1 . 一个人上台阶可以一次上1级台阶,也可以一次上3级台阶,或者一次上4级台阶.若这个人上
级台阶总共有
种走法,证明
为平方数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2bfe05ef6780e5f032ca3a7c96811a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b07f7a46323e7630dd8cd5cffcb11a5d.png)
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2 . 求集合
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b96ab3cc4f183937d0ecf24feeff9508.png)
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3 . 设
,其中,b为正奇数.定义数列
满足
,
.若正整数
,使得
为素数.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a29fca134ddbac482c9821e78e6ab842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1f83d1a6f169d2e621b366431729888.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91fc7b499561a3427ed1ef0354689f7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/772e2d1d7398351c44c55366756f16ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53500b98cf6e4620e77a8271c73549d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76a3ac74cdb64183be40d576145a9cd5.png)
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4 . 圆周上分布着2002 个点,现将它们任意地染成白色或黑色,如果从某一点开始,依任一方向绕圆周运动到任一点,所经过的(包括该点本身)白点总数恒大于黑点总数,则称该点为好点.为确保圆周上至少有一个好点,试求所染黑点数目的最大值.
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5 . 设
,其中
和
都是实数,且
.证明:若
,则对一切正整数
,均有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0f167a87f1dfceaf858d4dde6ee94ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3fb0c9c7e30bbc0ec8c3521577ee4fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38335830b93ac4d99c28a8e209eecb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/921c99563c54509e2ebf587dfdb26e1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/042235c88be02463c0b7bef603ae7e18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58edc69cbf7d0d5d1ce8a1dc6c125af5.png)
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6 . 求满足
的最小正整数
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3aae9e5309777a271677fd7b0bd4901e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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7 . 将边长为正整数m、n的矩形划分成若干边长均为正整数的正方形,每个正方形的边均平行于矩形的相应边,试求这些正方形边长之和的最小值.
![](https://img.xkw.com/dksih/QBM/2018/12/14/2096454455820288/2097419259199488/STEM/461fb579fa594c7385bcd8e7d61ce2e8.png?resizew=204)
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8 . 设
是n个不同质数,用这些质数作为项(允许重复),任意组成一个数列,使这个数列不存在某些相邻项的积是完全平方.证明:这种数列的项数有最大值(记为
),并求
的表达式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51d9cf284a6d151f05fc8fe80d36c4b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2837f8a4d91ca4202f69e39abf3f603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2837f8a4d91ca4202f69e39abf3f603.png)
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9 . 已知数列
、![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
满足
(1)证明:对一切的
,有
.
(2)求数列
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8403d042c0394bcd82b5d0c2ab49e8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6febbec171eb91cef5fe044734b1f3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feacd516ccd94571dd1600a44b77ff9a.png)
(1)证明:对一切的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8403d042c0394bcd82b5d0c2ab49e8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc4291b6f4399a55d0248f630fc7a77e.png)
(2)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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10 . 设
均为大于1的整数.证明:存在
个不被
整除的整数,若将它们任意分成两组,则总有一组有若干个数的和被
整除.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c847ac9cba2b914f01f85e07e0ccc5f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5631bc01b998a4b3fabd9e131699dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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