名校
1 . 由
个正整数构成的有限集
(其中
),记
,特别规定
,若集合M满足:对任意的正整数
,都存在集合M的两个子集A,B,使得
成立,则称集合
为“满集”.
(1)分别判断集合
与
是否为“满集”,请说明理由;
(2)若集合
为“满集”,求
的值:
(3)若
为满集,
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa99639ae86ed0eb77369d0ab16e062.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b85889b308ebb3923999a908aac92cbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebc22ec3dd9bd6aaca20dac58effc277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b44aed8cc107aecae26873891bfdc5f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea670041bfb683baa3544ae6b84e2674.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c929017a03ca7068dd9de323e1aafef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
(1)分别判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeb1d0526d8252e213e396c0ea497c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1ff3cf4b4399ea9e79f5b9f4cd285e1.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9221840cf541e2564ba035f43c7a408e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
解题方法
2 . 已知集合
.对集合A中的任意元素
,定义
,当正整数
时,定义
(约定
).
(1)若
,求
和
;
(2)若
满足
且
,求
的所有可能结果;
(3)是否存在正整数n使得对任意
都有
?若存在,求出n的所有取值;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91a02b715c992f112de2f9987c70a3df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8af8284494c4f58fa852984085b35603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3f3591879d4bacc70989aca85be2efa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12c54046ad997789682be07a1d47a3a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f88776c6c89617192c501dc7f5c9b46.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d64954772ca230b54f1072dc05fd1f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5977a625005a82318474380d6ede73e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df70ecf992ea644da971fe695ca36aa2.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8af8284494c4f58fa852984085b35603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b792bda20ea741c2507be50668572260.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43354369f1a232b2bcb18fffbc00b989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
(3)是否存在正整数n使得对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/733699269e98fab0eee3b98e95a7072a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd8bb7f8f7baedbe05c4a6c05f0b9e6d.png)
您最近一年使用:0次
2022-05-17更新
|
1506次组卷
|
4卷引用:北京市朝阳区2022届高三二模数学试题
名校
3 . 已知m,n,k为正整数,
,
,A是由
个不超过k的正整数组成的m行n列的数表,其第i行第j列为
,
,
,满足:
①对任意
,
,均有
,
,
互不相等;
②对任意
,不存在
,使得
且
;
③当
时,对任意
,存在
,使得
.
记
为所有这样的数表构成的集合.
(Ⅰ)写出
中的一个元素;
(Ⅱ)若
,则当n最大时,求m的最大值;
(Ⅲ)从问题(一)问题(二)中选择一个作答.
问题(一):求集合
的元素个数.
问题(二):求集合
的元素个数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/835c74bbb8c61dd2d2f008664a8c8810.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb74ca8fc86ddef279e33f31c1fedda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f55f10837591bacdcae5a9724082f73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a65bad3343f0e29460ead5b54771f362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68f01cfc245e926581bdb125e0fba733.png)
①对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a65bad3343f0e29460ead5b54771f362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/018bac8d691164179dd650e8e066a81e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9c2f3106360deeec6bf756c2edeacda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f55f10837591bacdcae5a9724082f73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f156d28b30a61f24915c73789be9c44a.png)
②对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a65bad3343f0e29460ead5b54771f362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04b6b088f50b09e446c001943b77da97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccbcfbf6fb3c3463f215c873384ca4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00b9e12355108f86a2c7b267b9f047d4.png)
③当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72ac49ab7c8001c209b8611b9ea40d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ebd42fc2af57877e06f485b50ae4d87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7601dbefa6836756e3d2731b79af0126.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2127bb2a84fd1935f668058d4a1515a6.png)
记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae9f52430f4c5d603e4370855970713a.png)
(Ⅰ)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d84e81a93d18e534727cc720abc0706.png)
(Ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21867b2e83d5014b07ab61183fd0efcb.png)
(Ⅲ)从问题(一)问题(二)中选择一个作答.
问题(一):求集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de081a5f6435a695ad71d6a539352334.png)
问题(二):求集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0f5b9ffcbd788692a3cc2e59648f3c9.png)
您最近一年使用:0次
4 . 设n为正整数,集合A=
,
,
,
,
,
.对于集合A中的任意元素
和
,记
.
(Ⅰ)当n=3时,若
,
,求
和
的值;
(Ⅱ)当
时,对于
中的任意两个不同的元素
,
,证明:
.
(Ⅲ)给定不小于2的正整数n,设B是A的子集,且满足:对于B中的任意两个不同元素
,
,
.写出一个集合B,使其元素个数最多,并说明由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18693093ebe22c307bd7a0e7546c583.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94b9e69fb1d4a48e9c24b04f86e496e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5095a28bb1b91bf6bed9e2cfbd76bb18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c93fedde9bb8f927c986094275598b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0350c23ae81d21f565637852d6056cee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ca0a6e55c27d82230a7341c7a9ff90a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1a7bce6e7acbd13e268551c7897f4d.png)
(Ⅰ)当n=3时,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc9ce0575abdf14f4403b3ddc460cafa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0addc2213ff0303a589dc71690ba43bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/272ba5e58ecd8f3369b5964590d834e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a193c5c4a0476eef3339aafa9bca0e61.png)
(Ⅱ)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b927eb0becc15f16353f5ca8a36d28c2.png)
(Ⅲ)给定不小于2的正整数n,设B是A的子集,且满足:对于B中的任意两个不同元素
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02c2e2b26fe4d99dbd55359ca13a82.png)
您最近一年使用:0次
2020-06-03更新
|
1539次组卷
|
7卷引用:2020届北京市密云区高三第二学期第二次阶段性测试数学试题
2020届北京市密云区高三第二学期第二次阶段性测试数学试题北京一零一中学2022届高三上学期统考(二)数学试题北京市中关村中学2022届高三下学期开学测试数学试题北京市第五中学2021届高三上学期10月月考数学试题北京市延庆区第一中学2024届高三上学期9月月考数学试题(已下线)专题04 集合中的压轴题(二)-【尖子生专用】2021-2022学年高一数学考点培优训练(人教A版2019必修第一册)(已下线)高一上学期第一次月考解答题压轴题50题专练-举一反三系列
名校
解题方法
5 . 如图,在等边三角形ABC中, AB=6.动点P从点A出发,沿着此三角形三边逆时针运动回到A点,记P运动的路程为x,点P到此三角形中心O距离的平方为f(x),给出下列三个结论:
![](https://img.xkw.com/dksih/QBM/2020/5/9/2458897091551232/2459212237316096/STEM/b4747df5-b1b9-475b-a696-c0a2be9628aa.png)
①函数f(x)的最大值为12;
②函数f(x)的图象的对称轴方程为x=9;
③关于x的方程
最多有5个实数根.
其中,所有正确结论的序号是____ .
![](https://img.xkw.com/dksih/QBM/2020/5/9/2458897091551232/2459212237316096/STEM/b4747df5-b1b9-475b-a696-c0a2be9628aa.png)
①函数f(x)的最大值为12;
②函数f(x)的图象的对称轴方程为x=9;
③关于x的方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a7606ddcfd176248b734eed06cf2e13.png)
其中,所有正确结论的序号是
您最近一年使用:0次
2020-05-09更新
|
3260次组卷
|
13卷引用:2020届北京市海淀区高三一模数学试题
2020届北京市海淀区高三一模数学试题北京景山学校远洋分校2020—2021学年高一上学期数学学科期中测试试题北京市中国人民大学附属中学朝阳学校2022届高三10月阶段检测数学试题北京市第四十三中学2022届高三上学期期中考试数学试题北京市一零一中学2022届高三下学期入学考试数学试卷题北京市海淀区北京交大附中2023-2024学年高一上学期12月月考数学试题2020届河北省衡水二中高三下学期二模数学(理)试题(已下线)【新东方】【2021.4.27】【宁波】【高一上】【高中数学】【00114】(已下线)【新东方】【2021.5.25】【NB】【高一上】【高中数学】【NB00091】(已下线)期末押题测试卷(一)-《聚能闯关》2021-2022学年高一数学提优闯关训练(人教A版2019必修第一册)(已下线)二轮拔高卷08-【赢在高考·黄金20卷】备战2022年高考数学(理)模拟卷(全国卷专用)安徽省合肥市第一中学、第六中学2021-2022学年高一下学期期末联考数学试题(已下线)专题11 函数的图象-2
6 . 已知集合
,且
中的元素个数
大于等于5.若集合
中存在四个不同的元素
,使得
,则称集合
是“关联的”,并称集合
是集合
的“关联子集”;若集合
不存在“关联子集”,则称集合
是“独立的”.
分别判断集合
和集合
是“关联的”还是“独立的”?若是“关联的”,写出其所有 的关联子集;
已知集合
是“关联的”,且任取集合
,总存在
的关联子集
,使得
.若
,求证:
是等差数列;
集合
是“独立的”,求证:存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10cdcb0e77b3ae3e701c6b51e15e2346.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d10449bc77d692a7270e0f20a68cdf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eeda5cef4846ef829069fe27f64e34e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9bf4032eb5a9ba68131b15182aa3491.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bf6c84731e5e1bd335ecfc2d36c3d81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9859aa908844a32c0e1e069a046727.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/918f1c94368c3a41177ff42cfedc0eb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f53190d6ead827a6338b9de847aeaf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6e0ad51c5541ec3dcca4a9845f8b7db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/498f92bf2e605cdbc91973e29b047566.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/ce4d89801d24aa43f47d6a366aad0571.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1ccce8225324817b0577551956464f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/874781ab5711bff6ee8c9cbad5b3b3dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f62295c36d2e2174908c2bec0eb5b30f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acfc595518cf752e1c7903dfff93dbda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8021f4f4c253a00360bf8f9425610e1.png)
您最近一年使用:0次
2020-02-09更新
|
1572次组卷
|
10卷引用:2020届北京市海淀区高三上学期期中数学试题
2020届北京市海淀区高三上学期期中数学试题(已下线)专题02 拿高分题目强化卷(第三篇)-备战2021年新高考数学分层强化训练(北京专版)北京市海淀区2021届高三模拟试题(一)北京市清华大学附属中学朝阳学校2021-2022学年高二5月月考数学试题北京市第五十七中学2021-2022学年高二下学期期末考试数学试题北京市第八中学2023届高三上学期12月测试数学试题北京市朝阳区中国人民大学朝阳分校2021-2022学年高三上学期开学考数学试题北京市日坛中学2023-2024学年高二下学期第三次月考(6月)数学试卷(已下线)考点47 推理与证明-备战2022年高考数学(文)一轮复习考点帮上海市上海中学2022届高三下学期高考模拟3数学试题