On Tue, 23 Jul 2013, 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. Accordingly rj = j.r1 is an arithmetic sequence.