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. No matter how he varies his travel speed within the two trips, there must indeed be such a point somewhere along the path.
An easy way to visualize this is to imagine, instead of one man making one trip and then making the return trip, two men making the trip at the same time. One man leaves the bottom at 1pm and heads toward the top. The other leaves the top at 1pm and heads toward the bottom. Regardless of their rate of travel over the course of the trip, they must pass each other on their respective journeys -- or, in other words, that at some point they must be at the same place at the same time.
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