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Do there exist two sequences of real numbers

Source: IMO Longlist 1989, Problem 108

September 18, 2008
algebra unsolvedalgebra

Problem Statement

Do there exist two sequences of real numbers {ai},{bi}, \{a_i\}, \{b_i\}, i∈N, i \in \mathbb{N}, satisfying the following conditions: 3⋅π2≤ai≤bi \frac{3 \cdot \pi}{2} \leq a_i \leq b_i and \cos(a_i x) \minus{} \cos(b_i x) \geq \minus{} \frac{1}{i} ∀i∈N \forall i \in \mathbb{N} and all x, x, with 0<x<1? 0 < x < 1?
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