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Functional equation with -1 missing from domain

Source: Swiss MO 2023/5

March 12, 2023
algebrafunctional equationfunctiondomain

Problem Statement

Let DD be the set of real numbers excluding −1-1. Find all functions f:D→Df: D \to D such that for all x,y∈Dx,y \in D satisfying x≠0x \neq 0 and y≠−xy \neq -x, the equality (f(f(x))+y)f(yx)+f(f(y))=x(f(f(x))+y)f \left(\frac{y}{x} \right)+f(f(y))=x holds.
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