真题
名校
1 . 给定有限个正数满足条件T:每个数都不大于50且总和
.现将这些数按下列要求进行分组,每组数之和不大于150且分组的步骤是:首先,从这些数中选择这样一些数构成第一组,使得150与这组数之和的差
与所有可能的其他选择相比是最小的,
称为第一组余差;然后,在去掉已选入第一组的数后,对余下的数按第一组的选择方式构成第二组,这时的余差为
;如此继续构成第三组(余差为
)、第四组(余差为
)、…,直至第N组(余差为
)把这些数全部分完为止.
(1)判断,
,
…
的大小关系,并指出除第N组外的每组至少含有几个数;
(2)当构成第
组后,指出余下的每个数与
的大小关系,并证
;
(3)对任何满足条件T的有限个正数,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af4ee184e4aa3dd89ebc05473e767517.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2858005b9ae89ae080d83dcc13cf8e81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2858005b9ae89ae080d83dcc13cf8e81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3e95410f3b4fcb0cba425b521d1f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f2cb48c0a69b8c420c0b64b2bfa1ef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4804e9b295d3b8de7f05e9c4e8e30a3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b768942c5e723cc71609c62c1919298f.png)
(1)判断,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2858005b9ae89ae080d83dcc13cf8e81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3e95410f3b4fcb0cba425b521d1f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b768942c5e723cc71609c62c1919298f.png)
(2)当构成第
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a72cd8c7b3d469bacee92ff4f9a98e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a135cb036833400f3fa1edc92d5ce410.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7199ce73cb1f7e661115e8cf022f7699.png)
(3)对任何满足条件T的有限个正数,证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9146abed0736e4cb89fbca640acadd7.png)
您最近一年使用:0次
2020-12-03更新
|
593次组卷
|
5卷引用:上海市虹口区复兴高级中学2020-2021学年高一上学期期中数学试题
上海市虹口区复兴高级中学2020-2021学年高一上学期期中数学试题2004 年普通高等学校招生考试数学(理)试题(北京卷)2004 年普通高等学校招生考试数学(文)试题(北京卷)(已下线)上海高一上学期期中【压轴42题专练】(2)(已下线)第六篇 数论 专题1 数论中的特殊数 微点1 数论中的特殊数
名校
2 . 已知函数
.
(1)设
,证明:
;
(2)设
,证明:
;
(3)设
都是正数,且
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f394beb581f75eb7efa71735b6f849.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/596004d85b69e05f9551dec3be53d7a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6c35b36213a897b6887b39ad0b971e3.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0adc67e0d7d3a7c501af506df484f23d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e29df1029ef1f3321e2301cd179618c.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ede3df8c9f8a40639cd9f8267414117.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a399a276e30510e557b42ee5db5510b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09b150975cb85995a57a7036481f476f.png)
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解题方法
3 . (1)解不等式
.
(2)设
表示
的解集;
表示不等式
对任意
恒成立的
的集合,求
;
(3)设关于
的不等式
的解集为
,试探究是否存在自然数
,使得不等式
与
的解都属于
,若不存在,说明理由.若存在,请求满足条件的
的所有的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/160f68000b902eef7c47c95f055c2d04.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dee96a17986a05fb4d8afac6ed25af10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa70964576558e912ac311efe800ebe6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4afabfd5b4885311f8d9a4bcf791b71.png)
(3)设关于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/789bcc64778a85418ac54d41d1798a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/700fa4dfbe1d291042d435778db55f5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecc21ccecbbb5acfff1ab104b6b8265f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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解题方法
4 . 定义:记
为
这
个实数中的最小值,记
为
这
个实数中的最大值,例如:
.
(1)求证:
;
(2)已知
,求
的最小值;
(3)若
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e824d15bf3a32ceafbf100e4bf3b6a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bee78f4561539faf2b6f382ce5f94f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b75c8ecbf2c10fd090d119238cbfbb1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bee78f4561539faf2b6f382ce5f94f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/883d52e4bbbec68682117456d078a2ca.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/543743dde06d90ad5da3632ef0792f59.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aede98f52c2cd9911b09d7faf08a0dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd44b30ab4f8d40eb4eb30e8f9f807fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
您最近一年使用:0次
名校
解题方法
5 . (1)若
,
,求证:
;
(2)设
,求函数
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/829c3146484a77b8597a60f806ce8ee3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d2aa5e48ef4cc10678706d9883e7a0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/031ede0c2bfeb8bfb8b347a2e7cd3bbc.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fecaefda8567646f10d76668293d845.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c7de8538ee943759e297736b8ac2845.png)
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解题方法
6 . 已知![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d85ab9b6287f810bfb1443ba4a98065.png)
(1)解不等式
;
(2)
恒成立,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d85ab9b6287f810bfb1443ba4a98065.png)
(1)解不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fbc901cbdb68130ddac3174583dd93c.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5931095eb29d9d6b55ed9fa32a4ef1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2020-03-02更新
|
149次组卷
|
2卷引用:广西南宁市第二中学2019-2020学年高一上学期期中数学试题
14-15高一上·上海徐汇·期中
名校
解题方法
7 . 若实数
满足
,则称
比
远离
,
(1)若
比
远离
,求
的取值范围;
(2)对于任意的两个不相等的正数
,
是否比
远离
?写出你的结论并加以证明;
(3)对于任意的
,是否存在
,使得
比
远离
?如果存在,求出
的范围;如果不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d32d4403d0e81eacfbe429dc51f07f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17d41b744a89e1a50c96ca75bf090830.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4e1fbe0fb49725cf6d1e689ee8986d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c95b6be4554f03bf496092f1acdfbb89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)对于任意的两个不相等的正数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632244ea6931507f8656e1cc3437d392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe2b05214c8b22507f0c36b110593d0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a977414a3ad65caf5eee28e0cd175de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c717d2811d58b258ce0b08ff602c027.png)
(3)对于任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73d22d807b8af8928aa2afc7172015f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73b5a9de8a72dda70e3eefeb0a408ffd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1d2e5599453f8d4c04369bc8f79962.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
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8 . 求出一个数学问题的正确结论后,将其作为条件之一提出与原来问题有关的新问题,我们把它称为原来问题的一个“逆向”问题.例如,原来问题是“若直角三角形的两条直角边长分别为3和4,求该直角三角形的面积”,求出面积6后,它的一个“逆向”问题可以是“若直角三角形的面积为6,一条直角边长为3,求另一条直角边的长”.试给出问题“已知
,若
,求
的取值范围”的一个“逆向”问题,并解答你所给出的“逆向”问题.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9678ae57ecdf808530b4cbc2ca0f12e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1492e429cb3b8309f733ea403bc8897.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
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名校
解题方法
9 . 若实数x﹑y、m
满足
,则称y比x接近m.
(1)若
比1接近0,求x的取值范围;
(2)对正实数a,b,如果
比
接近2,求证:当
时,
比
接近2;
(3)已知函数
等于
和
中接近0的那个值.写出函数
的解析式,并指出它的单调区间(结论不要求证明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed4a7dd81a499fbb8baf5a094c6c87e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9965755486b6b0887a2fcd0dc16e702b.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4e1fbe0fb49725cf6d1e689ee8986d6.png)
(2)对正实数a,b,如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b52159c4f340e243c243988c9469106.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd46e6f8f462d69e3b88c3245a7585d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d5750417af6a1d63b064c467414502.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82e778bb8ded9102f97acddffbc03130.png)
(3)已知函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b6894e8c345a035e89ec672503a01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da4fcad0fbcfd1c4811d5f84a1727119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65d0fbf9e5420e150263e457676c80e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8b6894e8c345a035e89ec672503a01f.png)
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10 . 已知函数
,
.
Ⅰ
记
在
上的最大值为M,最小值为m.
若
,求a的取值范围;
证明:
;
Ⅱ
若
在
上恒成立,求a的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e75613d81de6fe1309da3f046f8a366.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/980a8c4eb822aeb591ceacfe8a7aaa11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11d71379442f28c038d367d49422cf90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/987517758fad59f6f695761deb2a5ebd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d479a86a1711709b2d100fe4daf3e7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83a29726cb09e82862db881a3258c34b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/372470aee75717ec33c53c3434eb126d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41ea4f4e439a11a84a4445b4c8c0c96f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c18eca8193d91e13a240dec14be339cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2b8845bd03ac01a26ef57851a8764cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11d71379442f28c038d367d49422cf90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/987517758fad59f6f695761deb2a5ebd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c1472bd3540ad41abb5b0d8c4d8a66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35246b1f7e094e6e7ea093f2f53a1a3f.png)
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