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area of P_n >= 1/2 for tweaked a convex n-gon, starting with area 1

Source: Switzerland - 2005 Swiss MO Final Round p5

December 26, 2022
geometryGeometric Inequalitiescombinatorial geometryconvex

Problem Statement

Tweaking a convex nn-gon means the following: choose two adjacent sides ABAB and BCBC and replaces them with the line segment AMAM, MNMN, NCNC, where M∈ABM \in AB and N∈BCN \in BC are arbitrary points inside these segments. In other words, you cut off a corner and get an (n+1)(n+1)-corner. Starting from a regular hexagon P6P_6 with area 11, by continuous Tweaks a sequence P6,P7,P8,...P_6,P_7,P_8, ... convex polygons. Show that Area of ​​Pn​​P_n for all n≥6n\ge 6 greater than 12\frac1 2 is, regardless of how tweaks takes place.
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