# 1975 IMO Problems/Problem 1

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## Problem

Let $x_i, y_i$ $(i=1,2,\cdots,n)$ be real numbers such that $$x_1\ge x_2\ge\cdots\ge x_n \text{ and } y_1\ge y_2\ge\cdots\ge y_n.$$ Prove that, if $z_1, z_2,\cdots, z_n$ is any permutation of $y_1, y_2, \cdots, y_n,$ then $$\sum^n_{i=1}(x_i-y_i)^2\le\sum^n_{i=1}(x_i-z_i)^2.$$

## Solution

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 1975 IMO (Problems) • Resources Preceded byFirst Question 1 • 2 • 3 • 4 • 5 • 6 Followed byProblem 3 All IMO Problems and Solutions