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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

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