Three passengers $1,2,3$ starts out moving at constant speeds from $A$ to $B$. At the same time, a motorcyclist $M$ starts out from $B$ towards $A$ to pick the passengers up. As illustrated below:
$M$, however, can carry at most one passenger on board. He can drop his passenger off at anytime. When off the motorcycle, passengers just keep moving forward to $B$ at their respective speeds. $M$ can drive forward or backward, any way he wishes, at any speed not exceeding his top speed. Passengers will get on or off board at $M$'s bidding, and let $M$ drive them forward or backward with no complaint. We assume it takes no time to get on and off the motorcycle, or for $M$ to switch lanes to pick up different passengers.
$M$'s goal, or challenge, is to make all passengers arrive at $B$ simultaneously.
Now assume speeds for $1$ and $2$ are $60$ and $90$ respectively, and a top speed of $100$ for $M$.
Question: What is $3$'s speed range, if $M$ is able to accomplish his challenge?
Distance between $A$ and $B$ is irrelevant.
Here's a more involved one if you solved the above:
Instead of 3 passengers we now have 4, where $1,2,3$ have speeds $60,90,30$ respectively. Top speed for $M$ still is $100$. What is $4$'s speed range, if $M$ is able to accomplish his challenge?
In general, What relationship must the speeds for 3 passengers $s_1,s_2,s_3$ satisfy, given $s_m=100$ and $M$ is able to fulfill his challenge? What relationships must the speeds for 4 passengers satisfy?
If there are many passengers and the speed of $M$ is sufficiently large relative to the speeds of passengers. What is the most efficient way (aka requiring the least amount of time) for $M$ to accomplish his challenge?