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Inequality on quadratic trinomials

Source: St. Petersburg MO 2001, 10th grade, P1

April 24, 2023
inequalitiesquadraticsquadratic trinomialalgebra

Problem Statement

Quadratic trinomials ff and gg with integer coefficients obtain only positive values and the inequality f(x)g(x)≥2\dfrac{f(x)}{g(x)}\geq \sqrt{2} is true ∀x∈R\forall x\in\mathbb{R}. Prove that f(x)g(x)>2\dfrac{f(x)}{g(x)}>\sqrt{2} is true ∀x∈R\forall x\in\mathbb{R}
[I]Proposed by A. Khrabrov
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