Calc Problem about a Racetrack

[AnTiFreeze3]

Homework Statement

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.

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 $k\geq2$.

 

[Dick]

Homework Statement

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.

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 $k\geq2$.

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.

 

[AnTiFreeze3]

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.

Oh... wow That would explain my difficulty with the problem. Thanks for figuring that out!