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f (f (f (n))) = P (n), for all n \in N_0, all coefficients of P(x) are in N_0

Source: Germany Federal - Bundeswettbewerb Mathematik 2010, round 2, p4

April 14, 2020
polynomialfunctionalgebrafunctional equationfunctional

Problem Statement

In the following, let N0N_0 denotes the set of non-negative integers. Find all polynomials P(x)P(x) that fulfill the following two properties: (1) All coefficients of P(x)P(x) are from N0N_0. (2) Exists a function f:N0→N0f : N_0 \to N_0 such as f(f(f(n)))=P(n)f (f (f (n))) = P (n) for all n∈N0n \in N_0.
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