2017 USAMO Problems/Problem 4
Problem
Let ,
,
,
be
distinct points on the unit circle
, other than
. Each point is colored either red or blue, with exactly
red points and
blue points. Let
,
,
,
be any ordering of the red points. Let
be the nearest blue point to
traveling counterclockwise around the circle starting from
. Then let
be the nearest of the remaining blue points to
travelling counterclockwise around the circle from
, and so on, until we have labeled all of the blue points
. Show that the number of counterclockwise arcs of the form
that contain the point
is independent of the way we chose the ordering
of the red points.