power congruence mod $p^2$

[mutasimmim]

Prove that for any odd prime $p$, $1^{2p-1}+2^{2p-1}+....+(p-1)^{2p-1} \equiv \frac{p(p+1)}{2} (mod p^2)$

[*Mahi*]

As it's going unsolved for too long-

Prove $i^{2p-1}+(p-i)^{2p-1} \equiv - p \pmod {p^2} $