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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.

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    I removed the num$b$er theory tags, and added "inequalities" which seems like a no-brainer2012-04-12

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Take $n = 1$, $a_2 = 1$ and $a_1 \ll 0$, then $A = \frac{a_1}{2} \ll 0$ is unbounded.