ÖMO 2013 Problem 5

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?