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f(f(x))-2 f(x)+x = 0 for all x \in N, f(1989) wanted

Source: Romania IMO TST 1989 2.1

February 17, 2020
functionalfunctional equationalgebra

Problem Statement

Let FF be the set of all functions f:N→Nf : N \to N which satisfy f(f(x))−2f(x)+x=0f(f(x))-2 f(x)+x = 0 for all x∈Nx \in N. Determine the set A={f(1989)∣f∈F}A =\{ f(1989) | f \in F\}.
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