You walk up a mountain starting at 6 AM and get to the top at 6 PM. You spend the night at the top and on the next day, you start walking from the top at 6 AM and get to the bottom at 6 PM along the same path. You do not necessarily travel at the same speed on the two days. Show that there must be at least one clock time of day when, while going up and coming down, you were at the same height.
Solution is here