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n-gon's perimeter > 2n-gon's perimeter, perimeters ACE=ABCDEF

Source: 2023 South Russian Girls MO - Assara Juniors p8

February 28, 2024
geometryperimeter

Problem Statement

a) Given a convex hexagon ABCDEFABCDEF, which has a center of symmetry. Prove that the perimeter of triangle ACEACE is greater than half the perimeter of hexagon ABCDEFABCDEF.
b) Given a convex (2n)(2n)-gon PP having a center of symmetry, its vertices are colored alternately red and blue. Let QQ be an nn-gon with red vertices. Is it possible to say that the perimeter of QQ is certainly greater than half the perimeter PP? Solve the problem for n=4n = 4 and n=5n = 5.
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