1 . 设
为实数,
.证明:
(1)把
写成无穷乘积有唯一的表达式
其中,
为正整数,满足
;
(2)
是有理数,当且仅当它的无穷乘积具有下列性质:存在
,对所有的
,满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/948eb26dc27646285335664baad48ec0.png)
(1)把
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/947ce9ff5b37676e23feab8cbc331c8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/697e84689167caa36ac89184035aab56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6d2fa14ee14c2b79da6dbc875b61b8e.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef594556457f4989f78e8ad1038ff4ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f1d645832a371aaaf1a2720517f073f.png)
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2 . 将一枚棋子放在一个
的棋盘上,记
为从左、上数第
行第
列的小方格,求所有的四元数组
,使得从
出发,经过每个小方格恰一次到达
(每步为将棋子从一个小方格移到与之有共同边的另一个小方格).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2f92a1ac461e9b71247cd59e12cb2d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb4d5ca4e251ff0e503a26f9a7375326.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/c969c72b1466caaa73129f2ee968af3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3044d760ef9d263537c142478ac35c19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85f88bcbc7c997bd3f56170ce4c997c8.png)
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3 . 已知数列
满足
,
.给定奇质数
和正整数
满足
,证明:
的充分必要条件为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/909736dad505d81be43aef91e6309bf4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5108053cdc0b7d5d1813bbf96689ae5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c226ece1d984f7733996dbe669b6875.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6f95f1bdd88065ccc93d4f6410b9f3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/580fd18d9d137cf66ed9c24b91f6463c.png)
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4 . 设
为互不相等的正整数,它们的最小公倍数为
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f2b043b989216035c6fd985f1dd6a3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686ece75006ad358f23314dc8a246e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/525e5ce4e91a4738e90809b1b949d6df.png)
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5 . 已知正整数数列
满足对任意的正整数
均有
,证明:存在无穷多个正整数对
(
),使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22277c45c7203fd3d692d3fcc38fa194.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/698c4d4e50062b4a7dd70fe1b4ab4fd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d14e88b76e8fbfed5a6b57a9e708fc21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec6f8fe6bb79eca3ecb57816340d26bd.png)
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6 . 已知
,
.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac69e6db1df13ed64756b4f391ae9fac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e355789fe1363144c42e2c270e59e34.png)
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7 . 对给定的正整数
,定义
表示
的各个数位上的数字之和的平方,当
且
时,
表示
的各个数位上的数字之和的
次方,其中,当
为奇数时,
;当
为偶数时,
,试求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1580d1acb1ef184208d8a49694398cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e468312d09c6563c9094b710a35a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac69e6db1df13ed64756b4f391ae9fac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1622aec09c560184b0d253a7f53bf1bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66e90a6d7b5f61955a1d963892fc7bf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a135cb036833400f3fa1edc92d5ce410.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba3301f1a7b308916e5de17c38ebf1c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/029698978fcc20c10240e1dbcc1ec37b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef0030cefcfe3ce06aa02e80c4e2b351.png)
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8 . 由单位正方形组成的无限格阵的每个单位正方形内都写有一个整数.若每个方格内的整数等于其上方和左方与其相邻的两个方格内的整数之和,且存在一行
,其中,所有方格内的数都是正整数.记
下面一行为
,
下面一行为
,⋯⋯证明:对于每个正整数
,
上不能有
个方格内的整数都是0.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fb44db1dc864ff4901be1e10da79747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fb44db1dc864ff4901be1e10da79747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b104090ea2ac34be58a76a4e0e95cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b104090ea2ac34be58a76a4e0e95cb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df1d9b712b639c8b6809c9f3ae03706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a611aefd3eeb2e20980daae5af1970b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
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9 . 将
的方格表中的某些小方格染黑,使得不存在由三个黑色小方格构成的
形(共四种情形).求最多有多少个小方格被染色?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27c2e5f73ccb29008779df6f6b36f3e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c88d9142df6ba8e43c1a93bd04a1362.png)
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10 . 已知
,求证:存在无穷多个正整数
,使
除以
的余数互不相同.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2767a1fd94583584fa01f90dcbc9ec37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf5b296e1144d0e0398332a72bee14ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
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