2001 IMO 第 3 题

A3 (ROM) Let $x_{1}, x_{2}, \ldots, x_{n}$ be arbitrary real numbers. Prove the inequality $$\frac{x_{1}}{1+x_{1}^{2}}+\frac{x_{2}}{1+x_{1}^{2}+x_{2}^{2}}+\cdots+\frac{x_{n}}{1+x_{1}^{2}+\cdots+x_{n}^{2}}<\sqrt{n}$$

A3 (ROM) 令 $x_{1}, x_{2}, \ldots, x_{n}$ 为任意实数。证明不等式 $$\frac{x_{1}}{1+x_{1}^{2}}+\frac{x_{2}}{1+x_{1}^{2}+x_{2}^{2}}+\cdots+\frac{x_{n}}{1+x_{1}^{2}+\cdots+x_{n}^{2}}<\sqrt{n}$$