Question 6
Let $S$ be a nonempty set of positive integers such that for every $a,b\in S$ there exists $c\in S$ satisfying $c^2\mid a(a+b)$. Prove that some $a\in S$ divides every element of $S$.
Source-language clarification: the Chinese statement explicitly says that $S$ is nonempty; the source's English version omits that word. The nonempty hypothesis is retained here. Without it, the empty set would make the existential conclusion false.
Source-language clarification: the Chinese statement explicitly says that $S$ is nonempty; the source's English version omits that word. The nonempty hypothesis is retained here. Without it, the empty set would make the existential conclusion false.