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

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James Waldby

Posts: 545
Registered: 1/27/11
Re: regular n-gon runners problem
Posted: Jul 23, 2013 6:23 PM
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On Tue, 23 Jul 2013 02:33:27 -0500, 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.

If I haven't made a mistake, your easy proof of the converse leads
to an easy proof that pi/e is rational, which it's nice to have an
easy proof of. :)

Let q, r, s be the speeds of runners m, n, o, and let them be
equal to pi+5e, pi+6e, and pi+7e respectively. Let the track be an
equilateral triangle with unit sides. Let d, g, f be the distances
traveled at the first time t>0 when the locations of the n runners
are the triangle's vertices. So d, g, f are three different positive
integers. From d = t*q we have t = d/q. Let integer k = f - g > 0.
Now k = t*(s-r) = t*e = e*d/q so that d*e = k*q = k*(pi+5*e) whence
(d-5*k)*e = k*pi so that pi/e = (d-5*k)/k, which is rational.


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