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

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William Elliot

Posts: 1,449
Registered: 1/8/12
Re: regular n-gon runners problem
Posted: Jul 23, 2013 3:51 AM
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On Tue, 23 Jul 2013, 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.


r1 <= r2 <=..<= r_n
For j = 1,.. n, dj = rj.t, t = dj/rj, dj = j.d1

rj/dj = r1/d1; rj = r1.dj/d1 = j.r1

> Remark: It's easy to see that the converse holds.




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