Question 5
Let $n\gt2$ be an integer and let $x_1 \le x_2 \le \dots \le x_n$ be real numbers. Prove that $$
\left(\sum_{i=1}^{n}\sum_{j=1}^{n} |x_i - x_j|\right)^2
\le \frac{2(n^2-1)}{3}\sum_{i=1}^{n}\sum_{j=1}^{n} (x_i - x_j)^2
$$ with equality if and only if $x_1$, $x_2$, …, $x_n$ form an arithmetic sequence.
\left(\sum_{i=1}^{n}\sum_{j=1}^{n} |x_i - x_j|\right)^2
\le \frac{2(n^2-1)}{3}\sum_{i=1}^{n}\sum_{j=1}^{n} (x_i - x_j)^2
$$ with equality if and only if $x_1$, $x_2$, …, $x_n$ form an arithmetic sequence.