OSN 2016 Problem 7

Suppose that $p> 2$ is a prime number. For each integer $k = 1, 2,..., p-1$ , denote $r_k$ as the remainder of the division $k^p$ by $p^2$ . Prove that $r_1+r_2+r_3+...+r_{p-1}=\frac{p^2(p-1)}{2}$