RO-NMO 2010 Problem 94

Consider the set $\mathcal{F}$ of functions $f:\mathbb{N}\to\mathbb{N}$ (where $\mathbb{N}$ is the set of non-negative integers) having the property that $$f(a^2-b^2)=f(a)^2-f(b)^2,\ \text{for all }a,b\in\mathbb{N},\ a\ge b.$$ a) Determine the set $\{f(1)\mid f\in\mathcal{F}\}$ . b) Prove that $\mathcal{F}$ has exactly two elements.

Nelu Chichirim