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Polynomial Composition Inequality

Source: Indonesian Stage 1 TST for IMO 2022, Test 1 (Algebra)

December 11, 2021
algebrapolynomialinequalities

Problem Statement

Given a monic quadratic polynomial Q(x)Q(x), define Qn(x)=Q(Q(⋯(Q(x))⋯ ))⏟compose n times Q_n (x) = \underbrace{Q(Q(\cdots(Q(x))\cdots))}_{\text{compose $n$ times}} for every natural number nn. Let ana_n be the minimum value of the polynomial Qn(x)Q_n(x) for every natural number nn. It is known that an>0a_n > 0 for every natural number nn and there exists some natural number kk such that ak≠ak+1a_k \neq a_{k+1}. (a) Prove that an<an+1a_n < a_{n+1} for every natural number nn. (b) Is it possible to satisfy an<2021a_n < 2021 for every natural number nn?
Proposed by Fajar Yuliawan
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