MathNet 2016-part-03 — 184 problems (EN–VI) Question 81

Let $n$ be an odd positive integer, and let $x_1, x_2, \dots, x_n$ be non-negative real numbers. Show that $\min_{k=1,\dots,n} (x_k^2 + x_{k+1}^2) \le \max_{k=1,\dots,n} (2x_k x_{k+1})$, where $x_{n+1} = x_1$.