1984 USAMO 第 2 题

The geometric mean of any set of $m$ non-negative numbers is the $m$-th root of their product.

$\quad (\text{i})\quad$ For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer?

$\quad (\text{ii})\quad$ Is there an infinite set $S$ of distinct positive integers such that the geometric mean of any finite subset of $S$ is an integer?

任何 $m$ 个非负数集合的几何平均值是其乘积的 $m$ 次方根。

$\quad (\text{i})\quad$ 对于哪些正整数 $n$，存在 $n$ 个不同正整数的有限集合 $S_n$，使得 $S_n$ 的任何子集的几何平均值都是整数？

$\quad (\text{ii})\quad$ 是否存在不同正整数的无限集合 $S$，使得 $S$ 的任何有限子集的几何平均值都是整数？