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Bosnia and Herzegovina 2022 BMO TST P1

Source:

May 22, 2022

Sequencealgebra

Problem Statement

Let

a_1,a_2,a_3, \ldots

be an infinite sequence of nonnegative real numbers such that for all positive integers

k

the following conditions hold:

i)

a_k-2a_{k+1}+a_{k+2} \geq 0

;

ii)

\sum_{j=1}^{k} a_j \leq 1

. Prove that for all positive integer

k

holds:

0 \leq a_k - a_{k+1} < \frac{2}{k^2}