IMO Shortlist 2015 problem G2


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Triangle ABC has circumcircle \Omega and circumcenter O. A circle \Gamma with center A intersects the segment BC at points D and E, such that B, D, E, and C are all different and lie on line BC in this order. Let F and G be the points of intersection of \Gamma and \Omega, such that A, F, B, C, and G lie on \Omega in this order. Let K be the second point of intersection of the circumcircle of triangle BDF and the segment AB. Let L be the second point of intersection of the circumcircle of triangle CGE and the segment CA.

Suppose that the lines FK and GL are different and intersect at the point X. Prove that X lies on the line AO.

(Greece)

Izvor: https://www.imo-official.org/problems/IMO2015SL.pdf