Problem: Prove that for each $n\geq 2$,
there is a set $S$ of $n$ integers
such that $(a-b)^2$ divides $ab$ for every distinct $a,b\in S$.

Solution: This is a direct corollary of the more difficult USA TST 2015/2, reproduced below.
\begin{quote}
  Prove that for every positive integer $n$,
  there exists a set $S$ of $n$ positive integers
  such that for any two distinct $a,b \in S$, $a-b$ divides $a$ and $b$
  but none of the other elements of $S$.
\end{quote}