USAMO 2017 — Days 1 and 2 Question 4

There are $2n$ distinct points on the unit circle, none equal to $O=(1,0)$, colored with $n$ red and $n$ blue points. Order the red points arbitrarily as $R_1,\ldots,R_n$. Match $R_1$ to the first blue point encountered counterclockwise from it, call this $B_1$, then match $R_2$ to the first remaining blue point, and continue. Prove that the number of counterclockwise arcs $R_i\to B_i$ containing $O$ does not depend on the ordering of the red points.

Proposed by Maria Monks. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.