A Hungarian problem shortlisted for the 30th International Mathematical Olympiad, 1989:
Around a circular race track are n race cars, each at a different location. At a signal, each car chooses a direction and begins to drive at a constant speed that will take it around the course in 1 hour. When two cars meet, both reverse direction without loss of speed. Show that at some future moment all the cars will be at their original positions.
|
SelectClick for Answer |
Imagine that each car carries a flag, and that when two cars meet they exchange flags. Now each flag travels around the course with a constant direction and speed, and so after 1 hour each flag has returned to its starting position. Because each car reverses direction when it meets its neighbor, the order in which the cars are arranged around the track does not change. This means that the starting position has now been rotated, perhaps onto itself, and that after an integral number of hours the original position must be reached again.
See Measured Steps.
|
