Solution From Problem Solving Strategies by A Engel:
The theorem is obvious for (n=1 .) Suppose we have proven the theorem for (n). Let there be (n+1) cars. Then there is a car (A) which can reach the next car (B). (If no car could reach the next car, there would not be enough fuel for one lap.) Let us empty (B) into (A) and remove (B). Now we have (n) cars which, between them, have enough fuel for one lap. By the induction hypothesis, there is a car which can complete a lap. The same car can also get around the track with all ((n+1)) cars on the road. From (A) to (B,) there will be enough gas (from car (A) ) and, on the remaining road sections, this car has the same amount of gas as in the case of (n) cars.
.