# 2012 IMO - Problem 4

Find all functions f:\mathbb Z\to\mathbb Z such that, for all integers a , b , c with a+b+c=0 the following equality holds:
f^{2}(a)+f^{2}(b)+f^{2}(c)=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).
Here \mathbb Z is the set of all integers.