International Mathematical Olympiad 2011 — Official contest Question 5

Let $f \colon \mathbb{Z} \to \mathbb{Z}_{\gt 0}$ be a function such that $f(m-n) \mid f(m) - f(n)$ for $m,n \in \mathbb{Z}$. Prove that if $m,n \in \mathbb{Z}$ satisfy $f(m) \le f(n)$ then $f(m) \mid f(n)$.