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IMO ShortList 2001, algebra problem 3

Source: IMO ShortList 2001, algebra problem 3

September 30, 2004
inequalitiestrigonometryalgebraIMO Shortlist

Problem Statement

Let x1,x2,…,xnx_1,x_2,\ldots,x_n be arbitrary real numbers. Prove the inequality x11+x12+x21+x12+x22+⋯+xn1+x12+⋯+xn2<n. \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.
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