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f (f (x) f (y)) = xy and f (k) \ne k for k=0,1,-1

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 Shortlist F2 day1 (F= functions)

September 29, 2021
algebrafunctional equationfunctional

Problem Statement

Find all functions f:R→Rf: R \to R such that f(f(x)f(y))=xyf (f (x) f (y)) = xy and there is no k∈R−{0,1,−1}k \in R -\{0,1,-1\} such that f(k)=kf (k) = k.
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