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Let P\!\left(x\right) be a non-constant polynomial with integer coefficients. Prove that there is no function T from the set of integers into the set of integers such that the number of integers x with T^n\!\left(x\right) = x is equal to P\!\left(n\right) for every n \geqslant 1, where T^n denotes the n-fold application of T.

Proposed by Jozsef Pelikan, Hungary

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