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.
For every pair of income (gas station) and expense (length of track after the gas station) there is an associated net income (surplus or shortage of gas).
If you plot the net incomes there will be at least one maximum point and at least one minimum point.
I think the possible starting stations will coincide with with the minimum points (if there is more than one minimum).
Posted by Erik
on 2004-06-03 14:48:47