1975 IMO 第 2 题

${ }^{\mathrm{IMO} 1}$ Let $x_{1} \geq x_{2} \geq \cdots \geq x_{n}$ and $y_{1} \geq y_{2} \geq \cdots \geq y_{n}$ be two $n$-tuples of numbers. Prove that $$\sum_{i=1}^{n}\left(x_{i}-y_{i}\right)^{2} \leq \sum_{i=1}^{n}\left(x_{i}-z_{i}\right)^{2}$$ is true when $z_{1}, z_{2}, \ldots, z_{n}$ denote $y_{1}, y_{2}, \ldots, y_{n}$ taken in another order.

${ }^{\mathrm{IMO} 1}$ 设 $x_{1} \geq x_{2} \geq \cdots \geq x_{n}$ 和 $y_{1} \geq y_{2} \geq \cdots \geq y_{n}$ 为两个 $n$ 元组的数字。证明 $$\sum_{i=1}^{n}\left(x_{i}-y_{i}\right)^{2} \leq \sum_{i=1}^{n}\left(x_{i}-z_{i}\right)^{2}$$ 为真，当 $z_{1}, z_{2}, \ldots, z_{n}$ 表示 $y_{1} 时， y_{2}, \ldots, y_{n}$ 以另一个顺序获取。