Past exam of the options II course of the University of Oxford / 2010 / C / C2 / iv / Solution

Let $R$ be the sum of the nonzero quadratic residues. Each has two square roots by part (iii), sowhere part (ii) applies because $p>3$ makes $p-1$ not divide $2$. Since $2$ is invertible modulo $p$, $R\equiv 0$.

The sum of all nonzero classes is $p(p-1)/2\equiv 0(\mathrm{mod}p)$. Subtracting $R$ shows that the sum of quadratic nonresidues is also zero. Consequently both requested sums vanish modulo $p$.