CMO 2004 Problem 4

Let $p$ be an odd prime. Prove that

$$\sum_{k=1}^{p-1} k^{2 p-1} \equiv \frac{p(p+1)}{2} \left(\bmod p^{2}\right)$$

[Note that $a \equiv b(\bmod m)$ means that $a-b$ is divisible by $m$.]