Question 69
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
$$
A=\left\{(i, j)\left|1 \leqslant i\lt j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\} .\right.
$$
Prove that, if $A$ is not empty, then
$$
\sum_{(i, j) \in A} a_{i} a_{j}\lt 0 .
$$
$$
a_{1}+a_{2}+\cdots+a_{n}=0 .
$$
Define the set $A$ by
$$
A=\left\{(i, j)\left|1 \leqslant i\lt j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\} .\right.
$$
Prove that, if $A$ is not empty, then
$$
\sum_{(i, j) \in A} a_{i} a_{j}\lt 0 .
$$