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Inequality

Source: 1989 National High School Mathematics League, Exam Two, Problem 2

February 25, 2020
inequalities

Problem Statement

xi∈R(i=1,2,⋯ ,n;n≥2)x_i\in\mathbb{R}(i=1,2,\cdots,n;n\geq2), satisfying that ∑i=1n∣xi∣=1,∑i=1nxi=0\sum_{i=1}^n|x_i|=1,\sum_{i=1}^nx_i=0. Prove that ∣∑i=1nxii∣≤12−12n|\sum_{i=1}^n\frac{x_i}{i}|\leq\frac{1}{2}-\frac{1}{2n}
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