MathDB

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+\dots+a_n=0.

Define the set

A

A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}

Prove that, if

A

is not empty, then

\sum_{(i, j) \in A} a_{i} a_{j}<0.

A4