Let $ a_1,...,a_{2n}$ be real numbers, such that $|a_1|\geq...\geq |a_{2n}|$.
Let $A=\frac{\sum_{i\neq j} a_ia_j}{2n(2n-1)}$.
I would like to bound $A$ from below. (the ideal bound for me would be $a_{2n}^2$, but unfortunately it is not always true.