Problem:
Points , and lie in that order along a straight path where the distance from to is meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at and running toward , Paul starting at and running toward , and Eve starting at and running toward . When Paul meets Eve, he turns around and runs toward . Paul and Ina both arrive at at the same time. Find the number of meters from to .
Solution:
Let be the number of meters from to . Then the distance from to is meters. Because Paul runs times as fast as Eve, he covers of the distance from to while Eve covers of the distance, so Paul has run meters when he meets Eve. When Paul meets Ina he has run meters, and Ina has run meters. Because Paul runs twice as fast as Ina, he must have run meters in the time that he ran meters. Thus , and .
The problems on this page are the property of the MAA's American Mathematics Competitions