Question 72
Let $p$ be an odd prime. Prove that
$$
1^{p-2} + 2^{p-2} + 3^{p-2} + \dots + \left(\frac{p-1}{2}\right)^{p-2} \equiv \frac{2-2^p}{p} \pmod{p}.
$$
$$
1^{p-2} + 2^{p-2} + 3^{p-2} + \dots + \left(\frac{p-1}{2}\right)^{p-2} \equiv \frac{2-2^p}{p} \pmod{p}.
$$