IMO Shortlist 1989 problem 12


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2. travnja 2012.
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There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.
Izvor: Međunarodna matematička olimpijada, shortlist 1989