1 . 设
是正整数,整系数多项式
满足
.整系数多项式
,
,
满足
和![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4587ee6c046cf862cc0cd2426fad1197.png)
,其中
是一个不整除
的素数.求证:
的非常数项的系数均为
的倍数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/674c6027fe6ac3a582bced0c2cde8c8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c8a20b0ba8a4a7fb65339d045120f84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a723aebd8e4221c887b883733101e19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1803511584bb172d9445a4c49ab6fde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac390c9c4c3e09a82ebba66f01d9ae1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4587ee6c046cf862cc0cd2426fad1197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6ab259c01238737d6cec66506908dbd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7df298a9364b36e079a61caec815c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
您最近一年使用:0次
解题方法
2 . 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
3 . 春节将至,又是一年万家灯火的团圆之时.方方正正的小城里,住着
户人家,恰好构成了坐标平面上集合
的所有点.夜里,小城的人家挂上大红灯笼,交相辉映,将小城的夜晚编织成发光的大网.在坐标平面上看,A中的每个点均独立地以概率p被点亮,或以
的概率保持暗灭.若A中两个点的距离为1,则这两个点被称为是相邻的.若A中的n个被点亮的点
构成一依次相邻的点列
,则称这n个点组成的集合
是长度为n的“相邻灯笼串”.规定空集是长度为0的“相邻灯笼串”.
(1)给定A中3个依次相邻的点
,记随机变量X为集合
包含的“相邻灯笼串”的长度的最大值,试直接写出随机变量X的分布列(用p表示);
(2)若
,证明:存在长度为1000的“相邻灯笼串”的概率小于0.01;
(3)若
,证明:存在长度为1000的“相邻灯笼串”的概率大于0.99.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25ef6c862fe408c21f7779e4e8e82fe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e0c982180ad66af4330b1a8c43e4c07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ae7fb954b47cb67fdde891c3b9d8295.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce642b73be99b3c1a8c5dd38ec58eb28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3e59f96c54b4a41ed3c5b33b44320b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e69a66f2a66f4cd0fa05f5bbe185b6b.png)
(1)给定A中3个依次相邻的点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e85555fed049f8bb454c7569904bfed8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/718d05f8dd704256c99ac978e1ad5336.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bea306bdd00e500a305816f378060e4.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47938ad49863a8ff60ea48d0820e48f4.png)
您最近一年使用:0次
4 . 设
是正实数数列.
(1)若
收敛,求证:存在严格递增的无界正实数数列
满足
收敛.
(2)若
收敛,是否一定存在严格递增的正整数数列
,满足
收敛,且
?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc7376941fa463c63b1d4d4ea866b78c.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccecde965d7557d5ee35dea8ae7164a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60c988a3683540149b687486af0ed3a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/120dcd9c3adc5b08ab9d84f228cc4b90.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ad99ac2f9cbe69281dcdc7d4195d8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fba58d775c69de6d132c58581d614792.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/246e5563a2f86de45879b21393d814f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c69eef9b8e90f6a153b87738f759bcf.png)
您最近一年使用:0次
5 . 在锐角
中,
为
延长线上一点,过
分别作
,
平行线
,
,若
,
,且
的外接圆与
交于点
,证明:
(1)
;
(2)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5161dac60a307b5768018875063deaac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d61be62a066c7ad65f9fa64ba1de1ba4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9d172d77168b66134a5a852412d4f84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a6af45c7f8453cf711317ffe66dccec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/345f52e0835cc13f92b6801169105a17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e4c6b0869d7248b6c2f964718a67702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b0424af44604f489a804efb1e4b28a0.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/940730a19110cc363c871e0599388e56.png)
您最近一年使用:0次
2023-12-14更新
|
152次组卷
|
2卷引用:2023年第39届全国中学生冬令营(CMO)数学试题
6 . 已知复数
,
,…,
,
,
,…,
,满足
,
,…,
互不相同,
,
,…,
互不相同.已知对任意正整数
,均有
.求证:
,
.
![](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/dad7f66c97bfce4c00c53d86700c961b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54015ff5b49e3283901da1291b6b921d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f6872ffb1934339c53c2c2282d5889.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47233e81de2d9d728c80f02359f1af4a.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/dad7f66c97bfce4c00c53d86700c961b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54015ff5b49e3283901da1291b6b921d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f6872ffb1934339c53c2c2282d5889.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47233e81de2d9d728c80f02359f1af4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58c8214c80f9e949b1ac8d6e41b597e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cddf38600954700caa8a227dae4e8ced.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8389928ac57b99f195e6bef80d2a28d3.png)
您最近一年使用:0次
7 . 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
2024高三上·全国·专题练习
名校
解题方法
8 . 已知
,
,
(1)若
在
处取得极值,试求
的值和
的单调增区间;
(2)如图所示,若函数
的图象在
连续光滑,试猜想拉格朗日中值定理:即一定存在
,使得
,利用这条性质证明:函数
图象上任意两点的连线斜率不小于
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c566b6273b93a7231f891a0889579227.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2d60df31661ec394cdec5f0ad6bac38.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0843a602fe240e5798bcbc7d54b19ddb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)如图所示,若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ca6d68f1de3e70696f1d5d60affe6ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd7fc0ca8a82663b87fa36afb9c4ec09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f3fcc5073759c73c7a63c8818eca5c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a947d16d7293baf95e9274b9a0f5db78.png)
您最近一年使用:0次
9 . 如图,
内接于
,
是
的内心,过
作
的垂线交
于点
,交
于点
,
是
的中点,连接
,过
作
于点
.证明:
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/18/11efadd2-347e-4564-b671-6a3379f99d81.png?resizew=148)
(1)
;
(2)
、
、
、
四点共圆.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bb882b525043fb4602f598a9dfe5fbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42774a918f52ac8aa8b1f5b78a676f17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da701094ca9f9687f4b5af1825773a96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96a37f6963eefab01bd11be38cdbe0b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/18/11efadd2-347e-4564-b671-6a3379f99d81.png?resizew=148)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8174382753c87fee237ccf0a5f32551.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
您最近一年使用:0次
10 . 对任意满足
的非负实数组
,记
为
的元素个数,求证:
,并给出取等的充要条件.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b30a1123bc300b31b85509475669e9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6547f858c83bb153ba6828501dacb265.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/003789aa08a913801cba565a3d49ac75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/158c4fb001fcf161670b968be69f47c8.png)
您最近一年使用:0次
2023-12-15更新
|
163次组卷
|
2卷引用:2023年第39届全国中学生冬令营(CMO)数学试题