名校
1 . 我们称
为“花式集合”,如果它满足如下三个条件:
(a)
;
(b)
的每个元素都是包含于
中的闭区间(元素可重复);
(c)对于任意实数
中包含
的元素个数不超过1011.
对于“花式集合”
和区间
,用
表示使得
的对
的数量.求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(a)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4105c41b98ebc0e7144eff1ba792c76d.png)
(b)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
(c)对于任意实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb7bfa1a59b3451a4379f7cbc074ef60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
对于“花式集合”
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ebaa32f4f1f4f807ca9aeb7fb29951.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4fc1c4dd7cb41e3342eb79054ef1a67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0e6ceed58d644f9027ea60bd0f1f557.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aeb55594a33f7ac1d8d93dc5b13cb82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd1ac9030ad24307666928b511a0f45e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0e6ceed58d644f9027ea60bd0f1f557.png)
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2 . 设数列
满足
,且对任意整数
是最小的不同于
的正整数,使得
与
互质,但不与
互质.证明:每个正整数都在
中出现.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f79ae17a7a504d6b0998364c13a9e0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27cab8c94c52ac4170c6617790361246.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a7170836b85b2aad29b01f1af0e86d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7230de53663c75658c58bbf206a0085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bed25da42194b5a81d123933d5704f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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3 . 设实数
满足
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b5a34f44a823523a0a5ad8918e53754.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3af9e115914b3482a0983270a55c4d7.png)
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4 . 设
,
,…,
为曲线
上
个点,其横坐标为
,
,…,
,若正常数
使得存在
,
,
,则常数
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf9f50605db5d5f8f3a01ee8e474a112.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c904567c3b3734e1eca8d042ef7a7b2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0876215b2fd463d151523cd3c6b447.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11b6f3a094e746086715e1ab142aa727.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/75809a92693be16331ac76ece2afd8a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
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5 . 给定素数
.称1,2,…,
的排列
为“好排列”,如果对
,2,…,
均有
,并且
是
的倍数.求“好排列”的个数除以
的余数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20dd7cdef44777d41d74d699ffdd746d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85bbd4a74fa01b02d4604bf44e361089.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5095a28bb1b91bf6bed9e2cfbd76bb18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/175f3b6bd24295b2c60359318eaa4c03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e82c656aebb258b9b13459640a4ef99b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e839b03cd6ea4e9a2853344db2535a5.png)
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6 . 给定正整数
.记
,
,2,3,….证明:对任意素数
,存在无穷多个非负整数对
,满足
,
,…,
这100个数都能被
整除,并且都不能被
整除.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00823481ac7edf9e1d6f3ab00ab550f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c87b351f16728b0023fd63678f8103c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d7be8bc42c6f36ee9010656b918c46a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efa6e6840235cbfe76f9827fc755d4cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c596ae902e6408d14d78580c04267f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3d8af4e2b2c48c73e7897eb3da814c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a6c0cbe18032f93ac8799730c5fabb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5875afb0da6ef834c26e5898627713c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec3e532f6f9d9594b8055efb5c1d7db0.png)
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7 . 求最大的实数
,使得不等式
对任意正整数
以及任意实数
均成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2865bd92f42e2010f43ad17f5775fbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3979f4e3d834034bc48715afbb8f70c8.png)
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名校
8 . 如图,在
中,
分别为
上的点,且
,
,
.设
为四边形
内一点(
点不在边界上),若
,则实数
的取值范围为______
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/927456b0989846a2f1573844bbaa2105.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/746ee1515a178948b04f535705c6f738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01ef4036427f8a42a22bc5ed0ae26189.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19c2dcc2a3955011e0e503de6225fca5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e77686cf448ff6cea9bfc021581da83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48a61d8bedef61745ee4fd89a8c0c119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc3c25c92950e902bf3d79a41cb38e99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/17/3ad0d689-3d51-4bf1-9273-03aa043e42b5.png?resizew=218)
您最近一年使用:0次
2019-06-25更新
|
4335次组卷
|
11卷引用:2019年全国高中数学联赛浙江省预赛
2019年全国高中数学联赛浙江省预赛天津市耀华中学2021届高三下学期一模数学试题(已下线)【讲】 专题一 平面向量线性运算的最值问题(压轴大全)吉林省蛟河市一中2018-2019学年高一下学期第三次测试数学试题人教A版(2019) 必修第二册 突围者 第六章 第三节 课时1平面向量基本定理山东省新泰市第一中学2021-2022学年高一下学期期中考试数学试题苏教版(2019) 必修第二册 过关斩将 第9章 9.3.1 平面向量基本定理(已下线)6.2.1向量的加法运算(分层作业)-【上好课】2022-2023学年高一数学同步备课系列(人教A版2019必修第二册)(已下线)微专题02 平面向量的基本定理(1)-【微专题】2022-2023学年高一数学常考点微专题提分精练(人教A版2019必修第二册)上海市上海中学2022-2023学年高一下学期期末数学试题(已下线)重难点专题04 妙用等和线解决平面向量系数和、差、商问题-【帮课堂】(苏教版2019必修第二册)
9 . 设
为正整数.若数字
构成的排列
满足
(1)
;
(2)
;
(3)
,
则称此排列为“N型”的.记
为所有N型排列的个数.
(1)求
、
的值;
(2)证明:对任意正整数
,
均为奇数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3860f7f37da2dbf7b1df41b29503397.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c59221a068df563c785546c389be677.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2750b2fb577342d21ff3a2c58bf3f358.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bbf794832ea65fa70d1d2d12214575.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86fcdd09aae5c9a30d3b0bea09ef57d7.png)
则称此排列为“N型”的.记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8904e7018ec79c8b0efdcb3ba67cb7cb.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5edf900c810371fb21297c15f86d8743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b31ac1def558351e2e3ed1235c570530.png)
(2)证明:对任意正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8904e7018ec79c8b0efdcb3ba67cb7cb.png)
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10 . 设
满足
数列
是公差为
,首项
的等差数列; 数列
是公比为
首项
的等比数列,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4935544664f86edc57a0c3410fcf897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5217df154813a81ad37c406027e9f667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76c6099506bd60534ed57a71e3678b31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/affd21d3fc4f76dcc7fffa227541df28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f6cc218c568cc9d08e620696d1f61f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6e8e67f649bb2e18fc02d6118ff4e2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/decb5e8546e79397586cbbdf0fc2e085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1f5a2d53d857943074a092006e110d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50dedd2d9712979cae558023a3ae94b9.png)
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