Let $p$ be an odd prime. Prove that there is an equal number of quadratic residues and non-residues in the set $\{1, \ldots, p-1\}$.

If $n$ is an integer prime to $p$, let $m_{n}$ be an integer such that $n m_{n} \equiv 1 \bmod p$. Prove that

$$n(n+1) \equiv n^{2}\left(1+m_{n}\right) \bmod p$$

and deduce that

$$\sum_{n=1}^{p-1}\left(\frac{n(n+1)}{p}\right)=-1$$