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A hiker starts to climb up from base B to the summit S on sat 6am one day,spends the night at S and starts to climb down at 6am the next day.Prove that there is a point on the path B-S (there is only one path connecting B-S) which hiker crosses at the same time both days.

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I don't know a formal proof, but here is a way to think about it. Imagine that, instead of one man going up one day and coming down the next, there are two men, one going up and the other coming down on the same day, both starting at 6am. It is easy to see that they must cross paths, but this is really the same problem. Hope this helps.

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Note that this argument works because the man is climbing the mountain in a continuous way. – Amr Jan 5 '13 at 3:59

Do you know the intermediate value theorem? It states that if $f:[a,b] \rightarrow \mathbb{R}$ is a continuous function satisfying $f(a) = -1, f(b)=1$, then for any $x \in [-1,1]$ , there exists $c \in [a, b]$ such that $f(c) = x$.

Apply this to a suitable function describing your hiker's movement, and let $x=0$.

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@jim Ah yes! Thanks! – Calvin Lin Jan 5 '13 at 3:27

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