Problem of the Week

PROBLEM 209

Two Flatland mountain climbers start at opposite ends of a mountain range.  In Flatland, a mountain range consists of a sequence of alternating ascending and descending line segments having rational slope; see the figure on the left.  We can choose coordinates so that the first hiker starts at the origin and the second at the point (n,0), and the height of the points in the mountain range are non-negative.  They start hiking toward each other in such a way that their heights above the horizontal axis are always equal.  Will they always be able to meet at some point in the middle?  If so, at what point will that be; if not, why not?

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ã2005 Alberto L. Delgado