William Elliot 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. > >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