2019 IMO Shortlist A4

Let $n \geq 2$ be a positive integer and $a_{1}, a_{2}, \ldots, a_{n}$ be real numbers such that $$a_{1}+a_{2}+\cdots+a_{n}=0$$ Define the set $A$ by $$A=\left\{(i, j)\left|1 \leq i<j \leq n,\left|a_{i}-a_{j}\right| \geq 1\right\} .\right.$$ Prove that, if $A$ is not empty, then $$\sum_{(i, j) \in A} a_{i} a_{j}<0$$ (China)

设 $n \geq 2$ 为正整数，$a_{1}, a_{2}, \ldots, a_{n}$ 为实数，使得 $$a_{1}+a_{2}+\cdots+a_{n}=0$$ 通过 $$ A=\left\{(i, j)\left|1 \leq i<j \leq 定义集合 $A$ n,\left|a_{i}-a_{j}\right| \geq 1\右\} .\右。 $$证明，如果$A$不为空，则$$\sum_{(i, j) \in A} a_{i} a_{j}<0$$ (中国)