21-22高一·全国·课后作业
1 . 并集的概念
一般地,由___________ 属于集合A或属于集合B的元素所组成的集合,称为集合A与B的并集,记作:___________ (读作“A并B”),即
.用Venn图表示如图所示:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/20/7eadd36d-6b09-4283-a9cf-5b7c18234114.png?resizew=469)
由上述图形可知,无论集合A,B是何种关系,
恒有意义,图中阴影部分表示并集.
注意:并集概念中的“或”指的是只需满足其中一个条件即可,这与生活中的“或”字含义不同.生活中的“或”字是或此或彼,必居其一,而并集中的“或”字可以是兼有的.
一般地,由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af696aed52d8d4403811b14612d3592e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/20/7eadd36d-6b09-4283-a9cf-5b7c18234114.png?resizew=469)
由上述图形可知,无论集合A,B是何种关系,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3744e71abf4b43e128eabea9181b712.png)
注意:并集概念中的“或”指的是只需满足其中一个条件即可,这与生活中的“或”字含义不同.生活中的“或”字是或此或彼,必居其一,而并集中的“或”字可以是兼有的.
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2 . 设12元实数集合
满足:可将其划分为两个6元子集
和
,使得对每个
,均有
,则这样的
可以是______ .(写出一个即可)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab68ee11734bd89733c3064af16cea50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fe342a96adffebbb5d59c5fcfd138e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fde1b1e2b446a4db8650c9fdab83766.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff197132463fd76db478b76ed2cf653c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/908fbdc652b3cafc1f4d0b2e3ae2a094.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
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3 . 已知
,集合
,
,
.
(1)求
;
(2)若
且
,求实数
的取值范围;
(3)记
.当
时,若集合
中有且仅有一个元素
使得
0成立,试写出满足条件的
的表达式(只需写出一个即可).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0e7acdf79d526ab736473c01b92478f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4cdc10dc31aadc9212bfa553ab0b095.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3e0d7d4aea71fcf44c65ed792cbd421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25473c7106d6bd14c0917ba5e105364.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eb15ee27fd7ce9be1495232ac68948f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/952f1e0ce5bd53a6d5e8bb07ea2da5f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52854d0ead4737302f4b4706e1f80553.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(3)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce70d3efc3f0c7dc85b7d6be9421727a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa37a9a03564be3637c64117de07da8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
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4 . 对集合
,定义![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca578919b59d25a1bb2b5ea8e33da46.png)
①若
的元素个数为4,则
可以为:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f1e5d29de6e4d72bfed62d9c14dde5.png)
________ ,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f9fabbbe61a759e52ec975215e2e7c.png)
________ (写出一组即可)
②若集合
满足:存在
的子集
,使得
的元素个数不小于100,且对任意
,均有
,则集合
的元素个数的最小值是________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca578919b59d25a1bb2b5ea8e33da46.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81e37e656d05244fe3a5769cd1446725.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f1e5d29de6e4d72bfed62d9c14dde5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f9fabbbe61a759e52ec975215e2e7c.png)
②若集合
![](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/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81e37e656d05244fe3a5769cd1446725.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57a5235dcfd51f1ba4c1ee0037e167c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dfa843f0c5458e8b9ad3244680f7679.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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5 . 下列说法中正确的是( )
A.与定点A,B等距离的点不能构成集合 |
B.由“title”中的字母构成的集合中元素的个数为5 |
C.一个集合中有三个元素a,b,c,其中a,b,c是![]() ![]() |
D.高中学生中的游泳能手能构成集合 |
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解题方法
6 . 设所有能使
成立的
的集合为
,请写出一个集合
,使
,且
的元素有无限个,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f9fabbbe61a759e52ec975215e2e7c.png)
______ .(答案不唯一).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/011d3ad4f1d231dffbbf663013bb748d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caf22d7d1a965bda25168a233fb6290c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1f9fabbbe61a759e52ec975215e2e7c.png)
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7 . 对称变换在对称数学中具有重要的研究意义.若一个平面图形K在m(旋转变换或反射变换)的作用下仍然与原图形重合,就称K具有对称性,并记m为K的一个对称变换.例如,正三角形R在
(绕中心O作120°的旋转)的作用下仍然与R重合(如图1图2所示),所以
是R的一个对称变换,考虑到变换前后R的三个顶点间的对应关系,记
;又如,R在
(关于对称轴
所在直线的反射)的作用下仍然与R重合(如图1图3所示),所以
也是R的一个对称变换,类似地,记
.记正三角形R的所有对称变换构成集合S.一个非空集合G对于给定的代数运算.来说作成一个群,假如同时满足:
I.
,
;
II.
,
;
Ⅲ.
,
,
;
Ⅳ.
,
,
.
对于一个群G,称Ⅲ中的e为群G的单位元,称Ⅳ中的
为a在群G中的逆元.一个群G的一个非空子集H叫做G的一个子群,假如H对于G的代数运算
来说作成一个群.
(2)同一个对称变换的符号语言表达形式不唯一,如
.对于集合S中的元素,定义一种新运算*,规则如下:
,
.
①证明集合S对于给定的代数运算*来说作成一个群;
②已知H是群G的一个子群,e,
分别是G,H的单位元,
,
,
分别是a在群G,群H中的逆元.猜想e,
之间的关系以及
,
之间的关系,并给出证明;
③写出群S的所有子群.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ab1256702aef4e9f1a5eb6c12ecc96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77ab1256702aef4e9f1a5eb6c12ecc96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f8278c090ec35994a2300a2f6e03cd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2858005b9ae89ae080d83dcc13cf8e81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b9a0da1382342078b9b0bc326a0b58e.png)
I.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8362f15e544684164f38ff9ad7c38ac7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68f73696ca1660407be38423825ac579.png)
II.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/509a09a7391de2cc86e5e44ccccc981b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47512437070ec582249e3fe8a9422516.png)
Ⅲ.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27321be7cc5aec6555c61775f6638cea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebf00e8864c86c3ce8118ea76bf69773.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4a34726666c0499373270f6ca37136f.png)
Ⅳ.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebf00e8864c86c3ce8118ea76bf69773.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e78818e18abc456ae7a86110636386ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b2db6609d50b3b58c4c98ee07396606.png)
对于一个群G,称Ⅲ中的e为群G的单位元,称Ⅳ中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856b4ab24ff3b7d9e0b4d1c945232aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655c66701407d942ef38d482e6b3ffd7.png)
(2)同一个对称变换的符号语言表达形式不唯一,如
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/317369bcdd0bc35e2ca45ff7ee37ec09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7703f78bf42acd363d895107b6edae18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54ec72c22e432256b92c8c87f31f4bd2.png)
①证明集合S对于给定的代数运算*来说作成一个群;
②已知H是群G的一个子群,e,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3377b3f59d9c7ac048d59262ecbaf389.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c15c2fe2621766b6e71a4e61686f3bea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856b4ab24ff3b7d9e0b4d1c945232aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e90425090dfd36313d564a97289b3b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3377b3f59d9c7ac048d59262ecbaf389.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/856b4ab24ff3b7d9e0b4d1c945232aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e90425090dfd36313d564a97289b3b1.png)
③写出群S的所有子群.
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2024-03-20更新
|
1325次组卷
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5卷引用:安徽省芜湖市安徽师范大学附属中学2024届高三第二次模拟考试数学试题
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8 . 群的概念由法国天才数学家伽罗瓦(1811-1832)在19世纪30年代开创,群论虽起源于对代数多项式方程的研究,但在量子力学、晶体结构学等其他学科中也有十分广泛的应用.设
是一个非空集合,“
”是一个适用于
中元素的运算,若同时满足以下四个条件,则称
对“
”构成一个群:(1)封闭性,即若
,则存在唯一确定的
,使得
;(2)结合律成立,即对
中任意元素
都有
;(3)单位元存在,即存在
,对任意
,满足
,则
称为单位元;(4)逆元存在,即任意
,存在
,使得
,则称
与
互为逆元,
记作
.一般地,
可简记作
可简记作
可简记作
,以此类推.正八边形
的中心为
.以
表示恒等变换,即不对正八边形作任何变换;以
表示以点
为中心,将正八边形逆时针旋转
的旋转变换;以
表示以
所在直线为轴,将正八边形进行轴对称变换.定义运算“
”表示复合变换,即
表示将正八边形先进行
变换再进行
变换的变换.以形如
,并规定
的变换为元素,可组成集合
,则
对运算“
”可构成群,称之为“正八边形的对称变换群”,记作
.则以下关于
及其元素的说法中,正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655c66701407d942ef38d482e6b3ffd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655c66701407d942ef38d482e6b3ffd7.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655c66701407d942ef38d482e6b3ffd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4424f7a126daa000c5940787ee564521.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4424f7a126daa000c5940787ee564521.png)
A.![]() ![]() |
B.![]() ![]() |
C.![]() |
D.![]() |
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9 . 设A,B是两个非空集合,如果对于集合A中的任意一个元素x,按照某种确定的对应关系
,在集合B中都有唯一确定的元素y和它对应,并且不同的x对应不同的y;同时B中的每一个元素y,都有一个A中的元素x与它对应,则称
:
为从集合A到集合B的一一对应,并称集合A与B等势,记作
.若集合A与B之间不存在一一对应关系,则称A与B不等势,记作
.
例如:对于集合
,
,存在一一对应关系
,因此
.
(1)已知集合
,
,试判断
是否成立?请说明理由;
(2)证明:①
;
②
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42acae4bf2a6bead9d904b70d0480fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0915685a3eae67d5c6bc3bd722030876.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79aedd00413c6ff9b2696a63a854867.png)
例如:对于集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aac2c0e4c6fc7ae8950a38098cb062f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8794b3ea2ca1d6d2b70dcec2a991dd3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/210402b31fd895e4fd6921cb25c1ee88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0915685a3eae67d5c6bc3bd722030876.png)
(1)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cf4f47caab35fc473167ca17c7b5f4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eae2c499889a4619a5102a4b2e6b8129.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e386b0005c8f091434060361a07955d8.png)
(2)证明:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b06ec5553f5aeef37ec8ca6f0d9caba8.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/229c5c40da18cb86a81e709d802d4c1e.png)
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2024-04-18更新
|
963次组卷
|
4卷引用:浙江省台州市2024届高三下学期第二次教学质量评估数学试题
浙江省台州市2024届高三下学期第二次教学质量评估数学试题(已下线)压轴题01集合新定义、函数与导数13题型汇总 -1河北省名校联盟2024届高三下学期4月第二次联考数学试题 (已下线)情境10 存在性探索命题
10 . 对于给定的一个
位自然数
(其中
,
),称集合
为自然数
的子列集合,定义如下:
{
且
,使得
},比如:当
时,
.
(1)当
时,写出集合
;
(2)有限集合
的元素个数称为集合
的基数,一般用符号
来表示.
(ⅰ)已知
,试比较
大小关系;
(ⅱ)记函数
(其中
为
这
个数的一种顺序变换),并将能使
取到最小值的
记为
.当
时,求
的最小值,并写出所有满足条件的
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6be3d8ff4885ce8cf21ed3b7e4c9059.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff4ab89a62749697c6a67e4fe8e6f3c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12b0e3b00fe47801afb53ec56706c21a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5123cae73867329882792f626287b970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d688a7cacda715fc5c2fad9a2adaddee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab691edda624f588e85d493423b3e398.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76fb6d4810762396e3fbe728687a0794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba7a9e06bedb3aca590121cc47e64e52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e534ff8ca5451dce6629223e002d5878.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d407ed5fd8a5fd413426fc1fc118422.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eeaa1b4ec60977b69d48d3d023f5d731.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5123cae73867329882792f626287b970.png)
(2)有限集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
(ⅰ)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b43ea4161df6e6178c26c524935af465.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b53fe2a5d83d2e3e97f3a49d1f845370.png)
(ⅱ)记函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a2a7d5abc0e14bf1da403fba5b27644.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c558c7204d256c96b74b9c991c0e5c1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086eb439f6a1578fdba904825340772d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aad1bf0868f56ad3bda73d4ca5851cb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbf91726683a3963e941231877c8c6ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e731e11a03c0f5d2768e87a3442634d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c326b4a68d5148e8e5a5ebc15d3b132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aad1bf0868f56ad3bda73d4ca5851cb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e731e11a03c0f5d2768e87a3442634d.png)
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