IMO 2012 Problem 4

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

Proposed by Liam Baker, South Africa