MathDB
Clipping Polygons

Source: USAMO 1997

October 9, 2005
geometrymodular arithmeticgeometric transformationhomothetycombinatorial geometry

Problem Statement

To clip a convex nn-gon means to choose a pair of consecutive sides AB,BCAB, BC and to replace them by the three segments AM,MNAM, MN, and NCNC, where MM is the midpoint of ABAB and NN is the midpoint of BCBC. In other words, one cuts off the triangle MBNMBN to obtain a convex (n+1)(n+1)-gon. A regular hexagon P6{\cal P}_6 of area 1 is clipped to obtain a heptagon P7{\cal P}_7. Then P7{\cal P}_7 is clipped (in one of the seven possible ways) to obtain an octagon P8{\cal P}_8, and so on. Prove that no matter how the clippings are done, the area of Pn{\cal P}_n is greater than 13\frac 13, for all n≥6n \geq 6.
Back to ProblemsView on AoPS