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Topic: regular n-gon runners problem
Replies: 12   Last Post: Jul 25, 2013 4:26 AM

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quasi

Posts: 10,226
Registered: 7/15/05
Re: regular n-gon runners problem
Posted: Jul 23, 2013 3:42 AM
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quasi wrote:
>
>Suppose n runners, n >= 3, start at the same time and place
>on a circular track, and proceed to run counterclockwise along
>the track (forever), each at a distinct positive constant speed.
>
>Conjecture:
>
>If there is an instant where the locations of the n runners are
>the vertices of a regular n-gon, then the speeds of the runners,
>arranged in ascending order, form an arithmetic sequence.
>
>Remark:
>
>It's easy to see that the converse holds.


The converse holds but not the conjecture.

It's not that simple.

Here's an easy counterexample:

Take 3 runners on a track of circumference 1, with respective
speeds, expressed in revolutions per unit time, of 1,5,15.
Then at time t=1/6, the runners form an equilateral triangle.

quasi



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