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Part of 2022 Indonesia TST

Problems(4)

Polynomial Composition Inequality

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

12/11/2021
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
algebrapolynomialinequalities
Simple Inequality

Source:

12/22/2021
Let a,b,ca, b, c be positive real numbers such that abc=1abc = 1. Prove that (a+b+c)(ab+bc+ca)+3≥4(a+b+c).(a + b + c)(ab + bc + ca) + 3\ge 4(a + b + c).
InequalityAM-GMmathsimplealgebrainequalities
Asymmetric Weighted Inequality

Source: Indonesian Stage 1 TST for IMO 2022, Test 3 (Algebra)

12/25/2021
Let aa and bb be two positive reals such that the following inequality ax3+by2≥xy−1 ax^3 + by^2 \geq xy - 1 is satisfied for any positive reals x,y≥1x, y \geq 1. Determine the smallest possible value of a2+ba^2 + b.
Proposed by Fajar Yuliawan
algebraInequalityasymmetricminimuminequalities
Functional Inequality

Source: Indonesian Stage 1 TST for IMO 2022, Test 4 (Algebra)

12/25/2021
Determine all functions f:R→Rf : \mathbb{R} \to \mathbb{R} satisfying f(a2)−f(b2)≤(f(a)+b)(a−f(b)) f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) for all a,b∈Ra,b \in \mathbb{R}.
algebrafefunctionsinequalities