XXXVI OM - II - Problem 3

Let $ L $ be the set of all broken lines $ ABCDA $, where $ A, B, C, D $ are distinct vertices of a fixed regular 1985-gon. A broken line is randomly chosen from the set $ L $. Calculate the probability that it is the boundary of a convex quadrilateral.
Let $ W $ be the set of all vertices of a given regular polygon. Each broken line from the set $ L $ can be associated with a set of its vertices, i.e., a certain four-point subset of the set $ W $. Conversely, every four points from $ W $ are the set of vertices of exactly three different broken lines from the set $ L $ (Figure 6 shows three possible ways to connect these points with segments); of these three broken lines, one is the boundary of a convex quadrilateral. Thus, all broken lines in the set $ L $ can be grouped into triples such that in each triple one and only one is the boundary of a convex quadrilateral. The sought probability is therefore $ 1/3 $.