Here's a math puzzle. Taken from the 46th International Mathematical Olympiad, held in rainy Merida, Mexico:
In a mathematical competition 6 problems were posed to the contestants. Each pair of problems was solved by more than two-fifths of the contestants. Nobody solved all 6 problems. Show that there were at least 2 contestants who each solved exactly 5 problems.Give yourself four hours and thirty minutes. Incidentally, Romania won four gold medals, a silver, and a bronze at the Olympiad out of her six contestants. They're all high school students.