An IMO Problem Involving Concurrency
Let A, B, C, and D be distinct points on a line. Let the circles with diameters AC and BD intersect at points X and Y. Let the intersection of BC and XY be Z. Let P be a point on XY different from Z. Let CP intersect the circle with diameter AC at the point M different from C and let BP intersect the circle with diameter BD at the point N different from B. Then AM, DN, and XY are concurrent.
This is problem 1 from the 36-th International Mathematical Olympiad.