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Convex pentagon and equal areas

Source: Romanian TST 4 2007, Problem 2, BMO 2007 Shortlist

May 23, 2007
geometryparallelogramtrapezoidgeometry proposed

Problem Statement

Let A1A2A3A4A5 A_{1}A_{2}A_{3}A_{4}A_{5} be a convex pentagon, such that [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}]. Prove that there exists a point M M in the plane of the pentagon such that [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}]. Here [XYZ] [XYZ] stands for the area of the triangle ΔXYZ \Delta XYZ.
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