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IMO LongList 1987 - prove the existence of limit

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September 6, 2010

trigonometrylimitalgebra proposedalgebra

Problem Statement

Given two sequences of positive numbers

\{a_k\}

and

\{b_k\} \ (k \in \mathbb N)

such that:
(i)

a_k < b_k,

(ii)

\cos a_kx + \cos b_kx \geq -\frac 1k

for all

k \in \mathbb N

and

x \in \mathbb R,

prove the existence of

\lim_{k \to \infty} \frac{a_k}{b_k}

and find this limit.

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