Problem

Source: South African Mathematics Olympiad 2019, Problem 5

Tags: algebra, functional equation



Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $$ f(a^3) + f(b^3) + f(c^3) + 3f(a + b)f(b + c)f(c + a) = {(f(a + b + c))}^3 $$for all integers $a, b, c$.