IMO Shortlist 2012 problem G8

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Dodao/la: arhiva
3. studenoga 2013.
Let ABC be a triangle with circumcircle \omega and l a line without common points with \omega. Denote by P the foot of the perpendicular from the center of \omega to l. The side-lines BC,CA,AB intersect l at the points X,Y,Z different from P. Prove that the circumcircles of the triangles AXP, BYP and CZP have a common point different from P or are mutually tangent at P.
Izvor: Međunarodna matematička olimpijada, shortlist 2012