Suppose a race track in the shape of a simple closed curve (i.e. it does not intersect itself), and suppose that distributed along it, in any way whatsoever, are N gas stations. Suppose that a race car needs L liters of gas to go completely around the track, and that the sum of the gas available at the N stations is exactly L.

Consider that the car cannot move without gas (*assume it can't travel by momentum alone, thus it MUST need gas to move*), and that it has constant mileage (consumption of gas is directly proportional to the distance the car moves, and depends on nothing else).

Prove that there exists at least one gas station such that, starting from it, the car can do a full lap.

(In reply to re(2): solution - I don't think that proves the question by Charlie)

 I understand what you're saying now (after reading it again) but the question was "Prove that there exists at least one gas station such that, starting from it, the car can do a full lap." So basically your answer is that whatever station is the low point would be the only station that you can start from to make it all the way around. And if more than one station reaches the same low point then you can start from either of them. But there would always be at least one station that works because there has to be at least one lowest number given any set of numbers.

Posted by Danny on 2004-06-03 23:08:01