Virgil 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. >> >> Remark: >> >> It's easy to see that the converse holds. > >Suppose that the speeds of the first n-1 runners are in >proportion to 1:2:...:n-1 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 n-gon. > >So it would appear as if the conjecture is false.
Yes, it fails -- I realized as much soon after I posted.
Generalizing your example, using a track of circumference 1, the speeds, in revolutions per unit time, could just as well be any sequence v_1,v_2,...,v_n of positive integers such that v_i = i (mod n). Then at time t = 1/n, the runners are at the vertices of a regular n-gon.