Question 4
Find all functions $f \colon \mathbb{Z} \to \mathbb{Z}$ such that, for all integers $a$, $b$, $c$ that satisfy $a+b+c=0$, the following equality holds: $$
f(a)^2+f(b)^2+f(c)^2 = 2f(a)f(b)+2f(b)f(c)+2f(c)f(a).
$$
f(a)^2+f(b)^2+f(c)^2 = 2f(a)f(b)+2f(b)f(c)+2f(c)f(a).
$$