Problem

Source: Romania Junior TST 2022

Tags: romania, Romanian TST, geometry



Let $M,N$ and $P$ be the midpoints of sides $BC,CA$ and $AB$ respectively, of the acute triangle $ABC.$ Let $A',B'$ and $C'$ be the antipodes of $A,B$ and $C$ in the circumcircle of triangle $ABC.$ On the open segments $MA',NB'$ and $PC'$ we consider points $X,Y$ and $Z$ respectively such that \[\frac{MX}{XA'}=\frac{NY}{YB'}=\frac{PZ}{ZC'}.\] Prove that the lines $AX,BY,$ and $CZ$ are concurrent at some point $S.$ Prove that $OS<OG$ where $O$ is the circumcenter and $G$ is the centroid of triangle $ABC.$