Question 1
Find all the functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$$
f(a)^{2}+f(b)^{2}+f(c)^{2}=2 f(a) f(b)+2 f(b) f(c)+2 f(c) f(a)
$$
for all integers $a, b, c$ satisfying $a+b+c=0$.
$$
f(a)^{2}+f(b)^{2}+f(c)^{2}=2 f(a) f(b)+2 f(b) f(c)+2 f(c) f(a)
$$
for all integers $a, b, c$ satisfying $a+b+c=0$.