This one's right -- I'm sure of it. In fact, I can see how to prove it, but for now, I'll just state it as a conjecture.
The revision ...
Suppose n runners, n >= 3, start at the same time and place on a circular track of circumference 1, and proceed to run counterclockwise along the track (forever). Assume the speeds of the runners, expressed in revolutions per unit time, are pairwise distinct positive real numbers.
There is an instant of time where the locations of the runners are the vertices of a regular n-gon iff for some permutation v_1,v_2, ..., v_n of the n speeds, each of the n fractions
(v_i - v_1)/(v_2 - v_1)
for i = 1,2,...,n is a rational number, and moreover, when reduced to lowest terms, a_i/b say, we have b = 1 mod n and a_i = i-1 mod n (Thus, a_1,a_2, ..., a_n yield all possible residues mod n).