SOLUTION: I need help with this problem - Two cyclists ride at 8:00 am: one from A to B, the other from B to A. Each of them rode along a constant speed along the same road and when each arr

The Figure below illustrates the situation:


    +--------------X--------------------+

    A  --->        C             <----  B


You see the starting points A and B  and the meeting point C.


They started simultaneously at 8:00 am and move towards each other.


They met at the point C at 11:00 am spending 3 hours each.



OK and very good.


Now, there is one EXCLUSIVE case, when their moving rates are the same.  In this case, they will get the ending points A and B 
simultaneously after 3 hours (i.e. at 2:00 pm), each will turn back, and, OBVIOUSLY, will meet next time 
in 3 hours after 2:00 pm (i.e. at 5:00 pm) exactly at the midpoint between A and B.



    Memorize this case and this answer:  6 hours after 11:00 am, at 5:00 pm.



Now we consider the general case, when their moving rates are different.

    It is a miracle, but we will get THE SAME ANSWER (!)



So, now I consider the general case, when they have different moving rates . . . 


Again, I consider their movements from A to B and their meeting at the point C.


Now warp the segment AB into a (or "the") circle, by uniting points A and B.


Imagine that it is your ORIGINAL configuration and that cyclists started at the same point A=B at this circle, 
moving in opposite directions.


You will get an EQUIVALENT model, but now the points move not along the segment AB, but along the circle circumference.


So, they started from one single common point A=B at 8:00 am, move in opposite directions along the circle 
and met at 11:00 somewhere at the point C on the circle.


Fantastic !  Let's move forward . . . (and further . . . )


They covered the entire distance AB, which we treat now as the circumference of the circle.


So, next they started at the point C on the circle and move in opposite directions.  When they will meet again ?


But of course, the situation after 11:00 am is exactly as it was at 8:00 am: they started from the common point on the circle 
(now it is point C instead of A=B) and move in opposite directions.


When they will meet each other next time ?


-- But of source, in 3 hours, i.e. at 2:00 pm, when they cover the length of the circumference again.    


Because the situation is REPEATED.


But stop for a minute and be careful (!)


In my consideration, they met each other at 2:00 pm, but it is the meeting on the circle - not in the rectilinear segment AB.


They need to cover the total circumference ONE MORE time in 3 hours - then their next meeting at 5:00 pm will be REAL MEETING

on the segment AB after reflections at the ends (!) (!) (!)


So, the answer is: they will meet each other on the segment AB next time 6 hours after 11:00 am, i.e. at 5:00 pm.

    On the circle, they meet each other every 3 hours, by covering (together) the circumference of the circle.

    But on the segment AB, they meet each other (accounting for reflections) every 6 hours.