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Calc Problem about a Racetrack

  1. Jun 3, 2013 #1
    1. The problem statement, all variables and given/known data
    Alright, so I have this problem from a textbook I'm using:

    There is a track (of unit length) with ##k## runners on it. At the time ##t=0##, the runners start from the same position and take off simultaneously. Each runner has a distinct speed. A runner is said to be "lonely," at a time ##t##, if the runner is a distance of ##\frac{1}{k}## from any other runner. Prove that each runner will be lonely at some point in time.


    3. The attempt at a solution

    So I obviously need to use induction for this. The case for ##k=1## is trivial, because there are no other runners, so that runner will always be a distance of ##\frac{1}{k}## from other runners. However, I'm stuck at this point, and am having trouble for cases where ##k\geq2##.
     
  2. jcsd
  3. Jun 3, 2013 #2

    Dick

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    What kind of calculus course is this? k=1 is trivial and you should certainly be able to handle k=2. Beyond that it gets really hairy. http://en.wikipedia.org/wiki/Lonely_runner_conjecture Looks to me like it's actually an unsolved problem.
     
  4. Jun 4, 2013 #3
    Oh... wow :smile: That would explain my difficulty with the problem. Thanks for figuring that out!
     
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