Problem

Source: 5-th Hong Kong Mathematical Olympiad 2002

Tags: modular arithmetic, quadratics, number theory unsolved, number theory



Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.