Well done. I solved it the following way.
Suppose we have an imaginary car, that starts off with one lap worth of fuel. Let's put this imaginary car at exactly the same spot as one of our normal cars. So we run a lap along the track, taking fuel from each car we pass. Note that we cannot ever run out of fuel here, we start off with enough to finish the track. At the end of this, our imaginary car still has one lap of fuel in it. Now let us take a look at the graph of distance versus fuel in tank.
Our graph will start at 1, and fall downwards, and jump sharply up (when we meet a car and steal fuel), continue going down, and then up again, until we hit 1. Now we find the lowest point on this graph. This point must correspond to one of our original cars. Now we can discard our imaginary car, take this real car and drive. As we drive, we will not run out of fuel, becuause at no point in time will our fuel drop below what we started with. If it did, then we didn't pick the lowest point on our imaginary car graph.