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Part of 2013 Iran Team Selection Test

Problems(1)

f(m)+f(n)+f(f(m^2+n^2))=1

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

7/26/2013
The function f:Z→Zf:\mathbb Z \to \mathbb Z has the property that for all integers mm and nn f(m)+f(n)+f(f(m2+n2))=1.f(m)+f(n)+f(f(m^2+n^2))=1. We know that integers aa and bb exist such that f(a)−f(b)=3f(a)-f(b)=3. Prove that integers cc and dd can be found such that f(c)−f(d)=1f(c)-f(d)=1.
Proposed by Amirhossein Gorzi
functionalgebra proposedalgebrafunctional equation