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

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Virgil

Posts: 9,012
Registered: 1/6/11
Re: regular n-gon runners problem
Posted: Jul 23, 2013 3:21 AM
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In article <qlbsu89ktgnhgni2veeedjmfoe8m665nls@4ax.com>,
quasi <quasi@null.set> 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.
>
> quasi


Suppose that the speeds of the first n-1 runners are in proportion to
1:2:...:n-1 but the last speed is in proportion to 2*n instead of n.

THen when the fastest finishes his second full lap, the rummers will all
be at the vertices of a regular n-gon.

So it would appear as if the conjecture is false.
--





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