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Similar triangles in equiangular polygon

Source: 2022 Mexican Mathematics Olympiad P6

November 8, 2022
combinatorial geometrygeometrysimilar triangles

Problem Statement

Find all integers n≥3n\geq 3 such that there exists a convex nn-gon A1A2…AnA_1A_2\dots A_n which satisfies the following conditions: - All interior angles of the polygon are equal - Not all sides of the polygon are equal - There exists a triangle TT and a point OO inside the polygon such that the nn triangles OA1A2, OA2A3, …, OAn−1An, OAnA1OA_1A_2,\ OA_2A_3,\ \dots,\ OA_{n-1}A_n,\ OA_nA_1 are all similar to TT, not necessarily in the same vertex order.
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