2023高三·全国·专题练习
1 . 在
个不同数的排列
中,若
时
(即前面某数大于后面某数),则称
与
构成一个逆序. 一个排列的全部逆序的总数称为该排列的逆序数. 记排列
的逆序数为
,如排列21的逆序数
,排列4321的逆序数
.
(1)求
,
,并写出
的表达式;
(2)令
,证明
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2ea5126ca30d68cc8b70f03860ba2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ef8b6e83c3e5c996dbe676eaeaade54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ebd42fc2af57877e06f485b50ae4d87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba3799f12f1cd7361f0bce9fea0b7278.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59c709117ab1d3ef620883a732aed68b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f59061d84ba9db3cf6a114446549e75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ee5497ff422a53cab2a728381478ff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e8c45e4c4ab30665338dd87a2258f23.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daf464629fa321a6ff7401ab79f07083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f65fc200f10b97588a0c9896277c9c64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(2)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/806904740dfe923b069c7b37dbe74ece.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acde504126c73a1de25c2312e71b616b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7247567230a3bebb8fa497c2b22bb02.png)
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2 . 已知无穷数列
的各项均为正数,当
时,
;当
时,
,其中
表示
这
个数中最大的数.
(1)若数列
的前
项为1,4,3,8,写出
的值;
(2)是否存在
,使
,且
?请说明理由;
(3)设
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd015442628054692b8cc0a19c77d2bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42c7d66f6af0f1aa78f10b1567ac2412.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8a774a567b92c7fcdfa269e93743dcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba6e6aa0ef8f0628b65afb6403d73917.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/872fc6185291bde2743da17a2c7fe39b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a9675b993791e3a2a5df2e448ac0120.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7c8f0abd695b77d9c6228baa19828c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5873c01192b7d33b7483f444f90b5b0.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd015442628054692b8cc0a19c77d2bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a99113915b30fec4c7ec945a2bd2524.png)
(2)是否存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6b71956fd4fa11f337d254d47a8f074.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb8fa14a3edac65ef766229cede6efea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bddf3d994227213ffe4c8b16ca39d9d5.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd0e1e70b3a495393302ca73ce77cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28ed945b6599e934f7f86a5920ecd0c4.png)
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3 . 设集合
,集合
,如果对于任意元素
,都有
或
,则称集合
为
的自邻集.记
为集合
的所有自邻集中最大元素为
的集合的个数.
(1)直接判断集合
和
是否为
的自邻集;
(2)比较
和
的大小,并说明理由;
(3)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db8d9e00ef22cd220a6bbd291f280a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84cd2449f6ae27a72287be95a661d8f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/320a7c616f6f7207a0a38bb707ac2205.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bfbcd3d6b77c949be81a946ac9ed9d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a73707750f88b56101446fce394e0faf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f7f71b0119f257edb8d5060a810de92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
(1)直接判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4047b80385ef60ea5e9a1f184e7b948b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecde0085a473948c061942a1728a37c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5002f030017f6f0b34a61b2e15c5a9cb.png)
(2)比较
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64927a98d33b49dc5c6a0e65e5e8eb53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b41788e238eff245e567b58dea3a0003.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/293bd318a7a3796d3589db25148be688.png)
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