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Functional problem in a single variable f(f(x)) = xf(x) - x^2 + 2

Source: Peru Ibero+Cono Sur TST 2020 P2

September 11, 2023
number theoryfunctional equation

Problem Statement

Find all functions f:Z→Zf : \mathbb{Z} \to \mathbb{Z} that satisfy the conditions: i)f(f(x))=xf(x)−x2+2,∀x∈Zi) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z} ii)fii) f takes all integer values
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