EGMO 2024 Problem 5

Problem. Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that the following assertions are true for every pair of positive integers $(x, y)$ :

• $x$ and $f(x)$ have the same number of positive divisors.
• If $x$ does not divide $y$ and $y$ does not divide $x$, then

$$\operatorname{gcd}(f(x), f(y))>f(\operatorname{gcd}(x, y))$$

Here $\operatorname{gcd}(m, n)$ is the largest positive integer that divides both $m$ and $n$.