SIMON-MARAIS 2018 B1

For all positive integers $n$ and real numbers $x_1, x_2, \ldots, x_n$, prove that

$$\sum_{i=1}^{n} \sum_{j=1}^{n} \min(i, j) x_i x_j \geq 0.$$

(We define $\min(a, b) = a$ if $a \leq b$ and $\min(a, b) = b$ if $a > b$.)