¶ 2026 AMC8 Problem 19
Problem:
Miguel is walking with his dog, Luna. When they reach the entrance to a park, Miguel throws a ball straight ahead and continues to walk at a steady pace. Luna sprints toward the ball, which stops by a tree. As soon as the dog reaches the ball, she brings it back to Miguel. Luna runs times faster than Miguel walks. What fraction of the distance between the entrance and the tree does Miguel cover by the time Luna brings him the ball?
Answer Choices:
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Solution:
Let denote the distance between the park entrance and the tree. Note that the total distance that Luna and Miguel travel together is . Additional, note that for every unit of distance Miguel travels, Luna travels units of that distance. If we let and be the distances that Miguel and Luna traveled, respectively, then we have that
From here, we can substitute in the first equation to get that
which gives . Since , the fraction of the distance between the entrance and the tree is just .
The problems on this page are the property of the MAA's American Mathematics Competitions