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 \(R_2\) traveling 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.



Show that swapping \(R_i\) and \(R_{i+1}\) does not change the number of arcs crossing \((1,0)\).

post by Shen-Fu Tsai