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Pathological FE

Source: ISL 2022 A6

July 9, 2023

Problem Statement

Let R\mathbb R be the set of real numbers. We denote by F\mathcal F the set of all functions f ⁣:R→Rf\colon\mathbb R\to\mathbb R such that f(x+f(y))=f(x)+f(y)f(x + f(y)) = f(x) + f(y) for every x,y∈Rx,y\in\mathbb R Find all rational numbers qq such that for every function f∈Ff\in\mathcal F, there exists some z∈Rz\in\mathbb R satisfying f(z)=qzf(z)=qz.
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