1 . 已知数列
满足
,
.
(1)若
是递增数列,求实数
的取值范围;
(2)若
,且对任意大于
的正整数
,恒有
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31d0b399e3826e2721d683a357fe5dd4.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e144b442dc601367909266594699b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaacfaef44a654c0a1c283ef03fc0550.png)
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2 . 给定一个正99边形,将1,2,
,99放入99边形的99个顶点处,若两种放置方法在旋转之后可以重合,则称这两种方法是同一个.称交换某两个相邻顶点上的数为一次操作,求最小的
使得至多
次操作可以将一种放置方法变为任意另外一种放置方法.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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2023-12-15更新
|
150次组卷
|
2卷引用:2023年第39届全国中学生冬令营(CMO)数学试题
名校
3 . 我们称
为“花式集合”,如果它满足如下三个条件:
(a)
;
(b)
的每个元素都是包含于
中的闭区间(元素可重复);
(c)对于任意实数
中包含
的元素个数不超过1011.
对于“花式集合”
和区间
,用
表示使得
的对
的数量.求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(a)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4105c41b98ebc0e7144eff1ba792c76d.png)
(b)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e11f4ca0e7ace69f92130d0525bcdb3.png)
(c)对于任意实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb7bfa1a59b3451a4379f7cbc074ef60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
对于“花式集合”
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ebaa32f4f1f4f807ca9aeb7fb29951.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4fc1c4dd7cb41e3342eb79054ef1a67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0e6ceed58d644f9027ea60bd0f1f557.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aeb55594a33f7ac1d8d93dc5b13cb82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd1ac9030ad24307666928b511a0f45e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0e6ceed58d644f9027ea60bd0f1f557.png)
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4 . 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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5 . 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)
您最近一年使用:0次
6 . 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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7 . 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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8 . 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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9 . 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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10 . 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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