Subcontests
(6)Red and blue points
Let P1, P2, …, P2n be 2n distinct points on the unit circle x2+y2=1, other than (1,0). Each point is colored either red or blue, with exactly n red points and n blue points. Let R1, R2, …, Rn be any ordering of the red points. Let B1 be the nearest blue point to R1 traveling counterclockwise around the circle starting from R1. Then let B2 be the nearest of the remaining blue points to R2 travelling counterclockwise around the circle from R2, and so on, until we have labeled all of the blue points B1,…,Bn. Show that the number of counterclockwise arcs of the form Ri→Bi that contain the point (1,0) is independent of the way we chose the ordering R1,…,Rn of the red points.