IMC 2025 · Problem 7

Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all nonempty subsets $M \subseteq \mathbb{Z}_{>0}$ satisfying both of the following properties:

(a) if $x \in M$, then $2x \in M$,

(b) if $x, y \in M$ and $x + y$ is even, then $\dfrac{x+y}{2} \in M$.