1 . 已知
,求最大的实数
,使得对任意大于2022的正整数
及实数
,存在集合
的一个子集
满足
对所有
恒成立且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3c5cb4d381a7d02ed31e3d92ce4306e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce1132157a33c82610c2d5035493d024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6c6971463b9839ecc1c2352ae0afb7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0478ec4c7dcce9fa624789a6223eb53d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44c4dce884147e801b50675b9c0714e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e64c35495f9f68ad197ba30902915c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d81314852212a08f2e6d36338d0368a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a0a9f3aaedf6f256f10159eaf262a5b.png)
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2 . 设
为整数,数列
定义为
,
,且对任意
都有
.试求所有的
,使得这个数列的每一项都是完全平方数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c7bf0d4b73924e85ee5be8d6b170ee7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e18aa1cccf2f341a7886ca7a22d63513.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01fc0bf43fa7dcffcf1daebc1b0830c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97bccd30b914efbdf1bcedfb60d63b1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5575ee73675d9fb24280bb3d8d33361.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db0b397225aafbf5906a2b928c8f2947.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab4717e4827480f0f6f4ded85e52eab.png)
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3 . 如图,在钝角
中,
为钝角.设
的外角平分线与
过B和过C的高线分别交于点E,F,点M在线段EC上使得
,点N在线段BF上,使得
.证明:E,F,M,N四点共圆.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/854f480c60b88b546cb15d3b5622e212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d21fa3492cd1ac2b3f9278d3fe4cd4fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d21fa3492cd1ac2b3f9278d3fe4cd4fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/854f480c60b88b546cb15d3b5622e212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cd09bf3d43769b015a0a8d5b34cdefe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37595daefab7761a896f29f785d253d1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/9/bc6bb557-dd54-4bd6-8e9a-982625129d7d.png?resizew=217)
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4 . 给定正整数n,
,记
从
的一一映射f称为是可
划分的:若X可划分为k个非空子集
,
,…,
,且
(
,2,…,k)(即
,且
,
,…,
两两的交集为空集,
).已知f是一个X的
划分的一一映射,
,
,…,
是1,2,…,n的一个排列,则
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7863f720c810be5158d84a0e9b5e5be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b056db510d3e75cc12adc855015ad448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31556cfdfe2ebbf0dacfbb9e28f968ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307e85746fa628650484d89e3b9809ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9721059d158853671eaf19e39769b13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd8db4b168ddbcba90ac9b31d36a0432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7f2c72ab559a0615db4c51327b78d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd2781fa12b524b3b01f02d7b9d8f68a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a88501f72db91e5cf1a42114b4300ad6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb67fb9b5a8e475087ecc2a0aaca869a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9721059d158853671eaf19e39769b13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd8db4b168ddbcba90ac9b31d36a0432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d7f2c72ab559a0615db4c51327b78d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7187f7b32e990b39cf9df84428660a42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307e85746fa628650484d89e3b9809ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d04a8b7a7595251251b8e0b7e665e8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5453ec6a9e8b96357c888ea863ddcf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a10902a745902e85100175a0d7788fe.png)
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5 . 设锐角
,O,H分别为其外心和垂心,直线
分别交
,
于L,K.已知
与
的面积相等,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8db9f09c5f5d6cdbf2c48d1ab600488.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ff214d99784c6b23b7784bdaf3ed37a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4cd68cc82e90a5e2049a7ea3171b84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54435ea54b0214eb58ce6f4a4c53317d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc1eb72c3df18cdaa2ff9510b4742dfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8db9f09c5f5d6cdbf2c48d1ab600488.png)
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6 . 我们称正有理数n为“友好数”,当且仅当
化为最简分数
时,a,b为奇数.则在集合
中优好数的个数为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3836223819fb20bad051b2f3fa5c9ef0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2122e3f1e76a635e58e4d54aa594c552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f5d6982b8a897a3e8174360d4abe973.png)
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名校
解题方法
7 . 设函数
,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bcb1b4adc744507c0f6a08bbb170386.png)
A.若![]() ![]() ![]() | B.若![]() ![]() |
C.若![]() ![]() | D.若![]() ![]() |
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2022-09-21更新
|
718次组卷
|
3卷引用:2022年浙江省温州市摇篮杯高一数学竞赛试题
名校
解题方法
8 . 设函数
.
(1)证明:存在唯一的函数
,使得
;
(2)求所有的非负实数
使得
;
(3)
,
(i)证明:关于
的方程
与
都有唯一实根;
(ii)记
分别为方程
,
的实根,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041fc0e0ff542027634491e686a510b6.png)
(1)证明:存在唯一的函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2ee09653f053eccbba4b85fbf97c3e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aca8208f53be6a7033b4fd2a11c7964.png)
(2)求所有的非负实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f8bfb563f79688d136e0cb958b5153c.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb1523685c2d0a9ef21660908378ac90.png)
(i)证明:关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/713c59f274e9146b6d85375435315521.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3a4cb45ca6e42ced4a5c4026e2290f8.png)
(ii)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd8ca3aa2d1ba52e82613d0d65d800e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/713c59f274e9146b6d85375435315521.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3a4cb45ca6e42ced4a5c4026e2290f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3c414bc882fb3c5e364582886c3325c.png)
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名校
9 . 有三个盒子,每个盒子里有若干大小形状都相同的卡片.第一个盒子中有三张分别标号为
的卡片;第二个盒子中有五张分别标号为
的卡片;第三个盒子中有七张分别标号为
的卡片.现从每个盒子中随机抽取一张卡片,设从第
个盒子中取出的卡片的号码为
,则
为奇数的概率是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/680e9ef551b325387ab31dca1f893705.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5969a26e4b3c149f694f857c655b08be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a58f7fc045c0a8aec3aecaa6b287de84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2243513b880087c89c0aa4f662a4af26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35bacde908aec2c313978fc4309d82bc.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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名校
10 .
地发生地震时,相距
km的
两地都能感受到,已知
地位于A地的正东方向上,
地位于B地的东偏南
方向上,且
地距离
两地分别为
km和
km,则
的值是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ac09dc1ca2cdd7aef28c218763d3e4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b2a3c02083c62d1e61edbc2eef8dfcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58bbb8c432c2b551b6515fcea5e1b742.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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