Problem: On a weekend 2 young mountain climbers decided to climb and then descend a local mountain peak. Together they started off on Saturday morning and reached the peak Saturday night. They camped overnight on the summit and promptly started down the mountain on Sunday morning reaching their starting point Sunday afternoon. Prove that at some time of day they were at the exact same altitude on Saturday and Sunday. The two climbers always stay together.

Solutiuon: This isn't too difficult when you think it through. For any time t, let f(t) be the altitude of the climbers at time t on Sunday minus their altitude at the same time on Saturday. Now, as they climbers start their ascent on Saturday morning f(t) must be positive however, on Saturday night f(t) must be negative. Thus, by virtue of the intermediate value theorem, f(t) must be 0 at some point meaning that the climbers were at the same altitude on both Saturday and Sunday at some point in time. QED

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