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For which constants does this sequence exist?

Source: Brazilian Mathematical Olympiad 2023, Level U, Problem 6

October 21, 2023
Sequencesalgebrareal analysis

Problem Statement

Determine all pairs (c,d)∈R2(c, d) \in \mathbb{R}^2 of real constants such that there is a sequence (an)n≥1(a_n)_{n\geq1} of positive real numbers such that, for all n≥1n \geq 1, an≥c⋅an+1+d⋅∑1≤j<naj.a_n \geq c \cdot a_{n+1} + d \cdot \sum_{1 \leq j < n} a_j .
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