名校
1 . 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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2 . 设数列
满足
,且对任意整数
是最小的不同于
的正整数,使得
与
互质,但不与
互质.证明:每个正整数都在
中出现.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f79ae17a7a504d6b0998364c13a9e0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27cab8c94c52ac4170c6617790361246.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a7170836b85b2aad29b01f1af0e86d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7230de53663c75658c58bbf206a0085.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bed25da42194b5a81d123933d5704f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
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3 . 设实数
满足
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b5a34f44a823523a0a5ad8918e53754.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3af9e115914b3482a0983270a55c4d7.png)
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4 . 我们称正有理数n为“友好数”,当且仅当
化为最简分数
时,a,b为奇数.则在集合
中优好数的个数为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3836223819fb20bad051b2f3fa5c9ef0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2122e3f1e76a635e58e4d54aa594c552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f5d6982b8a897a3e8174360d4abe973.png)
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