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P(...(P(x))...) has all roots from 1,2,..., mb

Source: 2006 VMEO III Shortlist SL A9 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vme

November 24, 2021
polynomialalgebra

Problem Statement

Is there any polynomial P(x)P(x) with degree nn such that P(...(P(x))...)⏟m  times  P \underbrace{P(...(P(x))...)}_{m\,\, times \,\, P} has all roots from 1,2,...,mn1,2,..., mn ?
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