> 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.
If I haven't made a mistake, your easy proof of the converse leads to an easy proof that pi/e is rational, which it's nice to have an easy proof of. :)
Let q, r, s be the speeds of runners m, n, o, and let them be equal to pi+5e, pi+6e, and pi+7e respectively. Let the track be an equilateral triangle with unit sides. Let d, g, f be the distances traveled at the first time t>0 when the locations of the n runners are the triangle's vertices. So d, g, f are three different positive integers. From d = t*q we have t = d/q. Let integer k = f - g > 0. Now k = t*(s-r) = t*e = e*d/q so that d*e = k*q = k*(pi+5*e) whence (d-5*k)*e = k*pi so that pi/e = (d-5*k)/k, which is rational.