this problem was recently posed by BS in the number theory chat room. he thinks it may originate from the International Math Olympiad & he says he has a solution. has anyone seen it there? looking for a more precise reference/ history.
Let $S$ be a nonempty set of positive nonzero integers such that if $x$ is in $S$ then both $4x$ and $\lfloor \sqrt{x} \rfloor$ are in $S$. Prove that $S = \Bbb N \setminus \{0\}$