2009 IMO Shortlist C7

RUS Variant 1. A grasshopper jumps along the real axis. He starts at point 0 and makes 2009 jumps to the right with lengths $1,2, \ldots, 2009$ in an arbitrary order. Let $M$ be a set of 2008 positive integers less than 1005 - 2009. Prove that the grasshopper can arrange his jumps in such a way that he never lands on a point from $M$. Variant 2. Let $n$ be a nonnegative integer. A grasshopper jumps along the real axis. He starts at point 0 and makes $n+1$ jumps to the right with pairwise different positive integral lengths $a_{1}, a_{2}, \ldots, a_{n+1}$ in an arbitrary order. Let $M$ be a set of $n$ positive integers in the interval $(0, s)$, where $s=a_{1}+a_{2}+\cdots+a_{n+1}$. Prove that the grasshopper can arrange his jumps in such a way that he never lands on a point from $M$.

RUS 变体 1。蚱蜢沿着实轴跳跃。他从点 0 开始，使 2009 以任意顺序向右跳转，长度为 $1,2, \ldots, 2009$。令 $M$ 为一组小于 1005 - 2009 的 2008 个正整数。证明蚱蜢可以以这样的方式安排他的跳跃，使其永远不会落在 $M$ 的点上。变体 2. 令 $n$ 为非负整数。蚱蜢沿着实轴跳跃。他从点 0 开始，使 $n+1$ 以任意顺序以成对不同的正整数长度 $a_{1}、a_{2}、\ldots、a_{n+1}$ 向右跳转。令 $M$ 为区间 $(0, s)$ 内的一组 $n$ 个正整数，其中 $s=a_{1}+a_{2}+\cdots+a_{n+1}$。证明蚱蜢可以安排他的跳跃，使其永远不会落在 $M$ 的点上。