From the 2014 Mathcamp quiz: Hannah is about to get into a swimming pool in which every lane already has one swimmer in it. Hannah wants to choose a lane in which she would have to encounter the other swimmer as infrequently as possible. All swimmers, including Hannah, swim back and forth at constant speeds, never pausing at the ends of the pool. Hannah swims at speed $1$ (one pool length per minute).
a.Say Hannah chooses a lane with a swimmer who swims at speed $0<s<1$. Prove that, if they keep swimming long enough, eventually Hannah and the other swimmer will settle into a pattern where they pass each other (either in the same or in opposite directions, or at the edge of the pool) exactly N times every M minutes, where M and N are relatively prime integers. Find M and N. Do they depend on the other swimmer's speed and/or initial position when Hannah enters the pool?
b.What if Hannah chooses a lane with a swimmer whose speed is $s>1$?
c.From Hannah's point of view, what is the ideal nonzero speed that another swimmer can have? (Assume Hannah can time her entry into the pool with perfect precision, so she can make the other swimmer's initial position be whatever she wants.)
First I identified two functions to describe their positions with respect to time $S(t)$for the swimmer and $H(t)$ for Hannah(The graphs are triangular waves, thus allowing me to express the functions in closed forms). Then I encountered the following problem: If $s \in \mathbb Q$, the problem will be reduced to the discussion over the zeros of a periodic function $D(t)=H(t)-S(t)$, which is relatively trivial. If $s \in \mathbb R$, however I don't see a clear pattern of the distribution of zeros of $D(t)$ after playing around with Sketchpad. How would you deal with this case? Am I on the right track or should I consider this problem in some other perspective?