ZA-SAMO 2011 Problem 5

Let $\mathbb{N}_0$ denote the set of all nonnegative integers. Determine all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ with the following two properties:

• $0\le f(x)\le x^2$ for all $x\in\mathbb{N}_0$
• $x-y$ divides $f(x)-f(y)$ for all $x,y\in\mathbb{N}_0$ with $x>y$