1 . 设
为n个正整数,并且满足
,令
,并记
.求证:对于任意
,必存在正整数u、v,使得
,等于A或
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18f31fb58fdf7b1ca2b5678208adb645.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d705531a8bf4c29db4eb3babc7dd456e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/198b98fe39fc6cad96db9217d6bd5ae5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/570ae28768c5542147c0830214f1b4e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e50c6c7f5ebd4dadd5dd333953fd36c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53467a21f223db674eb9497021ea9c5c.png)
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2 . 求所有无穷正整数列
满足下列条件:
(1)
;
(2)不存在正整数(可以相同i、j、k)使
.
(3)有无穷多个正整数k,使
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ddd6fc72d06ab0baec942bb06269263.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd5889863c33a3a7a79c08a86b58ed51.png)
(2)不存在正整数(可以相同i、j、k)使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d169a02afabbe304cf64b355bf71742a.png)
(3)有无穷多个正整数k,使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f3e64a171a2e8f0143aefbbf8ccb0cb.png)
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3 .
的外接圆与内切圆分别为
、
,
为
旁切圆.
1.证明:存在唯一圆
,
与
内切、与
外切,并且与
内切于点A.
2.设圆
与
、
的切点分别为P、Q.如果
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0047f659c182291c84c224df6b5e993f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/843d593e8cb8219aad703d77d78ef2f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18a8ab9c2421408d202361aca2c944fb.png)
1.证明:存在唯一圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b55b59c92a868cc6f448e5d92d257401.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b55b59c92a868cc6f448e5d92d257401.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0047f659c182291c84c224df6b5e993f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/843d593e8cb8219aad703d77d78ef2f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
2.设圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b55b59c92a868cc6f448e5d92d257401.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/843d593e8cb8219aad703d77d78ef2f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0047f659c182291c84c224df6b5e993f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66b746a5add435fea2d4d75c7479f01e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
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4 . 在一张无限大的方格表上的每个方格中填有一个实数.已知任意一个由格线构成的正方形中的数之和的绝对值不超过1.证明:任意一个由格线构成的矩形中的数之和的绝对值不超过4.
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5 . 设
、
是无穷复数数列,满足对任意正整数n,关于x的方程
的两个复根恰为
、
(当两根相等时
).若数列
恒为常数,证明:
(1)
;
(2)数列
恒为常数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80289c798034033f2f7cfcd7590f2344.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3282e5fde4ae53fcb1bb072a685304c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3002f56900c2924bfd79fc3865b0a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52cabfa2464501decf05aed007cbaf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/841f4ea50fa0c2b4c6e47dc04597abba.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/561d594ed04e6652c75dac56259f4292.png)
(2)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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6 . 已知
是两个整数集合,且对于任意整数
,存在唯一的
使得
.记
.证明:对任意的
,存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20acd504c39502c6f09eb356b526de92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a6ec3553aa6ac48a909ad93263c174b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab09d975196e05808f811695b4f4af6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b8525cb251f2e9d98691873e9cd4de6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c41c277675b413bbff28387082c9785.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c2ba1ef1d54121f25edbe4f219b8941.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b2fcaf03b6fbfa8d111e8bb94687e63.png)
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7 . 设
为一个质数,且
也是一个质数,证明:
的小数表示形式中包含0至9的所有数码.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ca4c3cdbb63bf2e3c1c3d67d46d3acc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cbadc8f735cb509c99741c41f32d097.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1eccdf91bcfb339dee1e546ebcf17bc.png)
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8 . 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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9 . 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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