2002 IMO 第 5 题

N5 (IRN) Let $m, n \geq 2$ be positive integers, and let $a_{1}, a_{2}, \ldots, a_{n}$ be integers, none of which is a multiple of $m^{n-1}$. Show that there exist integers $e_{1}, e_{2}, \ldots, e_{n}$, not all zero, with $\left|e_{i}\right|<m$ for all $i$, such that $e_{1} a_{1}+e_{2} a_{2}+\cdots+e_{n} a_{n}$ is a multiple of $m^{n}$.

N5 (IRN) 设 $m, n \geq 2$ 为正整数，设 $a_{1}, a_{2}, \ldots, a_{n}$ 为整数，其中没有一个是 $m^{n-1}$ 的倍数。证明存在整数 $e_{1}、e_{2}、\ldots、e_{n}$，且不全为零，且所有 $i$ 的 $\left|e_{i}\right|<m$，使得 $e_{1} a_{1}+e_{2} a_{2}+\cdots+e_{n} a_{n}$ 是 $m^{n}$ 的倍数。