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2016-2017 Fall OMO Problem 26

Source:

November 16, 2016
geometryOmo

Problem Statement

Let ABCABC be a triangle with BC=9BC=9, CA=8CA=8, and AB=10AB=10. Let the incenter and incircle of ABCABC be II and γ\gamma, respectively, and let NN be the midpoint of major arc BCBC of the cirucmcircle of ABCABC. Line NINI meets the circumcircle of ABCABC a second time at PP. Let the line through II perpendicular to AIAI meet segments ABAB, ACAC, and APAP at C1C_1, B1B_1, and QQ, respectively. Let B2B_2 lie on segment CQCQ such that line B1B2B_1B_2 is tangent to γ\gamma, and let C2C_2 lie on segment BQBQ such that line C1C2C_1C_2 tangent to γ\gamma. The length of B2C2B_2C_2 can be expressed in the form mn\frac{m}{n} for relatively prime positive integers mm and nn. Determine 100m+n100m+n.
Proposed by Vincent Huang
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