Nemotron-IMO-Bench 2026 N38

Let $p$ be an odd prime and let $n$ be an even positive integer. How many ordered tuples $(x_1,\ldots, x_n)$ with entries in $\{1,\ldots, p\}$ have the property that $p$ divides both $x_1+\ldots+x_n$ and $x_1^2+\ldots+x_n^2$?