Virgil
Posts:
9,012
Registered:
1/6/11


Re: regular ngon runners problem
Posted:
Jul 23, 2013 3:21 AM


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 ngon, 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 n1 runners are in proportion to 1:2:...:n1 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 ngon.
So it would appear as if the conjecture is false. 

