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exists P\in R[x]$ of degree 2014^{2015} such that f(P)=2015 ?

Source: Balkan BMO Shortlist 2015 A6

August 5, 2019
algebrapolynomial

Problem Statement

For a polynomials P∈R[x] P\in \mathbb{R}[x], denote f(P)=nf(P)=n if nn is the smallest positive integer for which is valid (∀x∈R)(P(P(…P⏟n(x))…)>0),(\forall x\in \mathbb{R})(\underbrace{P(P(\ldots P}_{n}(x))\ldots )>0), and f(P)=0f(P)=0 if such n doeas not exist. Exists polyomial P∈R[x]P\in \mathbb{R}[x] of degree 201420152014^{2015} such that f(P)=2015f(P)=2015?
(Serbia)
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