International Mathematics Competition
for University Students
2025

IMC2025: Day 2, Problem 7

Problem 7. Let \(\displaystyle \mathbb{Z}_{>0}\) be the set of positive integers. Find all nonempty subsets \(\displaystyle M \subseteq \mathbb{Z}_{>0}\) satisfying both of the following properties:

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

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

Alexandr Bolbot, Novosibirsk State University

  Solution