2004高三·吉林·竞赛
1 . 设
,且
.求证:
.分析:为了证明结论中的不等式,可以先由已知条件,运用均值不等式证明以下的3个不等式
,
,
(其中
为常数).再将上述3个不等式相加即可得证.则分析过程中常数
的值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/876060b33593f5c1981e4c300506d882.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56667aabbe787eb1c3189d487d203e22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f564a7ca42b9fc2f0a21436046e06b89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ed65fb17fc6be31a10ae891c3485ef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c36cb0aa454c04efb1c44adb577d5353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf421508727cc6ec73edcde5e1eb6e77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
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2 . 阅读下面一道题目的证明,指出其中的一处错误.题目:平面上有六个点,任何三点都是三边互不相等三角形的顶点,则这些三角形中有一个的最短边又是另一个三角形的最长边.证明:第一步,对已知的六个点作两两连线,可以得出15条边,记为
,
,…,
.第二步,由于任何三点组成的都是“三边互不相等的三角形”,因此,15条边互不相等不妨设
.第三步,由于“任何三点都是三边互不相等三角形的顶点”,因此,任取三条边都可以组成三角形,则
、
、
组成的三角形的最长边
,也是
、
、
组成的三角形的最短边,命题得证.这三步中,第______ 步有错误,理由是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b42791b77924729f7e31712177b26af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cb5ca241bb7c313ef0366d3ddba93bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72b7d30383d4dc8ab21977ef04f0ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/077da57431b9c992d319f133be557416.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b42791b77924729f7e31712177b26af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cb5ca241bb7c313ef0366d3ddba93bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cbeed5324c432101be517dd6f5c735b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cbeed5324c432101be517dd6f5c735b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cbeed5324c432101be517dd6f5c735b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d199f9235e3bd36bfa78c3772e941896.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feaf542829b99205948fa97442c3db92.png)
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3 . 设四边形
内接于
.过
作
的切线交边
的延长线于点
,且点
位于点
和点C 之间;过
作
的切线交边
的延长线于点
,且点
位于点
和点
之间.已知
.证明:四边形
为梯形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67d822262ff00915910e5b87d81ad1ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb77cfdda1e367b8e72fd04282ebae11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
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