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Difference between revisions of "2017 USAMO Problems/Problem 4"

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==Problem==
 
==Problem==
 
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.
 
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==

Revision as of 15:02, 22 December 2017

Problem

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.

Solution

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