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JBMO Shortlist 2020 G2

Source: JBMO Shortlist 2020

July 4, 2021
JuniorBalkanshortlist2020geometry

Problem Statement

Let △ABC\triangle ABC be a right-angled triangle with ∠BAC=90∘\angle BAC = 90^{\circ}, and let EE be the foot of the perpendicular from AA to BCBC. Let Z≠AZ \neq A be a point on the line ABAB with AB=BZAB = BZ. Let (c)(c) and (c1)(c_1) be the circumcircles of the triangles △AEZ\triangle AEZ and △BEZ\triangle BEZ, respectively. Let (c2)(c_2) be an arbitrary circle passing through the points AA and EE. Suppose (c1)(c_1) meets the line CZCZ again at the point FF, and meets (c2)(c_2) again at the point NN. If PP is the other point of intersection of (c2)(c_2) with AFAF, prove that the points NN, BB, PP are collinear.
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