2019 AMC 10B Problems/Problem 18
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 kilometers from home. What is ?
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 substituting, we get . Adding these equations now gives . Multiplying by , we get , so .
Solution 2 (not rigorous)
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, .
Solution 3 (not rigorous; similar to 2)
Since Harry is very close to walking back and forth between two points, let us denote closer to his house, and closer to the gym. Then, let us denote the distance from to as . If Harry was at and walked of the way, he would end up at , vice versa. Thus we can say that the distance from to the gym is the distance from to his house. That means it is . This also applies to the other side. Furthermore, we can say + + = . We solve for and get . Therefore, the answer is .
For those who want a video solution: https://youtu.be/45kdBy3htOg
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