Nemotron-IMO-Bench 2026 A37

Let $n$ be an integer greater than $1$. Find the minimum value of $$\left( \sum_{i=1}^n x_i^2\right) \left( \sum_{i=1}^n x_i^{-2} \right)$$ over all families of nonzero real numbers $x_1,...,x_n$ for which $x_1+...+x_n=0$.