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# The holy mans walk up the mountain

## Question

A holy man starts up a mountain at 7:00 am ; Arrives at the top at 7:00 pm   He goes thru some worship ceremony  and the next morning ,and the next morning at 7:00 am he starts back down the mountain.   He arrives at the bottom at 7:00 pm .   Is there a place where going up and coming down he was at the place at the same time both days. Proof please

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Yes, but I can't tell you where (assuming there's only one path). Depending on how the holy man's speed varies between the two trips, there must be a place where the two trips are at the same point at the same time, but you couldn't say where without more information.

The proof is simple: if we imagine that there are two holy men, one going up and one coming down, there is no way that the two men do not meet each other (in order to get to the other end of the mountain, they must both take a continuous path from top to bottom or bottom to top, which is impossible to do without meeting each other). In order to meet each other, they have to be at the same point at the same time. Hence, the two different paths on the two different days must meet at some point.

I'm certain there's another way to do this via calculus theorems (maybe intermediate value or mean value, something like that), but I just can't think of it at the moment.

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23 minutes ago, flamebirde said:
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Yes, but I can't tell you where (assuming there's only one path). Depending on how the holy man's speed varies between the two trips, there must be a place where the two trips are at the same point at the same time, but you couldn't say where without more information.

The proof is simple: if we imagine that there are two holy men, one going up and one coming down, there is no way that the two men do not meet each other (in order to get to the other end of the mountain, they must both take a continuous path from top to bottom or bottom to top, which is impossible to do without meeting each other). In order to meet each other, they have to be at the same point at the same time. Hence, the two different paths on the two different days must meet at some point.

I'm certain there's another way to do this via calculus theorems (maybe intermediate value or mean value, something like that), but I just can't think of it at the moment.

Yes you cand do that. The elevation of the two men as they ascend and descend are continuous functions of time. If at one time one is greater and at a later time the other is greater there must be a time when they are equal. Since they both stay on the path, and if the the slope of the path never changes sign, then they meet at that time.

If the slope does change sign, then multiple points on the path will have the same elevation and there can be multiple times when the elevation of the two men are the same. They meet at one of those times.

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Two of you got the answer ,but seemingly struggled with the simplicity of the solution.   i.e. If a second man started up the mountain at the same time the other was coming down...regardless of their relative speeds  ,stopping to rest , at some point they must meet , and obviously at the same time.

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3 hours ago, Donald Cartmill said:

Two of you got the answer ,but seemingly struggled with the simplicity of the solution.   i.e. If a second man started up the mountain at the same time the other was coming down...regardless of their relative speeds  ,stopping to rest , at some point they must meet , and obviously at the same time.

hm, I thought that was more or less the proof I had used in my answer.

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