MathNet 2026 — Part 03 Question 45

Let $n$ be a strictly positive integer. Let $x_{1} \leq x_{2} \leq \ldots \leq x_{n}$ be real numbers such as $x_{1}+x_{2}+\ldots+x_{n}=0$ and $x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=1$. Show that $x_{1} x_{n} \leq -1 / n$.