There are n cars on a circular track. Among all of them, they have exactly enough fuel (in total) for one car to circle the track. Prove, using whatever method you want, that there exists at least one car that has enough fuel to reach the next car along the track.

Question

Mathematics

Wolfgang

11 March, 16:03

There are n cars on a circular track. Among all of them, they have exactly enough fuel (in total) for one car to circle the track. Prove, using whatever method you want, that there exists at least one car that has enough fuel to reach the next car along the track.

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Answers (1)

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Atticus · Answer 1

Based on mathematical induction, we conclude that for every n, there is at least one car that has enough fuel to reach the next car.

Step-by-step explanation:

We will solve the problem by mathematical induction.

For n=1. If we have one car, by assumption of the task, it will have enough fuel to cross the entire lane, and again reach where it was.

For n=2. If we have two cars, there will be fuel for the entire track in their tanks. As the track is circular, we can choose a shorter distance from the first car to the second car, and choose a car that has more fuel. So we will be sure to get from one car to the next.

Suppose this is true for n cars. We will show that this applies to the n + 1 cars.

If we have an n + 1 cars, we know from the assumption that there are at least one car in the n car that has enough fuel to reach the next car.

We conclude that in an n + 1 cars, there is at least one car that has enough fuel to reach the next car.

Based on mathematical induction, we conclude that for every n, there is at least one car that has enough fuel to reach the next car.