IMO Shortlist 2005 problem G6

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Let ABC be a triangle, and M the midpoint of its side BC. Let \gamma be the incircle of triangle ABC. The median AM of triangle ABC intersects the incircle \gamma at two points K and L. Let the lines passing through K and L, parallel to BC, intersect the incircle \gamma again in two points X and Y. Let the lines AX and AY intersect BC again at the points P and Q. Prove that BP = CQ.
Source: Međunarodna matematička olimpijada, shortlist 2005