Problem:
Ana, Bob, and Cao bike at constant rates of meters per second, meters per second, and meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point on the south edge of the field. Cao arrives at point at the same time that Ana and Bob arrive at for the first time. The ratio of the field's length to the field's width to the distance from point to the southeast corner of the field can be represented as , where , and are positive integers with and relatively prime. Find .
Solution:
Let be the length and the width of the field in meters with , and let be the distance in meters from point to the southeast corner of the field. According to the problem specifications, Ana bikes meters, Bob bikes meters, and Cao bikes meters, all in the same time. Thus it must be true that
Simplifying the first equality yields . Squaring both sides of the second equality and simplifying yields . The left side factors into , and so or . Substituting into the expression for yields
or
The second value for violates the requirement that , so and . The smallest value of for which all of , and are integers is . Thus the required ratio is , and .
The problems on this page are the property of the MAA's American Mathematics Competitions