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 solution
From what I gather, (and I'm not claiming to be anywhere near as smart as you, Charlie) but it sounded like what you said was if you start at the gas station (A, for ex.) where you would be at if the gas from the last station (Z) was barely enough to get you to that one (A) then you would you would have enough gas to go completely around the track without running out. But I don't see how you can be sure of that.
First of all it didn't say how many gas stations there were. Or how they were layed out on the course. Or how much gas there was at each one. So, there might not be enough gas at Z to get you to A.
This is how I figured it. First, pick any number for the number of stations. Say, 5. Represent each station with a %, for the amout of gas at that station. Now give each station another %, for the amount of track you have to drive to get to the next station from it. Both sets of numbers have to add up to 100, but any station can have a greater % for the amount of gas or a greater % for the amount of track to the next station. Or they could be equal.
If all the stations were spead out evenly along the track and each station had the same amount of gas then all the stations would have 20% for both numbers. But if you moved one station closer or farther to the next one, or put more or less gas at a certain station, then it would affect another station in the exact but opposite way. So, when you decrease any of the numbers, it would increase other numbers in that same set by the same amount.
So, the only way there would be NO station that you can start at to have enough gas to get you around the track is if the % for gas for EACH station is less than the % for the amount of track to the next station. But since both sets of numbers have to add up to 100, that's impossible.
Posted by Danny
on 2004-06-03 17:51:45