1 . 将正方形
分割成
个全等的小正方形(图1、图2分别给出了
的情形),在每个正方形的顶点各放置一个数,使位于正方形
的四边及平行于某边的任一直线上的数都分别依次成等差数列.若顶点
处的四个数互不相同且和为1,记所有顶点上的数和为
,则有
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/022205c82846dfa3420c67402a278fea.png)
______ ,…,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d64205bb7d470602fb525b02e72a81b.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/188ba12941ea122a7775a4999e129161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d15d4afcc5736bb747eeffcffd3f4464.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e345e86daf74312a6992e5d1c3f47f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cf8c7f5f6a35f7ed88271a077619aef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/022205c82846dfa3420c67402a278fea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d64205bb7d470602fb525b02e72a81b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/29/8a0dc8d3-051d-4666-9a45-8e1c699ed7bd.png?resizew=254)
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2 . 若
表示自然数
的最大奇因数,例如
,
,
,记
(
为自然数),则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72253d846d8750db2bf695df99c53f3e.png)
______ .,
的通项公式为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e91eb1a74ed4eb789a5cf6bf0d08900a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30ac85888a9b802aea6eb688edac8f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18443df3018a6806c7ded2bf11ed70a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9bf2837ae4419c83bb69ad10ab7ef14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/640faa4d58105334dafd1cf218f30ae9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72253d846d8750db2bf695df99c53f3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
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3 . 如图,将正整数按下表的规律排列,把行与列交叉处的那个数称为某行某列的元素,记作
,如第2行第4列的数是15,记作
,则有序数对
是____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acb574762e9dc1df3fc79fa957b1fd49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0485cc492f0f7fd84375fd26d4d12585.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49bd66006400c3413106caa2fc94fa2d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/14/8d8501a9-764d-4edf-86bc-031e42d95659.png?resizew=185)
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4 . 给定奇数
,设
是
的数阵.
表示数阵第
行第
列的数,
且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de075cbe45f637a11f53685a018e340a.png)
.定义变换
为“将数阵中第
行和第
列的数都乘以
”,其中
.设
.将
经过
变换得到
,
经过
变换得到
,
,
经过
变换得到
.记数阵
中
的个数为
.
(1)当
时,设
,
,写出
,并求
;
(2)当
时,对给定的数阵
,证明:
是
的倍数;
(3)证明:对给定的数阵
,总存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/767f5a4746f04db68386fac3970b1ed1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37a14c188b1c9d61aa237b137ba18023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/604a3f7f0c00236993c4659ff12fd63c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de075cbe45f637a11f53685a018e340a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40f4f7f26c112216d9a548b5ad082ea4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca81e328c01121c81869e7d304f01054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0393f387f255f7ade0e5c7bf1a8a9a0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39e524eaa9507cc5c8c81a0831d853ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f8e7d8908eb361e60af9da39fc1b1ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d2c0b8dcec87c9655bbb1a37d9884e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08c9bcf77059762fc46fc437ca8b060c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84221a34d3afe2b194c604aa642aa922.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b003c008551d85cda4fe287d0742216.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bd55ae46a41a37f90a3d745b9e8f879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88d6bc8e1dc170db94a6caf502b9b187.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90c0d07535b2eca42f44b8109c6ccd11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbf9bc3b31e16bd645b864a346187f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00442d96d695db2c58bf1fb7165fca94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d13fffd82b0a6b66580a17a6e0d2802.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae537905eee4d73c55298fa4280794b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84406a04370c0c0834550b1f22b49d50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
(3)证明:对给定的数阵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7356ec98b600ece41f3a6b4bc26a7d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/666a2f30e071ff37c5545baa18276fe7.png)
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5 . 对任意实数
,记
为不大于
的最大整数,再记
,由此可定义函数
,进而可定义递推数列
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/22/73191ebb-ef41-4fb1-8007-e0091a02c77e.png?resizew=231)
(1)求
的定义域,并判断
是否有反函数(只需写出判断结果,无需说明理由).
(2)求证:①
的每一项都是正有理数;②
的任意两项均不同.
(3)为进一步研究
各项的取值情况,有人把该数列排成了下述的“二分树状表”,并探究了图中由箭头连接的两数间的关系,进而猜想“
的各项取遍所有正有理数”.请你判断该猜想是否正确,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7db6fc42e0baf315ff7c5a30ff8ba73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb43db0a1162d7407114fb7efc74b79e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e121e47d3b2f0dc79f008fa9f215f69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8af3867356b63012dba362fa7267a333.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/22/73191ebb-ef41-4fb1-8007-e0091a02c77e.png?resizew=231)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(2)求证:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(3)为进一步研究
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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6 . 某投资公司评估一个需要投资980万的项目,该项目从第1年年末开始,每一年的净利润是
万元,而且收益可以持续50年.若年利率为8%,记第
年年末的收益现值为
(
,
),![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
___________ ;若该项目值得投资,则
的最小值为___________ 万元.(参考数据:
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d60a65136f3e00811ad1b401924fb10f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc4fdfde4a69599dbe9a97e83939e2ae.png)
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7 . 十九世纪下半叶集合论的创立,奠定了现代数学的基础,著名的“康托三分集”是数学理性思维的构造产物,具有典型的分形特征,其操作过程如下:将闭区间[0,1]均分为三段,去掉中间的区间段
,记为第1次操作;再将剩下的两个区间
,
分别均分为三段,并各自去掉中间的区间段,记为第2次操作...;每次操作都在上一次操作的基础上,将剩下的各个区间分别均分为三段,同样各自去掉中间的区间段:操作过程不断地进行下去,剩下的区间集合即是“康托三分集”,第三次操作后,依次从左到右第三个区间为___________ ,若使前n次操作去掉的所有区间长度之和不小于
,则需要操作的次数n的最小值为____________ .(
,
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f5788219e1b572a03b7453968ad25f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3721aa05c3bf03ee8e92c7fd7a0b48c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54cba28de35bd3365c48013aa2889a82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c130d40d975aba491541d1a823b509c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49f8340ff8ef6994abddab919418423b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b0deba532a99234201dd24b23a1b9fc.png)
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2022-05-11更新
|
1597次组卷
|
4卷引用:山东省德州市2022届高考二模数学试题
山东省德州市2022届高考二模数学试题(已下线)模块八 专题8 以数学文化新情景为背景的压轴题江苏省连云港市赣榆高级中学2022-2023学年高三上学期10月学情检测数学试题(已下线)压轴题03不等式压轴题13题型汇总-2
名校
8 . 定义
为与x距离最近的整数,令函数
,如:
,
.则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ab3bdcd0f5eb5bc4779bf5d539e5cdb.png)
______ ;![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3effa19895ed86815e16849f6283ab78.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5e710e4687e09d92d8a1e30f486eba2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0b586aaeee6b7d7eb3688b2d23b500f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ea0d11b623bc6b0d26e32cff0e004e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed023c308ad370de95787357e835621f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ab3bdcd0f5eb5bc4779bf5d539e5cdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3effa19895ed86815e16849f6283ab78.png)
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2022-05-11更新
|
625次组卷
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3卷引用:山东省菏泽市2022届高三二模考试数学试题
9 . 如图,画一个正三角形,不画第三边;接着画正方形,对这个正方形,不画第四边,接着画正五边形;对这个正五边形不画第五边,接着画正六边形;……,这样无限画下去,形成一条无穷伸展的等边折线.设第n条线段与第
条线段所夹的角为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59f8d31e3a15ed5f13d296cc02f8d2b1.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0876215b2fd463d151523cd3c6b447.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a4c4f7f44c074efc68583921c5cc912.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59f8d31e3a15ed5f13d296cc02f8d2b1.png)
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2022-04-28更新
|
892次组卷
|
5卷引用:上海市上海交通大学附属中学2022届高三下学期期中数学试题
上海市上海交通大学附属中学2022届高三下学期期中数学试题(已下线)文科数学-2022年高考考前20天终极冲刺攻略(二)(已下线)专题14数学知识的延伸必考题型分类训练-2福建省永春一中、培元中学、石光中学、季延中学2024届高三下学期第二次联合考试数学试题云南省昆明市第三中学2023-2024学年高二下学期第二次综合测试(4月)数学试题
10 . 已知
行
列
的数表
中,对任意的
,
,都有
.若当
时,总有
,则称数表A为典型表,此时记
.
(1)若数表
,
,请直接写出B,C是否是典型表;
(2)当
时,是否存在典型表A使得
,若存在,请写出一个A;若不存在,请说明理由;
(3)求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b7e250aad05323c9149dc8ad96e4f27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99648a1c5d4727be45896e91cccf3fc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39c9d0659cb8619fedee2eec1c5bb03f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddbd4cbb703eb9bd566273e8c680a49f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eebab31ed753d75177744c2ef33380a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19b98c229c9ae9b121c9f1cf5cfbe324.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63ba4a9285ee6ccdbdfc8b2d04de6b2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/968756d32748f1ee69e14c0944d6c3e0.png)
(1)若数表
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80dfa02ca1db6fd7d77f6ca9df5ffe75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ceb0bba344ccceaa98f18c9d9187bca.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c345907ebe27888332b1b44c666cc47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c1848d85c47d54c9b50093428004c47.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
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2022-01-12更新
|
673次组卷
|
4卷引用:北京市海淀区2022届高三上学期期末练习数学试题