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SAMO Problem 5: Functional equation based on expansion of $(a + b + c)^3$

Source: South African Mathematics Olympiad 2019, Problem 5

July 25, 2019
algebrafunctional equation

Problem Statement

Find all functions f:Zā†’Zf : \mathbb{Z} \to \mathbb{Z} such that f(a3)+f(b3)+f(c3)+3f(a+b)f(b+c)f(c+a)=(f(a+b+c))3 f(a^3) + f(b^3) + f(c^3) + 3f(a + b)f(b + c)f(c + a) = {(f(a + b + c))}^3 for all integers a,b,ca, b, c.
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