This problem was provided by Dr. Tony Bedenikovic. 

NOTE:  Due to the upcoming Thanksgiving Day Break, this problem will remain open for two weeks.  Happy Thanksgiving!

A tetrahedron is made of four equilateral triangles joined along their sides so as to form a pyramid, see the sketch on the left.  Imagine that on each of the four faces of the tetrahedron, a car is traveling in a clockwise direction at a constant rate of speed along the edges bounding the face.  (Each of the four cars may be traveling at different speeds, and may start anywhere along the boundary of the face.)  Can the cars always keep from crashing, or is there destined to be an accident along some edge at some time?