2007 IMO 第 1 题

Real numbers $a_{1}$ , $a_{2}$ , ..., $a_{n}$ are fixed. For each $1 \leq i \leq n$ we let $d_{i} = \max \{a_{j} : 1 \leq j \leq i\} - \min \{a_{j} : i \leq j \leq n\}$ and let $d = \max \{d_{i} : 1 \leq i \leq n\}$ .

(a) Prove that for any real numbers $x_{1} \leq \dots \leq x_{n}$ we have

$$\max \left\{|x_{i} - a_{i}|:1\leq i\leq n\right\} \geq \frac{1}{2} d.$$

(b) Moreover, show that there exists some choice of $x_{1} \leq \dots \leq x_{n}$ which achieves equality.

实数 $a_{1}$ 、 $a_{2}$ 、 ...、 $a_{n}$ 是固定的。对于每个 $1 \leq i \leq n$，我们令 $d_{i} = \max \{a_{j} : 1 \leq j \leq i\} - \min \{a_{j} : i \leq j \leq n\}$ 并令 $d = \max \{d_{i} : 1 \leq i \leq n\}$ 。

(a) 证明对于任何实数 $x_{1} \leq \dots \leq x_{n}$ 我们有

$$\max \left\{|x_{i} - a_{i}|:1\leq i\leq n\right\} \geq \frac{1}{2} d.$$

(b) 此外，证明存在一些可以实现相等的 $x_{1} \leq \dots \leq x_{n}$ 的选择。