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Problem 3 IMC 2004 Macedonia

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July 25, 2004
limitfunctioninequalitiesIMCcollege contests

Problem Statement

Let AnA_n be the set of all the sums ∑k=1narcsin⁡xk\displaystyle \sum_{k=1}^n \arcsin x_k , where n≥2n\geq 2, xk∈[0,1]x_k \in [0,1], and ∑k=1nxk=1\displaystyle \sum^n_{k=1} x_k = 1. a) Prove that AnA_n is an interval. b) Let ana_n be the length of the interval AnA_n. Compute lim⁡n→∞an\displaystyle \lim_{n\to \infty} a_n.
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