Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality \[\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2\]
Problem
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Tags: inequalities, n-variable inequality, IMO Shortlist, IMO Longlist
thien_tanphu
27.09.2010 17:41
if n = 2, $a_1 = a_2 = x$. The problem becomes $\frac{1}{x^2} \ge 4\frac{1}{x^4}$. It maybe not right
litongyang
29.09.2010 14:24
The dimension of the both side is not the same!
Amir Hossein
29.09.2010 14:33
Oh sorry, it was my mistake. Thanks for pointing it out. I've edited it.
Ovchinnikov Denis
29.09.2010 14:42
It's easy:)
by QM-AM
where $k$ -number of pairs $a_i,a_j$ for which $i<j$, equals $\binom{n}{2}$ so $QED$
Ritwin
02.01.2023 11:39
By Cauchy-Schwarz we have
\[ \binom n2 \sum_{i<j} \frac{1}{a_ia_j}
= \biggl(\sum_{i<j} 1\biggr) \biggl(\sum_{i<j} \frac{1}{a_ia_j}\biggr)
\geq \biggl(\sum_{i<j} \frac{1}{\sqrt{a_ia_j}}\biggr)^2.\]Hence we want to show
\[\sum_{i<j} \frac{1}{\sqrt{a_ia_j}} \geq \sum_{i<j} \frac{2}{a_i+a_j}.\]AM-GM, when reciprocated, becomes
\[\frac{1}{\sqrt{a_ia_j}} \geq \frac{2}{a_i+a_j},\]so applying this termwise finishes. \qed