1982 IMO 第 3 题

A3 (USS 4) ${ }^{\mathrm{IMO} 3}$ Consider the infinite sequences $\left\{x_{n}\right\}$ of positive real numbers with the following properties: $$x_{0}=1 \quad \text { and for all } i \geq 0, x_{i+1} \leq x_{i}$$ (a) Prove that for every such sequence there is an $n \geq 1$ such that $\frac{x_{0}^{2}}{x_{1}}+$ $\frac{x_{1}^{2}}{x_{2}}+\cdots+\frac{x_{n-1}^{2}}{x_{n}} \geq 3.999$. (b) Find such a sequence for which $\frac{x_{0}^{2}}{x_{1}}+\frac{x_{1}^{2}}{x_{2}}+\cdots+\frac{x_{n-1}^{2}}{x_{n}}<4$ for all $n$.

A3 (USS 4) ${ }^{\mathrm{IMO} 3}$ 考虑具有以下属性的正实数的无限序列 $\left\{x_{n}\right\}$： $$x_{0}=1 \quad \text { 且对于所有 } i \geq 0, x_{i+1} \leq x_{i}$$ (a) 证明对于每个这样的序列都有一个 $n \geq 1$ 使得 $\frac{x_{0}^{2}}{x_{1}}+$ $\frac{x_{1}^{2}}{x_{2}}+\cdots+\frac{x_{n-1}^{2}}{x_{n}} \geq 3.999$。 (b) 找到这样一个序列，对于所有$n$，$\frac{x_{0}^{2}}{x_{1}}+\frac{x_{1}^{2}}{x_{2}}+\cdots+\frac{x_{n-1}^{2}}{x_{n}}<4$。