名校
解题方法
1 . 已知f(x)是定义在R上的奇函数,
且对任意
均有
则 ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9eacff7f456194640de6801dc94799a.png)
_____
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3a2ffb7236b18cec72e965944a2ed75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e6f37ea158078072f8bd7771119d077.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa07570d61c2b610b8e5bbf93d944047.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9eacff7f456194640de6801dc94799a.png)
您最近一年使用:0次
2 . “让式子丢掉次数”—伯努利不等式(Bernoulli’sInequality),又称贝努利不等式,是高等数学分析不等式中最常见的一种不等式,由瑞士数学家雅各布.伯努利提出,是最早使用“积分”和“极坐标”的数学家之一.贝努利不等式表述为:对实数
,在
时,有不等式
成立;在
时,有不等式
成立.
(1)证明:当
,
时,不等式
成立,并指明取等号的条件;
(2)已知
,…,
(
)是大于
的实数(全部同号),证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30cdfc52dbd70827de9e15fffe39c321.png)
(3)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc98a4d9ae0580aa2c1152ffb770d4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c4fb8df3614557f13bdc68378437e90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3d4045366a437d4003259050718e244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f75f0daa973c8fc183b7d21bafd7e8cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c78998ba5f2665a1753c3fa84751716.png)
(1)证明:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65a40142c84be68ee2918c3a8303388c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bc98a4d9ae0580aa2c1152ffb770d4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5026dc5ead3b5adf0e5f4b3e7c4eca1d.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a1cc5cfec94bc5686b41b043acdc8ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30cdfc52dbd70827de9e15fffe39c321.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6b29215b2a741c01efc27199e6c6925.png)
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2024-05-30更新
|
290次组卷
|
3卷引用:2024年海南省海口实验中学高一学科竞赛选拔性考试(自主招生)数学试题
3 .
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fd8ec6fc551585f093d0a8848aace07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2829c117b291deabd43fdd524d253a26.png)
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4 . 将正整数
填入
方格表中,每个小方格恰好填1个数,要求每行从左到右10个数依次递减,记第
行的10个数之和为
. 设
满足:存在一种填法,使得
均大于第
列上的10个数之和,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deacb2d14b3b685334af74c9eb08e708.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23f3b42cd6069f0e461035e76459ee4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aba753aa5e77c45b0d328c036a954a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70e72518ba0d330df05786f6c48db9b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77efd8c62dacd2212c3ff5db6b02a5dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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5 . 校乒乓球锦标赛共有
位运动员参加.第一轮,运动员们随机配对,共有
场比赛,胜者进入第二轮,负者淘汰.第二轮在同样的过程中产生
名胜者.如此下去,直到第n轮决出总冠军.实际上,在运动员之间有一个不为比赛组织者所知的水平排序,在这个排序中
最好,
次之,…,
最差.假设任意两场比赛的结果相互独立,不存在平局,且
,当
与
比赛时,
获胜的概率为p,其中![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a998c92f966aae015d3e1e37c967e7b5.png)
(1)求最后一轮比赛在水平最高的两名运动员
与
之间进行的概率.
(2)证明:
,
为总冠军的概率大于
为总冠军的概率.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f31971306914638e5ceb1bbe437535d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9f1ad18371ec533aeac27cf1fad95c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49cc8f06c961b64b15a90b99f7adc604.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/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/519321dbfc38d9b89948762478f71d0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9454ddb2d570f884b15bd3ddf2a4545d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec6ba141730fd5aae78ada1a8eb17d21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ea8f47d8d8d9e1832d52b1c7425450.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a998c92f966aae015d3e1e37c967e7b5.png)
(1)求最后一轮比赛在水平最高的两名运动员
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eae64cb0b1c5e4f556e0ee0ca54fa9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ea8f47d8d8d9e1832d52b1c7425450.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5654866bd68198db845fb43c6b4c858.png)
您最近一年使用:0次
解题方法
6 . 给定
,若
,
满足
,均有
,则k的范围是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aff8d9b6533ff319420cdc5e8740b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/756cd04fb04459ef15b1d37c9bb88e9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2671f593186fa00f17ad26eba7b8f3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe383188576e4de78241ddaf8cf94144.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d3481d657e1d1ca13aa4ad8af6b4afd.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
您最近一年使用:0次
7 . 若数列
满足对任意
,数列
的前
项至少有
项大于
,且
,则称数列
具有性质
.若存在具有性质
的数列
,使得其前n项和
恒成立,则整数
的最小值是_____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b4a1fb2354b4716afb9191c880fcc11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ceef1abeeef220b4fe5f7d96feedd90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a400b5443af7580aa8f0fb7499fe362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e040f089ea13dbd9d9133d6934537f32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e040f089ea13dbd9d9133d6934537f32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0be2256f1cda4db7aa1e92b7372ee8ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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解题方法
8 . 对集合
,定义其特征函数
,考虑集合
和正实数
,定义
为
和式函数.设
,则
为闭区间列;如果集合
对任意
,有
,则称
是无交集合列,设集合
.
(1)证明:L和式函数的值域为有限集合;
(2)设
为闭区间列,
是定义在
上的函数.已知存在唯一的正整数
,各项不同的非零实数
,和无交集合列
使得
,并且
,称
为
和式函数
的典范形式.设
为
的典范数.
(i)设
,证明:
;
(ii)给定正整数
,任取正实数
和闭区间列
,判断
的典范数
最大值的存在性.如果存在,给出最大值;如果不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1304eb00ab95d664dc84385f602a8f09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aee6c8ae5004f2ffe7f8392b4d3c39b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/238908949859936af0e109ef684599b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/937c09d82c480e4d67f8a48d3f66c5f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a02da5d46478a54d279755a295d548f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b56da93ba7a2dec958070eb2666240.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05386869739fb11a190c637ba8a93174.png)
(1)证明:L和式函数的值域为有限集合;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab0fa51de98f090eda3e3f60a26475db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecfcda4333678bafacc4c676c2836977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee06844034f61cab7d421d55179ee367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/359a16305129aeea0953efd9100f4b9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7b4e32041b54703ade8e8c2cee01f13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ed82555c7d6fc6b449fbdb1f68fef1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
(i)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1462612f3654548c39489985987cb67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7870c36161f465fc992534b5fc3777f3.png)
(ii)给定正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81f69939291758b5eaa19146f76709e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b4010030e10725398b64d4dcc09429.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
解题方法
9 . 已知抛物线
的焦点
关于原点的对称点是
为
为圆心,
为半径的圆.直线
是过
上异于原点的一点
的
的切线,切点为
.
(1)求
的最大值;
(2)求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e6c830bfa9a1b979a1a9665166424bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/476e960b0c485b855bfcf5e17369f48d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06bbf1f42f0caeb7ab44281c07bc8c68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e00eb9ed89b33f218fbdff9b76a8cd17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fef27cb7cb1b666c1734c65a7aa9aa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b59b61e07af95ad8b7e34cadedc9ead.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7dbd0ecf66cbb83572ab4a9e5938ea3.png)
您最近一年使用:0次
解题方法
10 . 已知正七边形ABCDEFG的外接圆为
且A为该圆上距离坐标原点最远的点,则关于这七个点的回归直线方程为__________ ;设CG,AD交于Q,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d9132d70f647b0578d00678883d8c66.png)
___________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/175dd7ca22ea4f6a83530a746893624a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d9132d70f647b0578d00678883d8c66.png)
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