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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: 10,208
Registered: 7/15/05
Re: regular n-gon runners problem
Posted: Jul 23, 2013 5:14 AM
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William Elliot wrote:
>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 = r1.dj/d1 = 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.


No, that's not forced.

For one thing, the vertices of the regular n-gon can
be a rotation of the standard one.

Also, multiple revolutions are allowed, potentially
a different number of revolutions for each runner.

>Accordingly rj = j.r1 is an arithmetic sequence.

It seems you're trying to prove the conjecture.

But note -- several counterexamples have already been posted.

quasi



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