Difference between revisions of "2017 USAJMO Problems/Problem 6"

(Created page with "==Problem== Let <math>P_1, \ldots, 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...")
 
Line 3: Line 3:
  
 
==Solution==
 
==Solution==
 +
 +
{{MAA Notice}}
 +
 +
==See also==
 +
{{USAJMO newbox|year=2017|num-b=2|aftertext=|after=LastProblem}}

Revision as of 20:38, 20 April 2017

Problem

Let $P_1, \ldots, P_{2n}$ 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 $n$ of them red and exactly $n$ of them blue. Let $R_1, \ldots, R_n$ 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 the blue points $B_1, \ldots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \rightarrow B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \ldots, R_n$ of the red points.

Solution

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

See also

2017 USAJMO (ProblemsResources)
Preceded by
Problem 2
LastProblem
1 2 3 4 5 6
All USAJMO Problems and Solutions
Invalid username
Login to AoPS