BALTIC-WAY 2025 Problem 5

Let $n$ be a positive integer and let $x_1 \leq x_2 \leq \cdots \leq x_n$ be positive real numbers satisfying

$$\frac{x_1^3 + x_2^3 + \cdots + x_n^3}{x_1 + x_2 + \cdots + x_n} = \frac{x_1^2 + x_n^2}{2}.$$

Prove that

$$x_1^2 + x_2^2 + \cdots + x_n^2 \geq nx_1x_n.$$