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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: 2,637
Registered: 1/8/12
Re: regular n-gon runners problem
Posted: Jul 23, 2013 5:00 AM
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On Tue, 23 Jul 2013, quasi wrote:
> William Elliot 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

> The speeds were specified as distinct, so the inequalities are strict.

> >For j = 1,.. n, dj = rj.t, t = dj/rj, dj = j.d1
> >
> >rj/dj = r1/d1; rj = = j.r1

> I have no idea what you are trying to say above.

The j-th runner runs at a rate of rj for a distance dj.
At the time t, the runners form a regular polygon which
gives the equations dj = j.d1. Accordingly rj = j.r1
is an arithmetic sequence.

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