> 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 For j = 1,.. n, dj = rj.t, t = dj/rj, dj = j.d1
rj/dj = r1/d1; rj = r1.dj/d1 = j.r1
> Remark: It's easy to see that the converse holds.