IMO 1994- Problem 6

Problem: Show that there exists a set $A$ of positive integers with the following property: For any infinite set $S$ of primes there exists two positive integers $m\in A$ and $n\notin A$ each of which is a product of $k$ distinct elements of $S$ for some $k\ge 2$.
Solution: Let $A$ be the of all positive integers of the form $q_1q_2...q_{q_1}$ where $q_1<q_2<...<q_{q_1}$ are primes. For any infinite set $S=\left\{p_1,p_2,...,\right\}$ of primes with $p_1<p_2,...$, we can satisfy he requirement of the problem by taking $k=p_1,m=p_1p_2...p_k$ and $n=p_2p_3...p_{k+1}$.
Source: http://sms.math.nus.edu.sg/simo/IMO_Problems/94.pdf