2006 USAMO 第 4 题

Find all positive integers $n$ such that there are $k\ge 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \cdots + a_k = a_1\cdot a_2\cdots a_k = n$.

找到所有正整数 $n$，使得存在 $k\ge 2$ 个正有理数 $a_1, a_2, \ldots, a_k$ 满足 $a_1 + a_2 + \cdots + a_k = a_1\cdot a_2\cdots a_k = n$。