2006 AMC 10A Problems/Problem 15
Odell and Kershaw run for minutes on a circular track. Odell runs clockwise at and uses the inner lane with a radius of meters. Kershaw runs counterclockwise at and uses the outer lane with a radius of meters, starting on the same radial line as Odell. How many times after the start do they pass each other?
Since , we note that Odell runs one lap in minutes, while Kershaw also runs one lap in minutes. They take the same amount of time to run a lap, and since they are running in opposite directions they will meet exactly twice per lap (once at the starting point, the other at the half-way point). Thus, there are laps run by both, or meeting points.
We first find the amount of minutes, , until Odell and Kershaw's next meeting. Let be the angle in radians between their starting point and the point where they first meet, measured counterclockwise.
Since Kershaw has traveled meters at this point and the circumference of his track is , . Similarly, since Odell has traveled meters in the opposite direction and the circumference of his track is .
Solving for in the second equation, we get . Then, from the first equation, we have . Solving for , we get . After minutes, they are back at the same position, except rotated, so they will meet again in minutes. So the total amount of meetings is .
Solution 3 by Alcumus
Since Odell's rate is that of Kershaw, but Kershaw's lap distance is that of Odell, they each run a lap in the same time. Hence they pass twice each time they circle the track. Odell runs as does Kershaw. Because , they pass each other times.
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