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Problem of the Week 915

A Peculiar Pentagon

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Let ABCDE be a convex pentagon in which the sides have length 1, 2, 3, 4, and 5, but not necessarily in that order. Let F, G, H, I be the midpoints of AB, BC, CD, DE, respectively. Then let X be the midpoint of FH and Y the midpoint of GI, and suppose that the length XY is an integer. How long is AE?

Source: Gregory Galperin, in Mathematics and Informatics Quarterly, Volume 10, Jan 2000, p. 40.
© Copyright 2000 Stan Wagon. Reproduced with permission.

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12 September 2000