IMO Team Selection Test Problems
1993/Test 2/2.
Let AB and CD be distinct parallel chords of a circle and let P be a point on their common perpendicular bisector. The lines AP and CP meet the circle again at Q and R respectively. If S is the intersection of the lines BR and QD, prove that PS is parallel to AB.
1995/Test 2/6.
The King and the Queen invited n other married couples to the palace for a round-table dinner. In order that each person (including the King and the Queen) had a chance to sit beside every person except his/her spouse during the dinner, the King and the Queen decided to have a different sitting arrangement for each course of food served. Prove that they can achieve their purpose if n courses of food are served, but not fewer.