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nP >= 4d in convex polygon where d=\sum |\overrightarrow {A_kM}|,

Source: X All-Ukrainian Tournament of Young Mathematicians, Qualifying p12

May 26, 2021
geometric inequalityinequalitiesvectorgeometryUkrainian TYM

Problem Statement

Inside the convex polygon A1A2...AnA_1A_2...A_n , there is a point MM such that ∑k=1nAkM→=0→\sum_{k=1}^n \overrightarrow {A_kM} = \overrightarrow{0}. Prove that nP≥4dnP\ge 4d, where PP is the perimeter of the polygon, and d=∑k=1n∣AkM→∣d=\sum_{k=1}^n |\overrightarrow {A_kM}| . Investigate the question of the achievement of equality in this inequality.
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