IMO Shortlist 2005 problem G6

  Avg: 0,0
  Avg: 8,0
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.
Izvor: Međunarodna matematička olimpijada, shortlist 2005