A man decides to climb a mountain. He starts at sunrise from the bottom of the mountain and arrives on the top at sunset. He sleeps there and on the next day he goes back the same way, descending at higher speed. Prove that there is some point of his path, on which the man will be at the same time on both days.
Imagine a second man who starts climbing from the bottom of the mountain on the second day and following the first hiker's first day movements. At some point the first and the second hiker will meet each other, and this will be the point you are looking for.