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Functional Inequality Implies Uniform Sign

Source: 2023 ISL A2

July 17, 2024
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Problem Statement

Let R\mathbb{R} be the set of real numbers. Let f:R→Rf:\mathbb{R}\rightarrow\mathbb{R} be a function such that f(x+y)f(x−y)⩾f(x)2−f(y)2f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2 for every x,y∈Rx,y\in\mathbb{R}. Assume that the inequality is strict for some x0,y0∈Rx_0,y_0\in\mathbb{R}.
Prove that either f(x)⩾0f(x)\geqslant 0 for every x∈Rx\in\mathbb{R} or f(x)⩽0f(x)\leqslant 0 for every x∈Rx\in\mathbb{R}.
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