Problem

Source: Iran TST 2013:TST 3,Day 2,Problem 1

Tags: function, algebra proposed, algebra, functional equation



The function $f:\mathbb Z \to \mathbb Z$ has the property that for all integers $m$ and $n$ \[f(m)+f(n)+f(f(m^2+n^2))=1.\] We know that integers $a$ and $b$ exist such that $f(a)-f(b)=3$. Prove that integers $c$ and $d$ can be found such that $f(c)-f(d)=1$. Proposed by Amirhossein Gorzi