Find all surjective functions such that for all positive integers and , exactly one of the following equations is true: .
Remarks: denotes the set of all positive integers. A function is said to be surjective if for every there exists such that .
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:
$$f(a) = f(b),$$
$$f(a + b) = \min\{f(a), f(b)\}$$.
\emph{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 $f(x) = y$.
Izvor: Srednjoeuropska matematička olimpijada 2015, pojedinačno natjecanje, problem 1
Školjka 
such that for all positive integers 
and 
, exactly one of the following equations is true: 
.
denotes the set of all positive integers. A function 
is said to be surjective if for every 
there exists 
such that 
.