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
Just to show how to apply my cryptic method, let's do Charlie's example. Assume station 1 is at distance 0, with 40% of the gas; station 2, at 30 with 20%; C, at 45 with 5%; D, at 70 with 20%; and E at 95 with 15%.
From 1 you can reach 2, so we add the gas at 2 to the gas at 1, and forget about 2: we now have station 1 at 0 with 60%, and stations 3 to 5 as before.
From 1 you can now reach 3, so we add the gas at 3 to the gas at 1, and forget 3: we have station 1 with 65%, and stations 4 and 5 as before.
From station 5 we can reach station 1, so we add the gas at 1 to the gas at 5, forget 1, and we are left with station 4 at 70 with 20%, and station 5 at 95 with 80%.
Finally, we can reach 4 from 5, so the solution requires starting from 5, as Charlie found.