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| ==Problem==
| | #REDIRECT[[2017_USAJMO_Problems/Problem_6]] |
| Let <math>P_1</math>, <math>P_2</math>, <math>\dots</math>, <math>P_{2n}</math> be <math>2n</math> distinct points on the unit circle <math>x^2+y^2=1</math>, other than <math>(1,0)</math>. Each point is colored either red or blue, with exactly <math>n</math> red points and <math>n</math> blue points. Let <math>R_1</math>, <math>R_2</math>, <math>\dots</math>, <math>R_n</math> be any ordering of the red points. Let <math>B_1</math> be the nearest blue point to <math>R_1</math> traveling counterclockwise around the circle starting from <math>R_1</math>. Then let <math>B_2</math> be the nearest of the remaining blue points to <math>R_2</math> travelling counterclockwise around the circle from <math>R_2</math>, and so on, until we have labeled all of the blue points <math>B_1, \dots, B_n</math>. Show that the number of counterclockwise arcs of the form <math>R_i \to B_i</math> that contain the point <math>(1,0)</math> is independent of the way we chose the ordering <math>R_1, \dots, R_n</math> of the red points.
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| ==Solution==
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