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ROMANIA 2000 inequality

Source: Romanian TST 2000

January 6, 2004
inequalitiesinequalities solved

Problem Statement

Let n≥1n\ge 1 be a positive integer and x1,x2…,xnx_1,x_2\ldots ,x_n be real numbers such that ∣xk+1−xk∣≤1|x_{k+1}-x_k|\le 1 for k=1,2,…,n−1k=1,2,\ldots ,n-1. Prove that ∑k=1n∣xk∣−∣∑k=1nxk∣≤n2−14\sum_{k=1}^n|x_k|-\left|\sum_{k=1}^nx_k\right|\le\frac{n^2-1}{4}
Gh. Eckstein
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