I found this neat problem in Peter Winkler's excellent book, Mathematical Puzzles: A Connoisseur's Collection. I've dressed it up a little.
You have planned an expedition to travel in a 8000 mile loop around Antarctica. Your advance team has set up 20 fuel caches along the route, and has distributed 8000 miles worth of fuel among the caches. You know the amount of fuel at each cache, and the amount of fuel required to travel between any two consecutive caches. Prove that, regardless of the spacing of the caches or the amounts of fuel in each cache, you can complete the trip, assuming that you have an infinitely large fuel tank. Determine how to pick a cache you can start from.
(This might be an elementary problem, depending on how you look at it.)