11th European Girls' Mathematical Olympiad | EGMO 2022

Problem 2. Let $\mathbb{N}=\{1, 2, 3, \dots\}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$ , the following two conditions hold:
(1) $f(ab) = f(a)f(b)$ , and
(2) at least two of the numbers $f(a)$ , $f(b)$ , and $f(a+b)$ are equal.