BALKAN-SL 2014 N2

$\boxed{N2}$ Let $p$ be a prime numbers and $x_1,x_2,...,x_n$ be integers.Show that if $$x_1^n+x_2^n+...+x_p^n\equiv 0 \pmod{p}$$ for all positive integers n then $x_1\equiv x_2 \equiv...\equiv x_p \pmod{p}.$