Problem:
A foot long moving walkway moves at a constant rate of feet per second. steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person.
Solution:
Let be Al's travel time. Then is Bob's time, and is Cy's time, and . If is in the middle, then , which has no solution. If Bob is in the middle, then , which has solution . But , so this is impossible. If is in the middle, then , which has solution . In this case, is feet from the start and is feet from both Bob and . Thus the required distance is .
The problems on this page are the property of the MAA's American Mathematics Competitions