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Prove that a_k ≤ k(n+1-k)/2 for every k

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September 20, 2010
InequalityConvexitySequencerecurrence relationIMO Shortlist

Problem Statement

Let a0,a1,…,an,an+1a_0, a_1, \ldots, a_n, a_{n+1} be a sequence of real numbers satisfying the following conditions:
a0=an+1=0,a_0 = a_{n+1 }= 0, |a_{k-1} - 2a_k + a_{k+1}| \leq 1   (k = 1, 2,\ldots , n). Prove that |a_k| \leq \frac{k(n+1-k)}{2}   (k = 0, 1,\ldots ,n + 1).
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