Middle European Mathematical Olympiad 2016 Problem I-4

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$.

Remark: $\mathbb{N} = \{1, 2, 3, \ldots\}$ denotes the set of positive integers.