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Inequality with n(n-1)/2

Source: IMO Shortlist 1995, A6

August 10, 2008
inequalitiesn-variable inequalityIMO Shortlist

Problem Statement

Let n n be an integer,n≥3. n \geq 3. Let x1,x2,…,xn x_1, x_2, \ldots, x_n be real numbers such that x_i < x_{i\plus{}1} for 1 \leq i \leq n \minus{} 1. Prove that \frac{n(n\minus{}1)}{2} \sum_{i < j} x_ix_j > \left(\sum^{n\minus{}1}_{i\equal{}1} (n\minus{}i)\cdot x_i \right) \cdot \left(\sum^{n}_{j\equal{}2} (j\minus{}1) \cdot x_j \right)
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