1 . 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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2 . 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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3 . 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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4 . 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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5 . 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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6 . 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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7 . 设递增数列
满足
(
).
(1)求数列
的通项公式;
(2)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b177db870f81b9f26fd87fccda2177de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e167b43045b3297248e334c41c621b8f.png)
(1)求数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea639d7e3dae98ffb67642d2cb9b89b6.png)
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8 . 证明:对每个正整数
,存在正整数
,使得能将前
个正整数所排成的数列1,2,…,
顺次分成这样的
段,其中每一段的各数之和均是平方数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daab2e7f7f9ea6d1dcaf060c783c756b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daab2e7f7f9ea6d1dcaf060c783c756b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daab2e7f7f9ea6d1dcaf060c783c756b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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9 . 试求所有函数
:
,使得对一切
有
,且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8b6d9bc5c9e7c8f22b0993234415ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f02d5c8eec434a3f90348d770a2e2b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdb5d98a56a356ce186d6c8f9cd27043.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e61c9a7ed0961f8977a21dab37aab396.png)
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10 . 给定
个(
)互不相等的实数
,所有的
个和
(
,
,
)中互不相同的数恰好有
个的充分必要条件是
,
,…,
成等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5d0a73f50b3e4583f1c1b6d6bf0d18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f846d93390223b563770b676ee30b06c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/772dee834857b4fc72ed84cca530590b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/059a6c5a965c335b8da05e697da2c7c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9612cb52aacd0b66fc1ba1297267c9c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f97d4ad3c68d0a1b7ddad9b0a99ba48e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b71af6590f0f369c164a054a8b63bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/629455b181f187fe9b0edd9d66431ef0.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/96abfe2da27a63e6affb19a0c80236d9.png)
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