A monk starts to climb a mountain at 8:00 am and reaches the summit at noon. He spends the rest of the day and that night on the summit. The next morning he leaves the summit at 8:00 am and descends by the same route he used the day before, reaching the bottom at noon. Prove that there is time between 8:00 am and noon at which the monk was at exactly the same spot on the mountain on both days. Note that the monk can walk at different speeds, rest, or even go backward whenever he wants.
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