow degenerate segments, it's C(|M|, 2). However, if we require that every point is a midpoint, then |M| <= C(|M|, 2). Solving this inequality:|M| <= |M|(|M| - 1)/2Multiply both sides by 2:2|M| <= |M|² - |M|Simplify:0 <= |M|² - 3|M|Which simplifies to |M|(|M| - 3) >= 0Therefore, |M| >= 3 or |M| <= 0. But |M| cannot be <=0. So this tells us that if |M| is finite and >=3, then it's possible that |M| <= C(|M|,2). But this doesn't directly give a contradiction. For example, if |M| = 3, then C(3,2) = 3, so |M| = 3 <= 3, which is possible. However, we saw earlier that with three points, it's impossible to satisfy the midpoint condition. Therefore, this inequality is necessary but not sufficient. Hence, the counting argument alone isn't enough.But maybe considering that each midpoint can be generated by multiple pairs, but each pair generates only one midpoint. If the function f is surjective, then every point in M is achieved by at least one pair. However, if we have |M| points, each needing at least one pair, and the number of pairs is C(|M|,2). Therefore, we have |M| <= C(|M|,2), which as before, is true for |M| >=3, but this doesn't prevent the necessity of overlapping midpoints. However, even if multiple pairs generate the same midpoint, the structure required for all points to be midpoints might not be possible.Another approach: look at the parity. Suppose M is finite. Assign to each point in M a vector in ℝ². Then, for each point m, there exist p and q in M such that m = (p + q)/2. Therefore, 2m = p + q. So, for each m, 2m is the sum of two other points in M.If we consider the additive subgroup of ℝ² generated by M, then 2m is in the subgroup for each m. This suggests that the subgroup is divisible by 2, which in vector spaces over ℝ implies that the subgroup must be infinitely generated unless trivial. But since M is finite, the subgroup is finitely generated, but being divisible by 2 would require that each element can be halved within the subgroup. However, in a finitely generated subgroup of ℝ² (which is a free abelian group of finite rank), divisibility by 2 would imply that every element is divisible by 2, which is only possible if the group is trivial. But since M is non-empty, this is a contradiction. Wait, this might be too abstract.Alternatively, think of scaling. If we have a finite set M, then consider the set 2M = {2m | m ∈ M}. Each element of 2M must be expressible as the sum of two elements from M. Therefore, 2M ⊆ M + M, where M + M is the set of all p + q for p, q ∈ M. Since M is finite, M + M is also finite. Therefore, 2M is a subset of a finite set. However, if we iterate this, consider 4M = 2(2M) ⊆ 2(M + M) = M + M + M + M. Continuing this, we get that 2^k M ⊆ M^{2^k}, which grows in size but is constrained by the finiteness of M. However, unless all elements of M are zero, the norms of the points would grow without bound, contradicting the finiteness. Wait, but if M contains non-zero points, then scaling by 2 each time would produce points with arbitrarily large norms, but M + M can only contain points with norms up to twice the maximum norm in M. Therefore, unless all points in M are zero, which would make M a singleton set (but singleton set is invalid as discussed earlier), there's a contradiction. Therefore, M cannot contain non-zero points if it's finite. But a singleton set with the zero vector is invalid because it can't be the midpoint of two distinct points. Therefore, M must be infinite.This seems like a promising approach. Let me formalize it.Assume M is finite and non-empty. Let R = max{ ||m|| | m ∈ M }, where ||m|| is the Euclidean norm. Since M is finite, R is well-defined. Now, take any point m ∈ M. Then, 2m = p + q for some p, q ∈ M. Then, ||2m|| = ||p + q|| ≤ ||p|| + ||q|| ≤ R + R = 2R. Therefore, ||2m|| ≤ 2R ⇒ ||m|| ≤ R. However, this doesn't give new information. Wait, but if we iterate this process. Suppose we take a point m, then 2m = p + q. Then, 4m = 2p + 2q. But 2p = a + b and 2q = c + d for some a, b, c, d ∈ M. Therefore, 4m = (a + b) + (c + d). Continuing this, we see that 2^k m can be expressed as a sum of 2^k points from M. However, the norm ||2^k m|| would be 2^k ||m||, but the right-hand side, being a sum of 2^k points each with norm at most R, has norm at most 2^k R. Therefore, ||2^k m|| ≤ 2^k R ⇒ ||m|| ≤ R. Again, this doesn't lead to a contradiction unless m is non-zero.Wait, perhaps if we start with a non-zero point m. Then, ||2m|| = ||p + q||. If p and q are such that p + q = 2m, then the maximum possible norm ||p + q|| is 2R (if p and q are colinear with m). However, if m is non-zero, then 2m must be expressed as the sum of two points in M. If all points in M have norm ≤ R, then ||2m|| ≤ 2R. Therefore, ||m|| ≤ R. But this doesn't prevent ||m|| from being up to R. Hmm, maybe this approach isn't sufficient.Wait, but suppose there exists a point m in M with maximal norm R, i.e., ||m|| = R. Then, 2m = p + q. Then, ||2m|| = 2R = ||p + q|| ≤ ||p|| + ||q|| ≤ R + R = 2R. Therefore, equality holds, which implies that p and q are colinear with m and in the same direction. That is, p and q must both be equal to m. Because in the triangle inequality, equality holds if and only if the vectors are non-negatively scalar multiples of each other. Since we are in a Euclidean space, p and q must be scalar multiples of m with positive scalars. But since ||p|| ≤ R and ||q|| ≤ R, and p + q = 2m with ||2m|| = 2R, the only way this can happen is if p = q = m. However, p and q must be distinct points in M. Therefore, this leads to a contradiction because we cannot have p = q = m if they need to be distinct. Hence, our assumption that there exists a point m with ||m|| = R leads to a contradiction. Therefore, M cannot contain a point with maximal norm, which is impossible if M is finite and non-empty. Therefore, M must be infinite.Ah, this seems like a solid argument. Let me recap:1. Assume M is finite and non-empty.2. Define R as the maximum norm of points in M.3. Take a point m ∈ M with ||m|| = R.4. Since m is a midpoint, there exist distinct points p, q ∈ M such that m = (p + q)/2 ⇒ p + q = 2m.5. Then, ||p + q|| = ||2m|| = 2R.6. But ||p + q|| ≤ ||p|| + ||q|| ≤ R + R = 2R.7. For equality to hold in the triangle inequality, p and q must be parallel and in the same direction. Hence, p = k*m and q = l*m for some scalars k, l ≥ 0.8. But since p and q are in M, their norms are ≤ R. Therefore, ||p|| = |k|*||m|| = k*R ≤ R ⇒ k ≤ 1. Similarly, l ≤ 1.9. Then, p + q = (k + l)m = 2m. Therefore, k + l = 2.10. But since k ≤ 1 and l ≤ 1, the only solution is k = l = 1. Hence, p = q = m.11. However, p and q must be distinct points in M, leading to a contradiction.12. Therefore, our assumption that M is finite must be false.Therefore, M must contain infinitely many points.This seems to be a rigorous proof. Let me check if there are any loopholes or missed cases.First, the step where equality in the triangle inequality requires p and q to be parallel and in the same direction is correct in Euclidean space. The Cauchy-Schwarz inequality equality condition states that p and q must be linearly dependent, i.e., parallel. Since their sum has norm equal to the sum of their norms, they must point in the same direction.Second, the conclusion that p = q = m is forced by the previous steps. However, p and q are required to be distinct, which contradicts them both being m. Therefore, the existence of a point with maximal norm R is impossible, implying that no finite set M can satisfy the condition, as every finite set in Euclidean space has a point with maximal norm.Therefore, the proof holds.

✅:To prove that the set ( M ) must contain infinitely many points under the given conditions, we proceed by contradiction:Assumption: Suppose ( M ) is a finite non-empty set where every point is the midpoint of a line segment connecting two distinct points in ( M ).1. Maximal Norm: Let ( R = max{ |m| mid m in M } ), which exists since ( M ) is finite.2. Contradiction via Maximal Point: Consider a point ( m in M ) with ( |m| = R ). By the midpoint property, there exist distinct points ( p, q in M ) such that ( m = frac{p + q}{2} ), implying ( p + q = 2m ).3. Triangle Inequality: ( |p + q| = |2m| = 2R ). However, ( |p + q| leq |p| + |q| leq R + R = 2R ).4. Equality Condition: Equality in the triangle inequality requires ( p ) and ( q ) to be parallel and in the same direction. Thus, ( p = k cdot m ) and ( q = l cdot m ) for ( k, l geq 0 ).5. Norm Constraints: Since ( |p| leq R ) and ( |q| leq R ), we have ( k leq 1 ) and ( l leq 1 ). From ( p + q = 2m ), ( k + l = 2 ). The only solution is ( k = l = 1 ), forcing ( p = q = m ), contradicting distinctness.Conclusion: The existence of a point with maximal norm ( R ) leads to a contradiction. Hence, ( M ) cannot be finite. Therefore, ( M ) must contain infinitely many points.(boxed{M text{ has infinitely many points.}})

🔑:1. Assume, for the sake of contradiction, that the set ( M ) consists of a finite number of points.2. Select two points ( A ) and ( B ) such that ( A, B in M ). By the problem's condition, there exists a line segment ( CD ) with ( C, D in M ), such that point ( B ) is the midpoint of ( CD ).3. Case 1: If ( C ) lies on the line ( AB ). * Because ( B ) is the midpoint of ( CD ), point ( D ) must also lie on the line ( AB ). * Furthermore, ( C ) and ( D ) must be on opposite sides of ( B ) on the line segment ( AB ). Without loss of generality, assume ( D ) is on the opposite side of ( B ) relative to ( A ). Thus, the distance ( AD ) is greater than the distance ( AB ).4. Case 2: If ( C ) does not lie on the line ( AB ). * Extend the line segment ( AB ) to point ( A' ), such that ( AB = BA' ). Therefore, ( A' ) lies on the extension of ( AB ) past point ( B ). * Connect ( A' ) to ( D ), forming the segment ( A'D ). Given that ( B ) is the midpoint of ( CD ), point ( C ) lies on the line through ( A ) parallel to segment ( A'D ), and the distance ( A'C ) is equal to ( AD ). * Hence, we observe that ( AC + AD > 2AB ), which means one of ( AC ) or ( AD ) is longer than ( AB ).5. Combining the above cases, we can always find a line segment ( AD ) that is longer than ( AB ) under the assumption that ( A, B, D in M ).6. Given that ( M ) was assumed to be finite, we can continue finding increasingly longer line segments. This is a contradiction because, in a finite set, there must be a maximum distance between any two points.7. Since assuming a finite set ( M ) leads to a contradiction, we conclude that ( M ) must be an infinite set.Conclusion: ( boxed{M text{ must contain an infinite number of points.}} )