2010 APMO 第 2 题

For a positive integer $k$, call an integer a pure $k$-th power if it can be represented as $m^{k}$ for some integer $m$. Show that for every positive integer $n$ there exist $n$ distinct positive integers such that their sum is a pure 2009-th power, and their product is a pure 2010-th power.

对于正整数 $k$，如果某个整数 $m$ 可以表示为 $m^{k}$，则称该整数为纯 $k$ 次幂。证明对于每个正整数 $n$ 都存在 $n$ 个不同的正整数，使得它们的和是纯 2009 次方，它们的乘积是纯 2010 次方。