6

Part of 2023 Brazil Undergrad MO

Problems(1)

For which constants does this sequence exist?

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

Determine all pairs

(c, d) \in \mathbb{R}^2

of real constants such that there is a sequence

(a_n)_{n\geq1}

of positive real numbers such that, for all

n \geq 1

a_n \geq c \cdot a_{n+1} + d \cdot \sum_{1 \leq j < n} a_j .

Sequencesalgebrareal analysis