ISL 2019 A4

Let $n \geqslant 2$ 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 $$A=\left\{(i, j)\left|1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\} .\right.$$ Prove that, if $A$ is not empty, then $$\sum_{(i, j) \in A} a_{i} a_{j}<0$$ (China)