﻿﻿

# [Dr.Lib]Note:Math – Elegant Proof I

$0 \leq\frac{x – y}{1+x y} \leq\frac{\sqrt{3}}{3}$

### 练习：

50th Moscow MO 1987 求证：从任意三个正数或四个实数中总能取两个数x,y满足 $0 \leq \frac{x – y}{1+x y} \leq 1$

## 3th CMO WC 1988

（1）设正数a,b,c满足 $$(a^{2}+b^{2}+c^{2})^{2}>2(a^{4}+b^{4}+c^{4})$$

（2）设$$(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+ … +a_{n}^{2})^{2}>(n-1)(a_{1}^{4}+a_{2}^{4}+…+a_{n}^{4}) n\geq3$$，求证：$${a_{n}}$$中任意三个数是三角形的三边。

（1）

（2）

$$n=3$$时由（1）立即可得。

$$n>3$$时

$(n-1)(a_{1}^{4}+a_{2}^{4}+…+a_{n}^{4})$
$<(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+ … +a_{n}^{2})^{2}$
$=(\frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}{2}+\frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}{2}+ … +a_{n}^{2})^{2}$
$\leq (n-1)(\frac{(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})^{2}}{4}+\frac{(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})^{2}}{4}+a_{4}^{4}+…+a_{n}^{4}) (柯西不等式)$
$=(n-1)(\frac{(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})^{2}}{2}+a_{4}^{4}+…+a_{n}^{4})$

$$(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})^{2}>2(a_{1}^{4}+a_{2}^{4}+a_{3}^{4})$$

## 【To be continued】

Posted in Geek and tagged , , by .

## 2 thoughts on “[Dr.Lib]Note:Math – Elegant Proof I”

• 正切和角公式证存在的题目至少在四个大考中看到过……见过原题的话题目还是很水的