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Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y

Source: BMO 2023 Problem 1

May 10, 2023
algebrafunctional equationBMO Shortlist

Problem Statement

Find all functions f ⁣:R→Rf\colon \mathbb{R} \rightarrow \mathbb{R} such that for all x,y∈Rx,y \in \mathbb{R}, xf(x+f(y))=(y−x)f(f(x)).xf(x+f(y))=(y-x)f(f(x)).
Proposed by Nikola Velov, Macedonia
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