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.
It ocurred to me, while I was figuring this out, that this is eaxctly the same problem as the one where you figure out a yearlong budget when you have sporadic income and sporadic expenses. Provided that your income exactly equals your expenses, there is always at least one point in the year when you will have $0 in your account.
If you ever have less than $0 in your account it's time to ask for a raise.

Posted by Erik
on 20040603 14:56:56 