A mountaineer starts rapidly climbing up a mountain trail at 6 A.M. He makes frequent and irregular stops to rest or eat. He reaches the summit at 6 P.M. At 6 A.M. the next day, he starts his way back following the same route, stopping only once to eat. He reaches the starting point at 6 P.M. Is there a point on the way where he passes at exactly the same time on both days?
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There is such a spot. Let’s imagine that on the same day that one person was climbing down, the other one was climbing up. They must have met at a certain point of the trip. This is the spot we are looking for.