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f(x+f(y))<=xf(y)+x over R, prove no solutions

Source: IMOC 2017 A4

August 12, 2021
fefunctional equationFunctional inequalityalgebra

Problem Statement

Show that for all non-constant functions f:Rā†’Rf:\mathbb R\to\mathbb R, there are two real numbers x,yx,y such that f(x+f(y))>xf(y)+x.f(x+f(y))>xf(y)+x.
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