2006 IMO Shortlist Problems/A4
Revision as of 11:46, 29 December 2007 by Boy Soprano II (talk | contribs) (New page: == Problem == Prove the inequality <cmath> \sum_{i<j} \frac{a_ia_j}{a_i+a_j} \le \frac{n}{2(a_1 + a_2 + \dotsb a_n)} \sum_{i<j} a_i a_j </cmath> for positive real numbers <math>a_1, \dots...)
Problem
Prove the inequality
![\[\sum_{i<j} \frac{a_ia_j}{a_i+a_j} \le \frac{n}{2(a_1 + a_2 + \dotsb a_n)} \sum_{i<j} a_i a_j\]](http://latex.artofproblemsolving.com/5/a/d/5ad7bedff1e97d683345dcb2942d2aef77246574.png)
for positive real numbers 
.
Solution
Note that
![\[\sum_{i<j} \frac{a_ia_j}{a_i+a_j} = 1/2 \sum_{i\neq j} \frac{a_ia_j}{a_i + a_j} = 1/2 \sum_{i=1}^n \sum_{j\neq i} \frac{1}{1/a_i + 1/a_j} .\]](http://latex.artofproblemsolving.com/5/f/6/5f6d1d8c1480071cbd41bc760efaf3196161dee4.png)
Suppose that 
. Note that 
is an increasing function of both 
and 
. It follows that if 
, then
![\[\sum_{j \neq k} \frac{1}{1/a_k + 1/a_j} \le \sum_{j\neq \ell} \frac{1}{a_\ell + a_j},\]](http://latex.artofproblemsolving.com/5/4/5/5457135c2a4cb283bfb9de2ab68894f6c37d7e32.png)
i.e., 
is an increasing function of 
.
