Question 92
Let $p$ be an odd prime number. Find all $\frac{p-1}{2}$-tuples $(x_1, x_2, \dots, x_{\frac{p-1}{2}}) \in \mathbb{Z}_{p^2}^{\frac{p-1}{2}}$ such that
$$
\sum_{i=1}^{\frac{p-1}{2}} x_i \equiv \sum_{i=1}^{\frac{p-1}{2}} x_i^2 \equiv \dots \equiv \sum_{i=1}^{\frac{p-1}{2}} x_i^{\frac{p-1}{2}} \pmod{p}.
$$
$$
\sum_{i=1}^{\frac{p-1}{2}} x_i \equiv \sum_{i=1}^{\frac{p-1}{2}} x_i^2 \equiv \dots \equiv \sum_{i=1}^{\frac{p-1}{2}} x_i^{\frac{p-1}{2}} \pmod{p}.
$$