2008 CMO 第 3 题

Given a positive integer $n$ and $x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n$ , satisfying
$$\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i$$
Show that for any real number $\alpha$ , we have
$$\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]$$

Here $[\beta]$ denotes the greastest integer not larger than $\beta$ .

给定一个正整数 $n$ 和 $x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n$ ，满足

$$\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i$$

证明对于任何实数 $\alpha$ ，我们有

$$\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]$$

这里 $[\beta]$ 表示不大于 $\beta$ 的最大整数。