quasi
Posts:
12,012
Registered:
7/15/05


Re: regular ngon runners problem
Posted:
Jul 23, 2013 5:14 AM


William Elliot wrote: >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 ngon, 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 jth 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.
No, that's not forced.
For one thing, the vertices of the regular ngon can be a rotation of the standard one.
Also, multiple revolutions are allowed, potentially a different number of revolutions for each runner.
>Accordingly rj = j.r1 is an arithmetic sequence.
It seems you're trying to prove the conjecture.
But note  several counterexamples have already been posted.
quasi

