Middle European Mathematical Olympiad 2015 Problem I-1

Find all surjective functions $f: \mathbb{N} \to \mathbb{N}$ such that for all positive integers $a$ and $b$, exactly one of the following equations is true: $$f(a) = f(b),$$ $$f(a + b) = \min\{f(a), f(b)\}.$$

Remarks: $\mathbb{N}$ denotes the set of all positive integers. A function $f: X \to Y$ is said to be surjective if for every $y \in Y$ there exists $x \in X$ such that $f(x) = y$.