A man starts walking up a narrow mountain path at 6:00 a.m. He walks at different speeds, stopping to eat, sometimes going back a few steps to look at a flower, but never leaving the path. He eventually arrives at the top at 10:00 p.m. the same day. He camps out over night and starts down the same path at 6:00 a.m. the next day. After stopping to eat and so forth he arrives at the bottom at 10:00 p.m.
True or False: There is some point on the path that the man occupied at exactly the same time on the two different days. Explain your answer.
Imagine that on the first day a clone of the man starts down the mountain at 6:00 a.m. and arrives at the bottom by 10:00 p.m. The man and the clone have to pass each other at some point.
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