MathDB
Cutting a polygon

Source: Cono Sur 2010 #3

November 17, 2015
combinatorial geometrycono surgeometry

Problem Statement

Let us define cutting a convex polygon with nn sides by choosing a pair of consecutive sides ABAB and BCBC and substituting them by three segments AM,MNAM, MN, and NCNC, where MM is the midpoint of ABAB and NN is the midpoint of BCBC. In other words, the triangle MBNMBN is removed and a convex polygon with n+1n+1 sides is obtained. Let P6P_6 be a regular hexagon with area 11. P6P_6 is cut and the polygon P7P_7 is obtained. Then P7P_7 is cut in one of seven ways and polygon P8P_8 is obtained, and so on. Prove that, regardless of how the cuts are made, the area of PnP_n is always greater than 2/32/3.
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