MathDB
Turkey NMO 2009 Q4

Source:

August 31, 2010
geometrygeometry proposed

Problem Statement

Let HH be the orthocenter of an acute triangle ABC,ABC, and let A1, B1, C1A_1, \: B_1, \: C_1 be the feet of the altitudes belonging to the vertices A, B, C,A, \: B, \: C, respectively. Let KK be a point on the smaller AB1AB_1 arc of the circle with diameter ABAB satisfying the condition ∠HKB=∠C1KB.\angle HKB = \angle C_1KB. Let MM be the point of intersection of the line segment AA1AA_1 and the circle with center CC and radius CLCL where KB∩CC1={L}.KB \cap CC_1=\{L\}. Let PP and QQ be the points of intersection of the line CC1CC_1 and the circle with center BB and radius BM.BM. Show that A, K, P, QA, \: K, \: P, \: Q are concyclic.
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