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Another modulus inequality

Source: Romanian IMO TST 2006, day 2, problem 4

April 22, 2006
inequalitiesPutnaminequalities proposed

Problem Statement

Let xix_i, 1≤i≤n1\leq i\leq n be real numbers. Prove that ∑1≤i<j≤n∣xi+xj∣≥n−22∑i=1n∣xi∣. \sum_{1\leq i<j\leq n}|x_i+x_j|\geq\frac{n-2}{2}\sum_{i=1}^n|x_i|. Discrete version by Dan Schwarz of a Putnam problem
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