2009 IMO 第 6 题

Let $a_{1}$ , $a_{2}$ , ..., $a_{n}$ be distinct positive integers and let $M$ be a set of $n - 1$ positive integers not containing $s = a_{1} + \dots +a_{n}$ . A grasshopper is to jump along the real axis, starting at the point 0 and making $n$ jumps to the right with lengths $a_{1}$ , $a_{2}$ , ..., $a_{n}$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $M$ .

令 $a_{1}$ , $a_{2}$ , ..., $a_{n}$ 为不同的正整数，并令 $M$ 为一组不包含 $s = a_{1} + \dots +a_{n}$ 的 $n - 1$ 个正整数。蚱蜢沿着实轴跳跃，从点 0 开始，使 $n$ 向右跳跃，长度为 $a_{1}$ 、 $a_{2}$ 、 ...、 $a_{n}$ 按某种顺序。证明可以选择顺序以使蚱蜢永远不会落在 $M$ 中的任何点上。