2003 IMO Shortlist S06

Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:

(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;

(ii) for any nonempty subset of $A$ , the product of its elements is not a perfect $p$ -th power.

What is the largest possible number of elements in $A$ ?

设$p$为素数，$A$为满足以下条件的正整数集合：

(i) $A$ 中元素的素因数集合由 $p-1$ 元素组成；

(ii) 对于 $A$ 的任何非空子集，其元素的乘积不是完美的 $p$ 次方。

$A$ 中最大可能的元素数是多少？