IMO 1975 Q1 #

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Let x₁, x₂, ... , xₙ and y₁, y₂, ... , yₙ be two sequences of real numbers, such that x₁ ≥ x₂ ≥ ... ≥ xₙ and y₁ ≥ y₂ ≥ ... ≥ yₙ. Prove that if z₁, z₂, ... , zₙ is any permutation of y₁, y₂, ... , yₙ, then ∑ (xᵢ - yᵢ)^2 ≤ ∑ (xᵢ - zᵢ)^2

Solution #

Firstly, we expand the squares withing both sums and distribute into separate finite sums. Then, noting that ∑ yᵢ ^ 2 = ∑ zᵢ ^ 2, it remains to prove that ∑ xᵢ * zᵢ ≤ ∑ xᵢ * yᵢ, which is true by the Rearrangement Inequality