名校
1 . 如图,
是曲线
上的
个点,点
在
轴的正半轴上,且△
是正三角形
是坐标原点).
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/2/86311efd-f7c4-4a46-8bfb-5e6933f9905a.png?resizew=249)
(1)求
、
、
的值及数列
的递推公式;
(2)猜想点
的横坐标
关于
的表达式,并用数学归纳法证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/857d97c35c872937e4172dbc2ede1af2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/956620959792f8c3a28ed0fc4bf053d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3f94253ca8e33eca70b1a9ee6c2a0e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b27853b475a8bc2bfe297bbdbdaaa86d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a617f7ebacc3d380868cf6e002c0be3e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/2/86311efd-f7c4-4a46-8bfb-5e6933f9905a.png?resizew=249)
(1)求
![](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/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(2)猜想点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b25286f6029da66ce5270aacd05184f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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名校
2 . 设
,记
,
,
,3,…,集合
对所有正整数
,
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7600f6106deb991473530c04dd38d1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9ab7bb40f58f28c9799b20f91d15d33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dc2918652a71ff4f1f8455c7f36af2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d9f0d98d91c7cc5185459eb94298497.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68364db412ab3a5eb503ecdd9542915b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9559f82370d8b150774350d0552b0dde.png)
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名校
解题方法
3 . 在平面直角坐标系中,两点
、
的“曼哈顿距离”定义为
,记为
,如点
、
的“曼哈顿距离”为9,记为
.
(1)点
,
是满足
的动点
的集合,求点集
所占区域的面积;
(2)动点
在直线
上,动点
在函数
图像上,求
的最小值;
(3)动点
在函数
的图像上,点
,
的最大值记为
,请选择下列二问中的一问,做出解答:
①求证:不存在实数
、
,使
;
②求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84bd36d19352628cb54c214436ee3322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e27bd43bc4af1e3b28d0de0cc561b879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c59185f3d9547cd9065d10dcbb4127d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73e66b64267481405cc49dad9d8916c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9404ad60dd25cb0df6c37032d50b72ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dafabc98a78486af4fbf346e7cfad11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31bd35a290bbcf999ec26930c747084b.png)
(1)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9404ad60dd25cb0df6c37032d50b72ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a7bce4bf9358998e26ff0715c909a19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b454cdb97c408300b50d945f002c2cb.png)
(2)动点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ea05a2396e436b4df62d6328dbeaddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73e66b64267481405cc49dad9d8916c7.png)
(3)动点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c064084f6326c8b994c2bcb80ad258da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae4a30a3210d0a8130d5a1183289c23f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73e66b64267481405cc49dad9d8916c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6d8a4cf957865fad1cb648fcd2cbaa0.png)
①求证:不存在实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d41cfe4280d2384c9dd4287c8f07954.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6d8a4cf957865fad1cb648fcd2cbaa0.png)
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4 . 已知一个数列的通项为
,再构成一个数列
,…,这个数列是否为常数列?证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76c78e05a9bc289e0024f9d67a504264.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad2672e6f1b9b66be09ba18bf78e07c7.png)
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16-17高二下·上海浦东新·期中
名校
5 . 已知z是实系数方程
的虚根,记它在直角坐标平面上的对应点为
,
(1)若
在直线
上,求证:
在圆
:
上;
(2)给定圆
:
(m、
,
),则存在唯一的线段s满足:①若
在圆C上,则
在线段s上;②若
是线段s上一点(非端点),则
在圆C上、写出线段s的表达式,并说明理由;
(3)由(2)知线段s与圆C之间确定了一种对应关系,通过这种对应关系的研究,填写表(表中
是(1)中圆
的对应线段).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7383c643d74bb787f8f101830c12fe4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52225f75cebeb64408d27837cec03b98.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f00fdb0b1dfb21a2e192990b79be37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f281814a940820e52ec332185871e22f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52225f75cebeb64408d27837cec03b98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e383fcc122f267043fbafe0972bfb900.png)
(2)给定圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5303146fa7877bef2e0bf9acaecaa962.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b23a85c638da170e789c173739da6bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45ba6b6aa6c3f9faba6b03bc193a6e61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52225f75cebeb64408d27837cec03b98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f00fdb0b1dfb21a2e192990b79be37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f00fdb0b1dfb21a2e192990b79be37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52225f75cebeb64408d27837cec03b98.png)
(3)由(2)知线段s与圆C之间确定了一种对应关系,通过这种对应关系的研究,填写表(表中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae9ab1621cd729a18b2173e95d557376.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
线段s与线段![]() | m、r的取值或表达式 |
s所在直线平行于![]() | |
s所在直线平分线段![]() |
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6 . 在数列
中,
,其前
项和
满足关系式
(
).
(1)求证:数列
是等比数列;
(2)设数列
的公比为
,作数列
,使
(
),求
.
(3)求
的值.
![](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/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/480c365820a675435b5fc66284868565.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7890769dab7cc862d638144bc584c262.png)
(1)求证:数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2709ca478fb15ea08e8aa55328eae8e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4a156f7b93cc187b48663a6cc758c17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3948be904275d925cbaae6f2dbc8ab2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e69567f3a83855a706aff0b02f4e9e5.png)
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2018-12-15更新
|
128次组卷
|
6卷引用:上海市上海外国语大学附中2016-2017学年高二上学期期中数学试题
上海市上海外国语大学附中2016-2017学年高二上学期期中数学试题(已下线)2012届江西省师大附中高三10月月考理科数学(已下线)2013届湖南省浏阳一中高三暑假自主学习检测理科数学试卷2016-2017学年河北省武邑中学高一下学期期中考试数学试题卷【全国百强校】四川省南充高级中学2017-2018学年高一下学期期中考试数学试题(已下线)2004年全国高中数学联赛山东赛区预赛试题
7 . 对正整数
,记
为数
的十进制表示的数码和(如
是
的数码和,即
).
(1)证明:对任意的正整数
,
,且
;
(2)试找出一个正整数
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c0b21f55cd270d3053b332f07d7ebe7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9347bb4ffedcbea2f4c16d047a138d75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96c5fa9174d2bebafef71fbfd6f13217.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3359f013aac2768a0c54cae1a95a158c.png)
(1)证明:对任意的正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3a4e64c215e1a1b6e9084478559fcb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1388d5cdcb41e02e3fb134fa80cfe0bd.png)
(2)试找出一个正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/baa582cb215456700aec63492cb1cf05.png)
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8 . 将,1,2,
,2007这2007个数任意排列可得2007!个不同数列.问其中是否存在4个数列:
,
,
,
;
,
,
,
;
,
,
,
;
,
,
,
,使得
?证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.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/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/482c46ff6edb17f9bc1bf98009ea6a9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b715e7842b95f654f16056a7c7f2abe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13423c094861baf4b759b7f3d8c3c226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec3004865ff45025358d80c2e051eb0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7936359df4c926b72b48c6fdae55f12d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b76f79be89b8c6227b68eded6b675546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d04e39f02602b9a6c0f858a619a3aee6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5edf900c810371fb21297c15f86d8743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b31ac1def558351e2e3ed1235c570530.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/873751bc22588e552694be9c1e24f4e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7fb5985aff76fbc234fb8e46ac29cab.png)
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9 . 对集合
(
),若对于任何的
不能整除
的任何一个非空子集的元素和,则称
是“好集”(
表示集合
除去元素
后的集合).
(1)若
是好集,求
的最小值
;
(2)证明:集合
不是好集,其中,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2344e0bdfa0fc62283a93d4fe67ea89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b58b34e73f045f54f9b2ee7daad5fc98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5805e7b270434d58308046727e963ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5805e7b270434d58308046727e963ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3bb14e6163867fe0acaffbb05ad2f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7e9f86738335a22298559db41037a4.png)
(2)证明:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ff7ed34e0562a4542fe4b860753995f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4806dc988e47163bc2178983196b9bff.png)
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10 . (1)求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/591048ee5a3480eedec656dc37282407.png)
;
(2)求值:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/591048ee5a3480eedec656dc37282407.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00a1114d967e7135cc2a10a844e032eb.png)
(2)求值:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3dd035738fddfe5e82a1e6202864b0e3.png)
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