ÖMO 2013 Problem 6

Consider a regular octahedron $ABCDEF$ with lower vertex $E$ , upper vertex $F$ , middle cross-section $ABCD$ , midpoint $M$ and circumscribed sphere $k$ . Further, let $X$ be an arbitrary point inside the face $ABF$ . Let the line $EX$ intersect $k$ in $E$ and $Z$ , and the plane $ABCD$ in $Y$ . Show that $\sphericalangle{EMZ}=\sphericalangle{EYF}$ .