RO-TST 2014 Problem 6

Let $ABC$ be a triangle and let $X$ , $Y$ , $Z$ be interior points on the sides $BC$ , $CA$ , $AB$ , respectively. Show that the magnified image of the triangle $XYZ$ under a homothety of factor $4$ from its centroid covers at least one of the vertices $A$ , $B$ , $C$ .