USAMO 2017 Problem 4
source link: http://siongui.github.io/2017/04/20/usamo-2017-problem-4/
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USAMO 2017 Problem 4
April 20, 2017
Let P1,P2,…,P2nP1,P2,…,P2n be 2n2n distinct points on the unit circle x2+y2=1x2+y2=1, other than (1,0)(1,0). Each point is colored either red or blue, with exactly nn red points and nn blue points. Let R1,R2,…,RnR1,R2,…,Rn be any ordering of the red points. Let B1B1 be the nearest blue point to R1R1 traveling counterclockwise around the circle starting from R1R1. Then let B2B2 be the nearest of the remaining blue points to R2R2 traveling counterclockwise around the circle from R2R2, and so on, until we have labeled all of the blue points B1,…,BnB1,…,Bn. Show that the number of counterclockwise arcs of the form Ri→BiRi→Bi that contain the point (1,0)(1,0) is independent of the way we chose the ordering R1,…,RnR1,…,Rn of the red points.
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Solution:
Show that swapping RiRi and Ri+1Ri+1 does not change the number of arcs crossing (1,0)(1,0).
post by Shen-Fu Tsai
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