1 . 证明:存在无穷多个棱长为正整数的长方体,其体积恰等于对角线长的平方,且该长方体的每一个表面总可以割并成两个整边正方形.
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2 . 正整数数列
满足:
,
.试求通项公式
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/527bd6adefbb15deb6ad829d7584d072.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/191d7c7739907a08a2bbf5c23b6b3386.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
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3 . 设
是定义在自然数集合
上并在
上取值的函数,满足:对任何两个不相等的自然数
,有
.
(1)求
;
(2)假设
是100个两两不相等的自然数,求
;
(3)是否存在符合题设条件的函数
,使
,证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/145b9b7961cf1f359862430bbca15465.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f74081cfd8d2457ff2d575d439f13b5.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e38fffbc7ab9882480f4faa72390e23.png)
(2)假设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc7c76c1ceb8cfc22b108c55565d2b61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/977b29dc3c945f0177201359056566d0.png)
(3)是否存在符合题设条件的函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/524ee808469860f15659f9353b3d647e.png)
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4 . 给定正整数
,对于正整数
,集合
.集族
满足如下条件:
(1)
的每个集合都是
的
元子集;
(2)
中的任意两个集合至多有一个公共元素;
(3)
的任意一个元素恰出现在
中的两个集合中.
试求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92ee628911994243c46467361fd30098.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/950b2b2d555ce0f0adbdb85a63b30778.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a279eed3fea6a07ffc2e7784b9dfcfd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a279eed3fea6a07ffc2e7784b9dfcfd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
试求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
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5 . 求满足下列条件的最小正整数t,对于任何凸n边形
,只要
,就一定存在三点
,使
的面积不大于凸n边形
面积的
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd76ce38a05b798d45147dd2fe05294e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0166ef16246534081188fce28684b49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75245070263484fc737f7bacc16a807e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846c13953ba2a1c0a2113ad4c067ca78.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd76ce38a05b798d45147dd2fe05294e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee6a2eba56d4f2d1670b0256b8d86b92.png)
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6 . 设
是给定的正整数,
.记
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad89b2acebe155e869364acd8d464d77.png)
.证明:存在正整数
,使得
为一个整数,其中,
表示不小于实数
的最小整数(如
,
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ee72bf6c307a2d83a30c65893ebe90b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f721367c14f3dd608be3cbfe750eeb8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad89b2acebe155e869364acd8d464d77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbcf152cc8102e3472504d02355acde6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2890e1cf994207875837def2ca3e1ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/898a6c355155c992f5e44bb86684225a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a3f52f0ecb7ce48378c464edf66c40d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31dbd410162b646ba0d5951fdb9cb0ea.png)
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7 .
的反函数是
,
,设
,且对于
,有
.求
的解析表达式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec11283c9e3c6ea5379370135fcd6818.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bc6c4e5bb6e207512630b32f27db1c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f66a88cbaed58dcf9671ba9240359b9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0316344ece953782b3e3ed27185ba53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7afeb1c6e47de3f6eb82fd4a0234b557.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e338b64edbf3cab8609f51af26c1dd9.png)
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8 . 已知数列
满足
.
(1)若
,证明:
(i)当
时,有
;
(ii)当
时,有
.
(2)若
,证明:当
时,有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a229ee29685c7ec745dcb07717478885.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86fc336b4a83bf6d66c4afcc431597f8.png)
(i)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/030f5b94af92e8995f3ba989f1a03d03.png)
(ii)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e167b43045b3297248e334c41c621b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5930c0f0a7a529f37be1952d976401bd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5d0a73f50b3e4583f1c1b6d6bf0d18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e78a219a67a184a4307954aeeb33b4b.png)
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9 . 数列
满足:
,
,(
表示不大于x的最大整数,
).试求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/489340c9a2d70c00bae13b7018cad448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b62f3fd059c5c33802cb98123374f9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f73940d577a0bf4cfd373f0312b75c54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f8ced5ff7dfbe12a7fc3378a5e3c3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17f78969d9ac865772e51b9b5514868e.png)
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10 . 将长为
、宽为
的矩形划分为
个小正方形.一粒子不重复不遗漏连续地通过每个小正方形的一条对角线.这件事能否办到?若办不到,请说明理由;若能办到,请给出一种行走路线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e70354d6ca5ad9f6b4592fac0b5e559.png)
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