Starting at 9 a.m. on Monday, a hiker walked at a steady pace from the trailhead up a mountain and reached the summit at exactly 3 p.m. The hiker camped for the night and then hiked back down the same trail, again starting at 9 a.m. On this second walk, the hiker walked very slowly for the first two hours, but walked faster on other parts of the trail and returned to the starting point in exactly six hours. Prove that there is some point on the trail that the hiker passed at exactly the same time on the two days.
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