VN-TST 2005 Problem 5

Let $p\in \mathbb P,p>3$ . Calcute:

a) $S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 4$

b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 8$