1 . 冒泡排序是一种计算机科学领域的较简单的排序算法.其基本思想是:通过对待排序序列
从左往右,依次对相邻两个元素
(
,2,
,
)比较大小,若
,则交换两个数的位置,使值较大的元素逐渐从左移向右,就如水底下的气泡一样逐渐向上冒,重复以上过程直到序列中所有数都是按照从小到大排列为止.例如:对于序列
进行冒泡排序,首先比较
,需要交换1次位置,得到新序列
,然后比较
,无需交换位置,最后比较
,又需要交换1次位置,得到新序列
,最终完成了冒泡排序.同样地,序列
需要依次交换
,
完成冒泡排序.因此,
和
均是交换2次的序列.现在对任一个包含n个不等实数的序列进行冒泡排序(
),设在冒泡排序中序列需要交换的最大次数为
,只需要交换1次的序列个数为
,只需要交换2次的序列个数为
,则下列说法正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ed7b442e78e34e20513eda80b994057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7d2b73d53e55ed235678b902b04b5f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5095a28bb1b91bf6bed9e2cfbd76bb18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aadf9ab510510120699c5eee39ab18b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ef7fb37eb0663328147e890fe3743ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c11df207bfbfecfeda5b0dedff71986.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b30511e6903e1c1f9a8fedbcf916ca5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5363def6ab70faf774f1fc601977ccf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ec69d27edd7577262f2d23a26ef858b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/104431dedcf68e8bee516d4d14de765d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36bf49f0fea361cb1e0d5fd9fb304003.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f920d9bb6f755983c74df6ace9b54b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7845a338b3b64ae887423611ec7301e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/104431dedcf68e8bee516d4d14de765d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c11df207bfbfecfeda5b0dedff71986.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f920d9bb6f755983c74df6ace9b54b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
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解题方法
2 . 已知数列
满足对任意的
,均有
,且
,
,数列
为等差数列,且满足
,
.
(1)求
,
的通项公式;
(2)设集合
,记
为集合
中的元素个数.
①设
,求
的前
项和
;
②求证:
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e6cbbec0f900da8864d00e396893c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18d8e8f821111de8075e5c3dfb22a5d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59dd6c97d2ee3e74ba5730f1cbcc1d43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e506112c35bdf08b18460d233eb6595.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87bc4ebb7c9c2323c75011db21226ae5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c59e7c7a84a4bdb959e95536d0404ceb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
①设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffc96f796c19909fe80d0da1cd1d7823.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/960b682f983b053dc9064cf29c97e250.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d51f9147b8265c0276c1f2c2659197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b255b2aa8a99d13c0756a87906675c7f.png)
②求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d12d0bd9afdd4e53ff37f5bfcaa1106c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bac6b8de4384ff605892e8de5263de13.png)
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3 . 对于每项均是正整数的数列P:
,定义变换
,
将数列P变换成数列
:
.对于每项均是非负整数的数列
,定义
,定义变换
,
将数列Q各项从大到小排列,然后去掉所有为零的项,得到数列
.
(1)若数列
为2,4,3,7,求
的值;
(2)对于每项均是正整数的有穷数列
,令
,
.
(i)探究
与
的关系;
(ii)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9a724b59c890095baa5cb73e267c44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e50dbc39a6a06eb8bd7b295f8cc95a6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f4d9c1555c406f37a85ff5ddd2275af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed3926a18ee86236e7cf53467ac04f73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca3c54bcc0bd04c304d6513732584b51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9275bd8ce17fcc4a786510b008414ab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9275bd8ce17fcc4a786510b008414ab0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29422ba63f79b2087f0dcce4879d8c78.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf9f50605db5d5f8f3a01ee8e474a112.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00061b36983bb547aa63f9a9f6222105.png)
(2)对于每项均是正整数的有穷数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf9f50605db5d5f8f3a01ee8e474a112.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/131cac59d4d568a27dbb220810dbb014.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ef96396caccbf2f959e9d233f060317.png)
(i)探究
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00061b36983bb547aa63f9a9f6222105.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c496ba1d329d5fd94f4fdb1da8c7cde.png)
(ii)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5de2726fe544faf83d4ab88e9116135.png)
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名校
4 . 已知等差数列
的公差与等比数列
的公比相等,且
,
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/414187fca31df508dbf88d7f2bb83662.png)
______ ;若数列
和
的所有项合在一起,从小到大依次排列构成一个数列
,数列
的前
项和为
,则使得
成立的
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/684b935a7274130d081bfa7b2b938023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a89b8821c758c29c8b02bd79425ecbd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13624221ac7e06fc5ebd48b21ed0de10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/414187fca31df508dbf88d7f2bb83662.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/055bd850286feebbeac5df481988b1d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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名校
5 . 已知数列A:a1,a2,…,aN
的各项均为正整数,设集合
,记T的元素个数为
.
(1)①若数列A:1,2,4,5,求集合T,并写出
的值;
②若数列A:1,3,x,y,且
,
,求数列A和集合T;
(2)若A是递增数列,求证:“
”的充要条件是“A为等差数列”;
(3)请你判断
是否存在最大值,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/485eb1ad5fd643e739e15a39c7922b4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8f081d422da2385bed320a2c3a52633.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f812114476c7a8f0219a412039d07c89.png)
(1)①若数列A:1,2,4,5,求集合T,并写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f812114476c7a8f0219a412039d07c89.png)
②若数列A:1,3,x,y,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c73a0da1caaab9022852df736dc9cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67c4c7fe993913d8731716ac796359f0.png)
(2)若A是递增数列,求证:“
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5dd1d57123aedb635394b03c8b4c461.png)
(3)请你判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f812114476c7a8f0219a412039d07c89.png)
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7卷引用:广东省惠州市第一中学2024届高三元月阶段测试数学试题
广东省惠州市第一中学2024届高三元月阶段测试数学试题(已下线)北京市海淀区2023届高三上学期期末练习数学试题变式题16-21(已下线)专题03 条件存在型【讲】【北京版】1(已下线)专题1 集合新定义题(九省联考第19题模式)练(已下线)高考数学冲刺押题卷01(2024新题型)北京市北京大学附属中学2021-2022学年高二上学期期中数学试题北京市第二十四中学2023-2024学年高二上学期期末数学模拟试卷
6 . 设数列
的前
项和为
.若对任意的正整数
,总存在正整数
,使得
,则称
是“
数列”.
(1)若数列
,
,判断
和
是否是“
数列”;
(2)设
是等差数列,其首项
,公差
.若
是“
数列”,求
的值;
(3)证明:对任意的等差数列
,总存在两个“
数列”
和
,使得
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3693c7c942afef5517a3c18997c878df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e903c3dc6bdb559fd173f8d4e930f78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ff259bff098430a6512d0e4f6fb2d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2d8bbb4a09e0ac86bbae46222a90841.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
(3)证明:对任意的等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a13fcd18316e035cdc08901073672e.png)
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4卷引用:北京市海淀区教师进修学校附属实验学校2024届高三上学期12月练习数学试题
7 . 已知数列
是首项为1的等差数列,数列
是公比不为1的等比数列,满足
,
,
.
(1)求
和
的通项公式;
(2)求数列
的前
项和
;
(3)若数列
满足
,
,记
.是否存在整数
,使得对任意的
都有
成立?若存在,求出
的值;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84672a737e1ba65228ffd2f0064a8c9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40751e69baead4a0d5bea384aedfa6c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dbe8fa82ab04f0a4ba4ad1c570c9aa1.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec4bdc2a6d4fc387dc621f0b5a268c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(3)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dea6578afabc23f5d7041b88c3790dd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/614bac93e838d86d18422bed438368df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b02b03f064cffd092bba6be3bfc95ecc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1505d56f0b35fe7f2de1fe1888036e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/414847d018e36898bde4a88772c1d2c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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名校
解题方法
8 . 若函数
存在连续四个相邻且依次能构成等差数列的零点,则实数k的可能取值有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41c9917441658555b8cb9fd4c2c25d1e.png)
A.![]() | B.![]() | C.0 | D.![]() |
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9 . 已知数列
满足
(
且
),则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bec805491b68bcd47219f79e69e26b63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37a59558292ad6b3d0978bfd7484990.png)
A.![]() ![]() |
B.若数列![]() ![]() |
C.数列![]() ![]() ![]() |
D.当n是奇数时,![]() |
您最近一年使用:0次
2023-07-08更新
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1049次组卷
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5卷引用:云南省昆明市第一中学2024届高三新课标第四次一轮复习检测数学试题
10 . 已知数列
的项数均为m
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34919e3b413417f8fcc06fbfbca9bfe0.png)
的前n项和分别为
,并规定
.对于
,定义
,其中,
表示数集M中最大的数.
(1)若
,求
的值;
(2)若
,且
,求
;
(3)证明:存在
,满足
使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e567d7e9761951a266953c8d5042ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c53fc8ddaa412b237ecb095cf1c65335.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34919e3b413417f8fcc06fbfbca9bfe0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62e567d7e9761951a266953c8d5042ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b64f109cde567dc5750276a16a6b774.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7d230d1915653fb876373f882ca81b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cd13665a47f5548727c599936b32dc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e2d6df455d7702a81bdbc86f17e8c81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/698f45c9ed5bb04924f1037107e76988.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15dc21f6a796961cc506633a4fe32563.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd374d21bbdff3c6f8e69b557a86e2ce.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/295f2712a68800672db5c617713eedf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9de2f1a28584f093949cc0b854dfb3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a135cb036833400f3fa1edc92d5ce410.png)
(3)证明:存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a23a3f55b2eb456a65b9788574437678.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/363d7ed2c067c37fb1dfc5e2a50ba573.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1eedada19441233cfac2f4e4322cf85.png)
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2023-06-19更新
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10452次组卷
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19卷引用:北京市第八十中学2023-2024学年高三上学期10月月考数学试卷
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