Question 95
Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that, for all integers $a$, $b$, and $c$ satisfying $a + b + c = 0$, the following equality holds:
$$
f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a).
$$
$$
f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a).
$$