# A quiz for bike riders

After riding the bike many days on the same roads, I came to the following interesting exercise: find a speed, such that you can reach your destination with this speed without having to stop at a red light. Of course we assume, that we always stop at red light.

So the exercise is the following. Consider a finite number of traffic lights, $a_0, \ldots,a_n$, and suppose that the bike rider starts when $a_0$ becomes green. Then let $T_1,\ldots,T_n$ be the times for which the respective lights are green, and $F_1,\ldots,F_n$ be the times for which the lights are red. Further, let $d_1,\ldots,d_n$ be the distances between the lights, and suppose that we know at t = 0 the remaining green or red time for every light. Let these times be $R_1,\ldots,R_n$ with positive times for remaing green and negative for remaining red time. Determine the set of all possible constant speeds (also unrealistic) for which the rider can pass through without having to stop.

P.S. $(R_1,\ldots,R_n)$ could be any n-tuple from $\{(x_1,\ldots,x_n)|x_i \in [-F_i,T_i] \}$

P.S.2 He he, it is easy to find some trivial funny unrealistic solutions. But we want all solutions!