Problem:
Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running in the east/west direction. Jon rides east at miles per hour, and Steve rides west at miles per hour. Two trains of equal length, traveling in opposite directions at constant but different speeds, each pass the two riders. Each train takes exactly minute to go past Jon. The westbound train takes times as long as the eastbound train to go past Steve. The length of each train is miles, where and are relatively prime positive integers. Find .
Solution:
Let the length of each train be miles. In passing each rider, each train travels miles relative to that rider. Because the trains each go past Jon in minute, their speed relative to Jon is miles per minute. Jon and Steve are each riding at a speed of mile per minute in opposite directions. Therefore, relative to Steve, the speed of the eastbound train is miles per minute, and the speed of the westbound train is miles per minute. The times required for the trains to go past Steve are and , respectively. Thus , from which . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions