Nemotron-IMO-Bench 2026 N33

Let $p$ be a prime number and let $A$ be a set of positive integers such that:\ a) the set of prime divisors of the elements of $A$ consists of $p-1$ elements and\ b) 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 of $A$?