2003 IMO 第 4 题

A4 (IRE) ${ }^{\mathrm{IMO} 5}$ Let $n$ be a positive integer and let $x_{1} \leq x_{2} \leq \cdots \leq x_{n}$ be real numbers. (a) Prove that $$\left(\sum_{i, j=1}^{n}\left|x_{i}-x_{j}\right|\right)^{2} \leq \frac{2\left(n^{2}-1\right)}{3} \sum_{i, j=1}^{n}\left(x_{i}-x_{j}\right)^{2}$$ (b) Show that equality holds if and only if $x_{1}, \ldots, x_{n}$ is an arithmetic progession.

A4 (IRE) ${ }^{\mathrm{IMO} 5}$ 设 $n$ 为正整数，设 $x_{1} \leq x_{2} \leq \cdots \leq x_{n}$ 为实数。 (a) 证明 $$\left(\sum_{i, j=1}^{n}\left|x_{i}-x_{j}\right|\right)^{2} \leq \frac{2\left(n^{2}-1\right)}{3} \sum_{i, j=1}^{n}\left(x_{i}-x_{j}\right)^{2}$$ (b) 证明等式成立，如果并且仅当 $x_{1}, \ldots, x_{n}$ 是算术级数时。