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Logic puzzle A sage stays atop a mountain in the Himalayas. Every morning he starts at 5:00 and reaches the foothills at 9:00. He travels non-uniformly, walking leisurely at times and at times breaking into a trot. He also takes rest at some points, but never digresses from his path. In the evening he starts at 5:00 to climb up in the same manner on the exact same path and reaches the top exactly at 9:00. On any given day, will the sage be at any point on his route at the same time in the morning and evening? Explain your answer.

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Yep. If you ask another person to go the same route in the evening as the sage does in the morning but he starts from the top, they will meet regardless of how they travel.
I've read it a few times and unless I'm misunderstanding the question, I believe the answer is no. The question states he travels non-uniformly: It always takes him 4 hours to get there but his speed varies throughout the neither his velocity or acceleration are constant.
It's possible however very unlikely

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Other answers:

Looks like an IVT problem

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