名校
1 . 约数,又称因数.它的定义如下:若整数a除以整数m(
)除得的商正好是整数而没有余数,我们就称a为m的倍数,称m为a的约数.
设正整数a有k个正约数,即为
,
,⋯
,
,(
).
(1)当
时,是否存在
,
,…,
构成等比数列,若存在请写出一个满足条件的正整数a的值,若不存在请说明理由;
(2)当
时,若
,
,⋯
构成等比数列,求正整数a.
(3)当
时,若
,
,…,
是a的所有正约数的一个排列,那么
,
,
,⋯,
是否是另一个正整数的所有正约数的一个排列?并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0baedc4d7e690ab3f7d80d30ba0a9efe.png)
设正整数a有k个正约数,即为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68c0cd13ec90e5697013e59d73d3e82c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df21efb81bd9f5ec47c8ad705a2272ad.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/835c74bbb8c61dd2d2f008664a8c8810.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbeaed9ec21e090defafcfeefe0059c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe164d8a8a4049e01565b576007651de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01416ee1d48b17f889e444b7eda99740.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/177e4374fb738c4f13dc58e9025c88e4.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/835c74bbb8c61dd2d2f008664a8c8810.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f255d0395fba51ca2d44293cca42e0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e6f19b84484b5480ea2100165abfd81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f9d2e152db0845ff23e4ea0cd00974d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15f30fd21924e7bcf368854ef38af82e.png)
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2 . 设
为正整数,集合
对于
,设集合
.
(1)若
,写出集合
;
(2)若
,且
满足
令
,求证:
;
(3)若
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb39983c5c3fc32d3f0a2c98f04cdb3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ee7981aba05ee1d4945062913c8db8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08897069d8b3e4c9f42a768d7ae2d416.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40deecdcc73cfee7fe7624a813af16ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeff5c5f7214a5f9b920af89af5b3802.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ee7981aba05ee1d4945062913c8db8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c5c31ea128ff2a390979894f53486f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/695e27822adf2901dd6f48a1a29e68d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764928f3bec8c40cbff944c0b36ffb57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18bf5b3e05fdd0bdb9597f5b9d2386cf.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ee7981aba05ee1d4945062913c8db8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f53037f80f8cf5faf2f861a3859c3166.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8aa4b0e70b5c191c6391263d905c52b0.png)
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3 .
,
,则不超过2024的正整数中可以作为
函数值的个数为______
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb63478132d4c1fef3c17e591919da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95749241ac8431d2e731e69769c1cdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
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解题方法
4 . 复平面
与
交点个数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87432634f161e669f23527a5fc813ecb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b70c2519610d6d1d6d0855b0f27dfc5e.png)
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5 . 设
,
,则
=______
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6123443832507911255a4378124b8d87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12d0a8dfc3620c557571e954abea5627.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25da493a41219082d08558a7dd41bcde.png)
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名校
6 . 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)
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7 . 已知点集
.设非空点集
,若对
中任意一点
,在
中存在一点
(
与
不重合),使得线段
上除了点
外没有
中的点,则
中的元素个数最小值是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51b22251511d66ad0c49513dae61f2bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55fcd91711251a8bfe28b22c870dad80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1fb56e714978b3dd418996c1f75768c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
A.1 | B.2 | C.3 | D.4 |
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8 . 三个复数的模分别为,且这三个复数实部虚部均为整数,则这三个复数的积有多少个可能值?
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9 . 数列满足
,求使该数列
有极限的
的最大值.
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10 . 复数
,则以下正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd85688279a50d3d6e7091d88e70a2af.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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