2021 IMO Shortlist A4

Show that for all real numbers $x_{1}, \ldots, x_{n}$ the following inequality holds: $$\sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}-x_{j}\right|} \leq \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}+x_{j}\right|}$$

证明对于所有实数 $x_{1}, \ldots, x_{n}$，以下不等式成立： $$\sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}-x_{j}\right|} \leq \sum_{i=1}^{n} \sum_{j=1}^{n} \sqrt{\left|x_{i}+x_{j}\right|}$$