USAMO 2017 Problem 4

Let $P_1$ , $P_2$ , $\dots$ , $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$ , other than $(1,0)$ . Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$ , $R_2$ , $\dots$ , $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$ . Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$ , and so on, until we have labeled all of the blue points $B_1, \dots, B_n$ . Show that the number of counterclockwise arcs of the form $R_i \to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \dots, R_n$ of the red points.