IR-TST 2013 Problem 16

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