Question 4
Let $n$ be a positive integer and let $x_{1} \leq x_{2} \leq \cdots \leq x_{n}$ be real numbers.
(1) Prove that
$$ \left(\sum_{i,j=1}^{n}\left|x_{i}-x_{j}\right|\right)^{2}\leq\frac{2(n^{2}-1)}{3}\sum_{i,j=1}^{n}(x_{i}-x_{j})^{2}. $$
(2) Show that the equality holds if and only if $x_{1}, \ldots, x_{n}$ is an arithmetic sequence.
(1) Prove that
$$ \left(\sum_{i,j=1}^{n}\left|x_{i}-x_{j}\right|\right)^{2}\leq\frac{2(n^{2}-1)}{3}\sum_{i,j=1}^{n}(x_{i}-x_{j})^{2}. $$
(2) Show that the equality holds if and only if $x_{1}, \ldots, x_{n}$ is an arithmetic sequence.