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Sequence inequality - a_n - 2a_{n+1} + a_{n+2} ≤ 0

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September 21, 2010
inequality systemSequenceConvexityalgebraIMO Shortlist

Problem Statement

Let a1,a2,…,an,…a_1, a_2, \ldots , a_n, \ldots be a sequence of real numbers such that 0≤an≤10 \leq a_n \leq 1 and an−2an+1+an+2≥0a_n - 2a_{n+1} + a_{n+2} \geq 0 for n=1,2,3,…n = 1, 2, 3, \ldots. Prove that 0≤(n+1)(an−an+1)≤2 for n=1,2,3,…0 \leq (n + 1)(a_n - a_{n+1}) \leq 2 \qquad \text{ for } n = 1, 2, 3, \ldots
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