Ah, the question was, can you prove that there is
a place where the monk is
at the same time of day on the way up as on the way down - not to say what that place is. Now, there are mathematical solutions to the problem involving
Brouwer's Fixed Point Theorem, which are virtually useless for actually understanding the solution. Much better is to use your intuition - imagine that a monk starts from the bottom of the mountain at sunrise, and another monk starts from the top of the mountain at sunrise. Now you can simply "see" that these two monks will pass each other and, hence, be at the same place on the path at the same time - though you don't know where or when, (but that's not what you are asked). From this you can also simply "see" that if the journeys take place on different days this makes no difference to whether the monks pass the same place at the same time of day.
Hope that helps. Apparently Arthur Koestler is responsible for this puzzle.