IMO Shortlist 2015 problem G7

  Avg: 0,0
  Avg: 9,0

Let ABCD be a convex quadrilateral, and let P, Q, R, and S be points on the sides AB, BC, CD, and DA, respectively. Let the line segment PR and QS meet at O. Suppose that each of the quadrilaterals APOS, BQOP, CROQ, and DSOR has an incircle. Prove that the lines AC, PQ, and RS are either concurrent or parallel to each other.