名校
1 . 若函数
满足:对任意实数
以及定义中任意两数
、
(
),恒有
,则称
是下凸函数.
(1)证明:函数
是下凸函数;
(2)判断
是不是下凸函数,并说明理由;
(3)若
是定义在
上的下凸函数,常数
,满足:
,
,且
,求证:
,并求
在
上的解析式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4799629218b4b62ffa4082b96888e3c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33bd24e647a626899a243a3f3984f90a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1538a4b84a99b2da4de9600fc5552c50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
(1)证明:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/681d6d27b23b1c41834d7516122f73f9.png)
(2)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f45afdf4d717bb03adac6b899c367acb.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304226ca50149b49702928e44d565964.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2da2ab1f8b5d3281efb94b763fa74081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9db2cae6cc39553ca2b984741630917.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ceb4645bb34156bfc57de16ec11300f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e61c9a7ed0961f8977a21dab37aab396.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d87cd4403487962c38c8707ba3ab3fa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304226ca50149b49702928e44d565964.png)
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真题
名校
2 . 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
|
594次组卷
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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(已下线)上海高一上学期期中【压轴42题专练】(2)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)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
3 . 已知集合
,且
中的元素个数
大于等于5.若集合
中存在四个不同的元素
,使得
,则称集合
是“关联的”,并称集合
是集合
的“关联子集”;若集合
不存在“关联子集”,则称集合
是“独立的”.
分别判断集合
和集合
是“关联的”还是“独立的”?若是“关联的”,写出其所有 的关联子集;
已知集合
是“关联的”,且任取集合
,总存在
的关联子集
,使得
.若
,求证:
是等差数列;
集合
是“独立的”,求证:存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10cdcb0e77b3ae3e701c6b51e15e2346.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d10449bc77d692a7270e0f20a68cdf2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eeda5cef4846ef829069fe27f64e34e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9bf4032eb5a9ba68131b15182aa3491.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bf6c84731e5e1bd335ecfc2d36c3d81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9859aa908844a32c0e1e069a046727.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/918f1c94368c3a41177ff42cfedc0eb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f53190d6ead827a6338b9de847aeaf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6e0ad51c5541ec3dcca4a9845f8b7db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/498f92bf2e605cdbc91973e29b047566.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4d89801d24aa43f47d6a366aad0571.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1ccce8225324817b0577551956464f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/874781ab5711bff6ee8c9cbad5b3b3dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f62295c36d2e2174908c2bec0eb5b30f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acfc595518cf752e1c7903dfff93dbda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8021f4f4c253a00360bf8f9425610e1.png)
您最近一年使用:0次
2020-02-09更新
|
1568次组卷
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10卷引用:上海市上海中学2022届高三下学期高考模拟3数学试题
上海市上海中学2022届高三下学期高考模拟3数学试题2020届北京市海淀区高三上学期期中数学试题(已下线)专题02 拿高分题目强化卷(第三篇)-备战2021年新高考数学分层强化训练(北京专版)北京市海淀区2021届高三模拟试题(一)(已下线)考点47 推理与证明-备战2022年高考数学(文)一轮复习考点帮北京市清华大学附属中学朝阳学校2021-2022学年高二5月月考数学试题北京市第五十七中学2021-2022学年高二下学期期末考试数学试题北京市第八中学2023届高三上学期12月测试数学试题北京市朝阳区中国人民大学朝阳分校2021-2022学年高三上学期开学考数学试题北京市日坛中学2023-2024学年高二下学期第三次月考(6月)数学试卷
名校
4 . 定义:对于任意
,满足条件
且
(M是与n无关的常数)的无穷数列
称为M数列.
(1)若等差数列
的前
项和为
,且
,判断数列
是否是M数列,并说明理由;
(2)若各项为正数的等比数列
的前
项和为
,且
,证明:数列
是M数列,并指出M的取值范围;
(3)设数列
,问数列
是否是M数列?请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b98ef143f8159f3a7dafa1fd2f2370.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68f165a34038d89623948dbe0a669df0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5612ce06759d0f77ca029d10083f7d1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
(1)若等差数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/998f5aef88cd5d583707464d3a11f187.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)若各项为正数的等比数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f617305d7343adb94241921816b264f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de777c4e44546bcfe26ad5b6bb418052.png)
(3)设数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdf1c8b23b5c5835f9775b1750976659.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
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5 . 若存在实数
使得
则称
是区间
的
一内点.
(1)求证:
的充要条件是存在
使得
是区间
的
一内点;
(2)若实数
满足:
求证:存在
,使得
是区间
的
一内点;
(3)给定实数
,若对于任意区间
,
是区间的
一内点,
是区间的
一内点,且不等式
和不等式
对于任意
都恒成立,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2d2919e5bd26b5e9be672a3ff7604cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/220070d9bef1244a81af87a13885d817.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1add6ea18092e4db74ff941591d8950.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85b7e150af2052a1664cde963273d905.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96dfaeb8281858eafc652223d9abecab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30277e0be448b4955903e81e8795e45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(2)若实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/663a61ad241d5d874c9a9362f0ee917c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a727f8c2f098bc3bce6719a9616c013e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2d2919e5bd26b5e9be672a3ff7604cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2f89a8b5cf6996a6455375e405bfb9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fe5f1f33b2085760378b8abc0f1a582.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(3)给定实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/958ca60e8f470ce1b747c7e6a8c5cc69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1add6ea18092e4db74ff941591d8950.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75b5dc876d7dcd3c971b36d26668b1e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28643dbf049ac0eab51b51f6c1c64646.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cd882dced6d048a704bfc678b8e7791.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cb0aa8c434bdadb3725591e5e49099d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83eb829e3338a9e4be598124855685e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a85ea4968343b0d94ed2fe01b535.png)
您最近一年使用:0次
2019-10-23更新
|
1276次组卷
|
4卷引用:上海市南模中学2019-2020学年高三上学期10月月考数学试题