If $X$ would contain a positive real number $x$, then it will also contain $2x < 4x < \cdots < 2^nx <\cdots$, so it will be infinite. On the other hand, if $x\in X$, $x\leq 0$, then that just forces $0\in X$. So the answer is $X = F \cup \{0\}$, where $F$ is any finite (maybe empty) set of negative real numbers. Quite an easy question for the National Stage of an Olympiad ...