AT-OEMO-REG 2006 Problem 1

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