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A monk climbs a mountain . He starts at 8AM and reaches the summit at 5 PM.He spends the night on the summit. The next morning, he leaves the submit at 8AM and descends by the same route that he used the day before, reading the bottom at 5PM.Prove that there is a time between 8AM and 5PM at which the monk was at exactly the same spot on the mountain on both days.(Notice that we do not specify anything about the speed that the monk travels. For example, he could face at 100 miles per hour for the first few minutes, then sit still for an hour, then travel backward,etc. Nor does the monk have to travel at the same speeds going up as going down.)

My friend gave me the problem but I don't know how to start.I don't know think. Please help me.

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    $\begingroup$ This is not about combinatorics. Try the intermediate value theorem $\endgroup$ Sep 3, 2018 at 6:29

2 Answers 2

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Note: Not a mathematical proof. A logical one.

Just visualise Monk I going up the hill and Monk 2 coming down the hill. There will be a time in the trip when both of them meet.

Or you can just draw position-time graph of both the trips and it is easy to prove that they must intersect.

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first I think you must mean 'at the same vertical height above sea level'. It is possible that the monk, may take totally different routes up and down and so may not cross the same actual spot on his return journey.

Then I think this result is a consequence of two things:

1) The fact that, however fast he goes, whether he jumps, hops, skips or falls over, the monk's position (or height above sea level) is a continuous function of time.

2) As @Hagen von Eitzen says, the intermediate value theorem https://en.wikipedia.org/wiki/Intermediate_value_theorem

And just to add one note, what we are really applying the intermediate value theorem to is the height above sea level after time $t$ on the ascent minus the height at time $t$ on the descent. This is continuous as the difference of two continuous functions is continuous.

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  • $\begingroup$ The exercise says that the monk "descends by the same route that he used the day before". So, he will cross the same exact spot on his return journey. $\endgroup$
    – user5826
    Nov 20, 2021 at 18:14

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