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TOT 334 1992 Spring A S2 sum of areas of 51 triangles >= 3S

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June 9, 2024
areascombinatoricscombinatorial geometrygeometrybroken line

Problem Statement

Let aa and SS be the length of the side and the area of regular triangle inscribed in a circle of radius 11. A closed broken line A1A2...A51A1A_1A_2...A_{51}A_1 consisting of 5151 segments of the same length aa is placed inside the circle. Prove that the sum of areas of the 51 51 triangles between the neighboring segments A1A2A3,A2A3A4,...,A49A50A51,A50A51A1,A51A1A2A_1A_2A_3, A_2A_3A_4,..., A_{49}A_{50}A_{51}, A_{50}A_{51}A_1, A_{51}A_1A_2 is not less than 3S3S.
(A. Berzinsh, Riga)
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