At precisely 7:00 a.m., a monk sets out to climb a tall mountain, so that he might visit a temple at its peak. The trail he walks is narrow and winding, but it is the only way to reach the summit. As he ascends the mountain, the monk walks the path at varying speeds. Though he stops occasionally to rest and eat, he never strays from the path, and he never walks backwards. At exactly 7:00 p.m., the monk reaches the temple at the summit, where he stays the night.
The following morning at 7:00 a.m. sharp, the monk departs the temple and begins his journey back to the bottom of the mountain. He descends by way of the same path, again walking slowly at times and quickly at others, stopping here and there to eat and drink and rest, but never deviating from the path and never going backwards. Twelve hours later, at 7:00 p.m. on the nose, the monk arrives back at the foot of the mountain.
Is there any point along the path that the monk occupied at precisely the same time on both days? How do you know?