ÖMO 2013

Let $n\geqslant3$ be an integer. Let $A_1A_2\ldots A_n$ be a convex $n$ -gon. Consider a line $g$ through $A_1$ that does not contain a further vertice of the $n$ -gon. Let $h$ be the perpendicular to $g$ through $A_1$ . Project the $n$ -gon orthogonally on $h$ . For $j=1,\ldots,n$ , let $B_j$ be the image of $A_j$ under this projection. The line $g$ is called admissible if the points $B_j$ are pairwise distinct. Consider all convex $n$ -gons and all admissible lines $g$ . How many different orders of the points $B_1,\ldots,B_n$ are possible?

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 $\sphericalangle{EMZ}=\sphericalangle{EYF}$ .