A set of fuel dumps on a circular racetrack have just enough gasoline for one car to make one round trip. Prove that there exists a fuel dump from which one car, starting with an empty gas tank, can complete the round trip.

From The Mathematical Gardner (1982), page 283. An article by Ross Honsberger attributes this puzzle to Laszlo Lovasz. Also discussed in Mathematical Puzzles: A Connoisseur's Collection (2003, 163 pages) by Peter Winkler.

There must exist a fuel dump that allows the car to reach the next fuel dump. Remove the second fuel dump and transfer its fuel to the first fuel dump. Continuing this way, we must be left with one fuel dump. An alternative solution at www.cut-the-knot.org.