A man leaves home for a mountain at 1pm and reaches the top at 3pm. The following day he departs from the top at 1pm and gets home at 3pm, by following the same path as the day before. Was he necessarily ever at the same point on the path at the same time on both days?

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Yes. Necessarily. It's called physics.

Replace the context with two people (one walking up, one walking down) on the same path. If they are on the same path, they must meet (i.e., by at the same place at the same time).

Yes. Draw a graph on a Cartesian coordinate plane, labeling the x-axis time from 1pm to 3pm and the y-axis distance from home. Draw two functions of where the man with respect to time. No matter how you draw it, there's always an intersection between the functions, where he's at the same point on the path at the same time.

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