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convex k-gon of largest area inscribed in convex n-gon

Source: VI All-Ukrainian Tournament of Young Mathematicians, Qualifying p18

May 25, 2021
geometryconvexconvex polygonmaxareageometric inequalityUkrainian TYM

Problem Statement

The convex polygon A1A2...AnA_1A_2...A_n is given in the plane. Denote by TkT_k (kn)(k \le n) the convex kk-gon of the largest area, with vertices at the points A1,A2,...,AnA_1, A_2, ..., A_n and by Tk(A+1)T_k(A+1) the convex k-gon of the largest area with vertices at the points A1,A2,...,AnA_1, A_2, ..., A_n in which one of the vertices is in A1A_1. Set the relationship between the order of arrangement in the sequence A1,A2,...,AnA_1, A_2, ..., A_n vertices: 1) T3T_3 and T3(A2)T_3 (A_2) 2) TkT_k and Tk(A1)T_k (A_1) 3) TkT_k and Tk+1T_{k+1}