2019 AMC 10 B
Complete problem set with solutions and individual problem pages
Henry decides one morning to do a workout, and he walks of the way from his home to his gym. The gym is kilometers away from Henry's home. At that point, he changes his mind and walks of the way from where he is back toward home.When he reaches that point, he changes his mind again and walks of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point kilometers from home and a point kiometers from home. What is ? (2019 AMC 10B Problem, Question#18)
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Let the two points that Henry walks in between be and , with being closer to home. As given in the problem statement, the distances of the points and from his home are and respectively.
By symmetry, the distance of point from the gym is the same as the distance from home to point .
Thus, . In addition, when he walks from point to home, he walks of the distance，
ending at point . Therefore, we know that . By subsituting, we
get . Adding these equations now gives .
Multipiying by , we get , so .
We assume that Henry is walking back and forth exactly between points and , with closer to Henry’s home than . Denote Henry’s home as a point and the gym as a point .
Then and , so .
Therefore, .
