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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: 9,903
Registered: 7/15/05
Re: regular n-gon runners problem
Posted: Jul 23, 2013 4:28 AM
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Virgil wrote:
>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.

>
>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.


Yes, it fails -- I realized as much soon after I posted.

Generalizing your example, using a track of circumference 1,
the speeds, in revolutions per unit time, could just as well
be any sequence v_1,v_2,...,v_n of positive integers such that
v_i = i (mod n). Then at time t = 1/n, the runners are at the
vertices of a regular n-gon.

quasi



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