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Functional inequality with size constraint

Source: 2018 Latvia BW TST P4

June 4, 2022
functioninequalitiesalgebraalgebra unsolved

Problem Statement

Let f:R→Rf:\mathbb{R}\rightarrow\mathbb{R} be a function that satisfies 2f(x)−2f(x)−f(2x)≥2\sqrt{2f(x)}-\sqrt{2f(x)-f(2x)}\ge 2 for all real xx. Prove for all real xx: (a) f(x)≥4f(x)\ge 4; (b) f(x)≥7.f(x)\ge 7.
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