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0 < f(x) < f(x + f(x)) <\sqrt2 x, x < f(x + f(x)) <\sqrt2 f(x)

Source: 2021 Swedish Mathematical Competition p4

November 17, 2022
algebrafunctionalFunctional inequality

Problem Statement

Give examples of a function f:R→Rf : R \to R that satisfies 0<f(x)<f(x+f(x))<2x0 < f(x) < f(x + f(x)) <\sqrt2 x, for all positive xx,
and show that there is no function f:R→Rf : R \to R that satisfies x<f(x+f(x))<2f(x)x < f(x + f(x)) <\sqrt2 f(x), for all positive xx.
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