1 . 设
为椭圆
:
的左准线与对称轴的交点,椭圆
的左焦点为
,过
任作一斜率不为0的直线与椭圆
交于点
、
,
关于
轴的对称点为
.
(1)求证:
;
(2)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dd54b9df3402ad91e2d34c40efe0c7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a42da28be159399514cc6179a96e34b.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0773c7a3457a6bb0704963b927dd218.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07907d140d36f97a216173f268eac0b6.png)
您最近一年使用:0次
2007高三·陕西·竞赛
2 . 已知
.求证:
(1)
;
(2)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2da182f0a8f04d8148283d25b2763e6d.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50f636183aacbefb064add006c4c3714.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/454af15c109ed6a508c6af26c86f87ab.png)
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3 . 设
四点均在双曲线
的右支上.
(1)若
(实数
),证明:
(O是坐标原点);
(2)若
,P是线段AB的中点,过点P分别作该双曲线的两条渐近线的垂线,垂足为M、N,求四边形
的面积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc4594a9943a8ab0ff38abef9af7ba37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e06289aa19b479a693110d0a6c22fb6.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcfde0625124e356c8df9f5c3a85786b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bd75b1ffdc667e1d9d37f8c73bc14d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b2d03e9a74ce2db2ad792f9ced3ab3d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7b0f0fef9c1b70d1d66b07e9af3891f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/013ab2607569ae5da929b434ef5dedc1.png)
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2002高三·湖南·竞赛
名校
4 . 设关于 x 的一元二次方程
的两个根为 α、β(α < β).
(1))若 x1、x2 为区间[ α, β] 上的两个不同的点,求证:
;
(2)设
,
在区间[ α, β] 上的最大值和最小值分别为
和
,
.求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58f247254d25c597e12c9f400820274f.png)
(1))若 x1、x2 为区间[ α, β] 上的两个不同的点,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385836628973f021654e20b9997b17d4.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ff175b22e5ef641b512f609318d40bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff2d8bdc2cded003dbec9822d1bb9e07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce86efd98b97c710761f8aa6983f25ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1759ff4cca81d97af125facee38e2e1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a82a1224e47f31ecdfffd328d5a3ab6d.png)
您最近一年使用:0次
2018-12-15更新
|
376次组卷
|
5卷引用:2002年湖南省高中数学奥林匹克
5 . 给定正整数
,且
,
,
,
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69d45e896826de39e9704cb71b508224.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29d17946784c8a818ebc9651ad19b938.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ac0d9c57fe35ae1ef24f466e982812b.png)
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2011高三·天津·竞赛
6 . 设
、
、
、
、
、
为实数,且
对任意的实数
成立.证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74dcc8b824bf93f5981bfd1fd3d1a71d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9723d411b2a0409c768eb8bbbe2f0bc0.png)
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2013高三·湖南·竞赛
7 . 已知函数
,且对任意的
均有
.
(1)求
的取值范围;
(2)若
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d67e8dd0ec306e7b9626681d5b1e401.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa18838a13fda4e45612c32cdf98b71.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68c2a93f134ec21d101bc0b5b856af57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95c0aa434c171c539d2e56cabcfea5f6.png)
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8 . 某厂第一天产品不超过
件,以后每天日产量都有所增加,但每日增产数量也不超过
件,且设
,
.证明:当日产量达到
件时,工厂生产产品总数不少于
件.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f6509f3e0cb2db3cd2e27c9784f8318.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bddb6dd373f212f25bb481c0ec4de2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc2301943aa6471cb3de884cf85b180a.png)
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9 . 设双曲线
的两支为
(如图),正三角形PQR的三顶点位于此双曲线上.
![](https://img.xkw.com/dksih/QBM/2018/12/5/2090247534321664/2092572359991296/STEM/7306340275c24081a48a46c38e1e1f8f.png?resizew=258)
(1)求证:P、Q、R不能都在双曲线的同一支上;
(2)设P(-1,-1)在
上,Q、R在
上.求顶点Q、R的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/377a2333ff8c63cbdb20b882d6d5a7ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/880248fa1259b2600a87f09a61287d44.png)
![](https://img.xkw.com/dksih/QBM/2018/12/5/2090247534321664/2092572359991296/STEM/7306340275c24081a48a46c38e1e1f8f.png?resizew=258)
(1)求证:P、Q、R不能都在双曲线的同一支上;
(2)设P(-1,-1)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
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10 . 设
均取正实数,且
.求三元函数
的最小值,并给出证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3fb0c9c7e30bbc0ec8c3521577ee4fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd24c686fbaaa68705d654b880481ffe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c051c8c5faed37949c0c8a91e2081728.png)
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