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Finite preimages, symmetry

Source: 2020 Israel Olympic Revenge P1

June 11, 2022
functional equationalgebraolympic revengefunctionsymmetry

Problem Statement

Find all functions f:R→Rf:\mathbb{R}\to \mathbb{R} such that for all x,y∈Rx,y\in \mathbb{R} one has f(f(x)+y)=f(x+f(y))f(f(x)+y)=f(x+f(y)) and in addition the set f−1(a)={b∈R∣f(b)=a}f^{-1}(a)=\{b\in \mathbb{R}\mid f(b)=a\} is a finite set for all a∈Ra\in \mathbb{R}.
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