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Functional equation over the integers

Source: Centroamerican 2020, problem 3

October 28, 2020
algebrafunctional equationfunction

Problem Statement

Find all the functions f:Zā†’Zf: \mathbb{Z}\to \mathbb{Z} satisfying the following property: if aa, bb and cc are integers such that a+b+c=0a+b+c=0, then
f(a)+f(b)+f(c)=a2+b2+c2.f(a)+f(b)+f(c)=a^2+b^2+c^2.
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