IN-TST 2001 Problem 11

Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$ , define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$ . Prove that : $$\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}$$