One morning, at crack of dawn, a monk began to climb a mountain to reach a shrine at its top. The narrow path, only wide enough for two or three persons to pass at a time, spiralled all the way to the top.
The monk ascended the path at varying rates of speed. He would stop to rest or to eat the dried fruits he had brought with him or to drink water. He reached the shrine shortly after sunset.
He spent several days at the shrine, praying and meditating. Then one day at the crack of dawn he began his journey down the mountain. He stopped several times along the way, but of course his average speed descending was greater than his average climbing speed.
Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.