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Stereometic inequality with vectors

Source: St. Petersburg MO 2000, 11th grade, P2

April 22, 2023
geometric inequalityinequalitiesgeometryVectorsvector geometry3D geometryanalytic geometry

Problem Statement

Point OO is the origin of a space. Points A1,A2,…,AnA_1, A_2,\dots, A_n have nonnegative coordinates. Prove the following inequality: ∣OA1→∣+∣OA2→∣+⋯+∣OAn→∣≤3∣OA1→+OA2→+⋯+OAn→∣|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|
[I]Proposed by A. Khrabrov
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