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Part of 2024 Middle European Mathematical Olympiad

Problems(2)

Iterative F ineq on non-negative integers

Source: MEMO 2024 I1

8/26/2024
Let N0\mathbb{N}_0 denote the set of non-negative integers. Determine all non-negative integers kk for which there exists a function f:N0→N0f: \mathbb{N}_0 \to \mathbb{N}_0 such that f(2024)=kf(2024) = k and f(f(n))≤f(n+1)−f(n)f(f(n)) \leq f(n+1) - f(n) for all non-negative integers nn.
algebraSequencesfunctionsmonotoneconstructionFunctional inequalityfunction
Two recursive sequences and an inequality

Source: MEMO 2024 T1

8/27/2024
Consider two infinite sequences a0,a1,a2,…a_0,a_1,a_2,\dots and b0,b1,b2,…b_0,b_1,b_2,\dots of real numbers such that a0=0a_0=0, b0=0b_0=0 and a_{k+1}=b_k,   b_{k+1}=\frac{a_kb_k+a_k+1}{b_k+1} for each integer k≥0k \ge 0. Prove that a2024+b2024≥88a_{2024}+b_{2024} \ge 88.
Sequencesalgebraalgebra proposedinequalitiesinequalities proposed