1 . 设数列
满足
,且对任意正整数
均有
.求
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13a6e1d671215fc96e4bee3541d1096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/280860dd039e1305a5ccc455f63e8223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6324f5c2c73bdcac4e9154429ac79baa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
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2 . 给定正整数m、k,有n个选手
参加一次测试,该测试由m个项目构成,每个项目完成后都会取得一个评分,没有两个人在一个项目取得相同的评分.求n的最小值,使得总存在k个选手
,在第j个项目中的k个得分要么单调递增,要么单调递减,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/252bab154aa5bdc9b4bce4c0d43aaf73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d559314445dc191250eed301d0f0b0ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fddec21d907fa4d0115be92d85c7a3dd.png)
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3 . 给定正整数
.求最大的实数
.使得
对任意正实数
恒成立,其中
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/526e037512bd10e65ebd99b73aa26341.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/897b606fdc64a88a0938d3d60c3ea3e9.png)
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4 . 某个会议有若干人(至少3人)参加,现要将这些人分组.分组前,每个人都选择两个人.若被选择的两个人同组.则选择他们的人不能在这组中.求最小的正整数
,使无论有多少人参加,且无论每人如何选择,都可以将他们按要求分成
组.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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5 . 设数列
满足
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df2577a0871cb3b018eb5e2840da1d7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48d5780c707bbf8fc91265df5f2093cb.png)
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6 . 已知
.证明:当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d1a4f6aed102bf6a4ac8b0f7dc63228.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/152b9780c93f0157f48c59a946354d37.png)
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7 . 求所有的函数
,满足
,且对于所有整数
,有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b32cabeba8b8adf0e787829153a8cf6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba8a114af9a47b5323510da90975657a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2e647c14561826ba9e396acc5a3792c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98818bef6625dfef58baf9c0730cb6f4.png)
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8 . 数列
满足:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70da88438143d70e067f8eff38b86d5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfd605736613503176c32d2daffb9100.png)
,求
的通项公式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70da88438143d70e067f8eff38b86d5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3f7fda69e2b32b9ced2239f915fa59b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfd605736613503176c32d2daffb9100.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f98839bbc91952387b4fd97ad5153e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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9 . 已知n个非负实数
和为1.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fe1c31a81f198c443e71b83ca662939.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/493f509c3370b2b26d46e2e46d038134.png)
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10 . 设
和
为两组复数,满足:
.求证:存在数组
(其中
),使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40b2f1f2faeb04505752c4c08b5d4a24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5b023415e27a14faac81fd5237fa1a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38af3182c396bef07007d4b04c92d32c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f094ba98f6c02216c714bd9ff85bd59f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/057d57358dd4fd992df26157efa5bd9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83c72aec1d89b300f5a069080cdd3e1a.png)
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