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Taiwan 2018 TST geometry

Source: 2018 Taiwan TST Round 2, Test 1, Problem 2

April 13, 2018
geometry proposedgeometry

Problem Statement

Let ABCABC be a triangle with circumcircle Ω\Omega, circumcenter OO and orthocenter HH. Let SS lie on Ω\Omega and PP lie on BCBC such that ∠ASP=90∘\angle ASP=90^\circ, line SHSH intersects the circumcircle of △APS\triangle APS at X≠SX\neq S. Suppose OPOP intersects CA,ABCA,AB at Q,RQ,R, respectively, QY,RZQY,RZ are the altitude of △AQR\triangle AQR. Prove that X,Y,ZX,Y,Z are collinear.
Proposed by Shuang-Yen Lee
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