Two Runners Start a Race at the Same Time and Finish in a Tie — How Do You Prove That at Some Time During the Race They Have the Same Speed?

Suppose two runners start a race simultaneously and finish with the same time. How can we demonstrate that, at some point during the race, the two runners must have identical speeds?

Let's consider the race divided into infinitesimal intervals. During each interval, either the first or second runner is faster than the other. Since they finish at the same time, there must be an equal number of intervals in which each runner is faster. This implies that the total time the first runner is faster than the second runner is equal to the total time the second runner is faster than the first.

During the intervals when the first runner is faster, they cover more distance than the second runner. Conversely, during the intervals when the second runner is faster, they cover more distance than the first runner. Since the total distance covered by both runners is the same, there must exist a moment when the distance covered by both runners is identical. At this moment, the speeds of the two runners must be equal.

Related Questions and Brief Answers

  1. Why is the proof only applicable to runners with the same finishing time? Because it relies on the assumption that the total time spent with each runner being faster than the other is equal.
  2. Can the runners have the same speed at multiple points during the race? Yes, multiple instances of identical speeds are possible.
  3. Does the proof require the runners to run at constant speeds? No, it allows for variations in speed throughout the race.
  4. How can we determine the exact moment when the runners have the same speed? The proof does not provide a way to pinpoint the exact moment.
  5. What if the runners start at different times? The proof does not apply in this scenario.

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