RO-TST 2005 Problem 12

Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that $$\sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 .$$

Călin Popescu