MathDB
Number of different orders of points

Source: Austrian Federal Competition 2013, part 2, problem 5

June 18, 2013
combinatorics proposedcombinatorics

Problem Statement

Let n⩾3n\geqslant3 be an integer. Let A1A2…AnA_1A_2\ldots A_n be a convex nn-gon. Consider a line gg through A1A_1 that does not contain a further vertice of the nn-gon. Let hh be the perpendicular to gg through A1A_1. Project the nn-gon orthogonally on hh. For j=1,…,nj=1,\ldots,n, let BjB_j be the image of AjA_j under this projection. The line gg is called admissible if the points BjB_j are pairwise distinct. Consider all convex nn-gons and all admissible lines gg. How many different orders of the points B1,…,BnB_1,\ldots,B_n are possible?
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