RMO 2010

Let $ABCDEF$ be a convex hexagon in which diagonals $AD, BE, CF$ are concurrent at $O$ . Suppose $[OAF]$ is geometric mean of $[OAB]$ and $[OEF]$ and $[OBC]$ is geometric mean of $[OAB]$ and $[OCD]$ . Prove that $[OED]$ is the geometric mean of $[OCD]$ and $[OEF]$ . (Here $[XYZ]$ denotes are of $\triangle XYZ$ )

Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.

Let $ABC$ be a triangle in which $\angle A = 60^\circ$ . Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ with $E$ on $AC$ and $F$ on $AB$ . Let $M$ be the reflection of $A$ in line $EF$ . Prove that $M$ lies on $BC$ .