2006 IMO Shortlist Problems/A5
(Hojoo Lee, South Korea) Let be the sides of a triangle. Prove that
This problem also appeared on the 2007 IMO TSTs of Italy and Bangladesh.
Lemma. For any positive reals ,
Proof 1. This is the Vornicu-Schur Inequality on the function .
Proof 2. Without loss of generality, suppose that . Evidently, so it suffices to show that or By assumption, and by AM-GM, , or Since all sides of both the previous two inequalities are positive, multiplication yields the desired bound, proving the lemma.
We note that in the problem statement, each denominator is greater than zero, for We now abbreviate , , . Then By the Power Mean Inequality, Also, by the lemma, so proving the desired inequality.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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