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 for positive real numbers .
Solution
Note that Suppose that . Note that is an increasing function of both and . It follows that if , then i.e., is an increasing function of .