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nice and hard inequality

Source: Romanian ROM TST 2004, problem 12, created by Harazi

May 3, 2004
inequalitiesinductionn-variable inequalityRomanian TST2004

Problem Statement

Let n≥2n\geq 2 be an integer and let a1,a2,…,ana_1,a_2,\ldots,a_n be real numbers. Prove that for any non-empty subset S⊂{1,2,3,…,n}S\subset \{1,2,3,\ldots, n\} we have (∑i∈Sai)2≤∑1≤i≤j≤n(ai+⋯+aj)2. \left( \sum_{i \in S} a_i \right)^2 \leq \sum_{1\leq i \leq j \leq n } (a_i + \cdots + a_j ) ^2 . Gabriel Dospinescu
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