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37th Austrian Mathematical Olympiad 2006

Source: round1, problem1 - arrange intervalls by their lenght

February 10, 2009
quadraticsinequalitiesinequalities unsolved

Problem Statement

Let 0<x<y 0 < x <y be real numbers. Let H\equal{}\frac{2xy}{x\plus{}y} , G\equal{}\sqrt{xy} , A\equal{}\frac{x\plus{}y}{2} , Q\equal{}\sqrt{\frac{x^2\plus{}y^2}{2}} be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of x x and y y. As generally known H<G<A<Q H<G<A<Q. Arrange the intervals [H,G] [H,G] , [G,A] [G,A] and [A,Q] [A,Q] in ascending order by their length.
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