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.
The converse holds but not the conjecture.
It's not that simple.
Here's an easy counterexample:
Take 3 runners on a track of circumference 1, with respective speeds, expressed in revolutions per unit time, of 1,5,15. Then at time t=1/6, the runners form an equilateral triangle.