1 . 已知数列
中,
,
.正项等比数列
的公比
,且满足
,
.
(1)证明数列
为等差数列,并求数列
和
的通项公式;
(2)如果
,求
的前n项和为
;
(3)若存在
,使
成立,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59dd6c97d2ee3e74ba5730f1cbcc1d43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1ac9d17bc8f166fc3894bf865ad6a53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d21cfd1ad0f4254be1d88c6577b6bbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e90d19bcd582ed71cde0f18027174459.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7e3824ddd2210bd1e6313ad11dc8b13.png)
(1)证明数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/102abb18c888eb23d40708b97de140ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034ba25825c13725931c483aa47c9363.png)
(2)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c39de50ed2f805d9c28aa7aadc46fe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57ef6d44448092ebdb9e4a49d866a749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1ae9a3b0b7aeb1545b65d91aa371b3c.png)
(3)若存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cea4ac187cbb465180e89f38250b3970.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/066141ba294715e4e725d59acfb85caa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
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名校
2 . 已知函数f(x)=xlnx-ax2,a∈R.
(1)证明:当1<x<3时,
;
(2)设函数F(x)=|f(x)|(x∈[1,e])有极小值,求a的取值范围.
(1)证明:当1<x<3时,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c7f197f4390616d2bfdee94fb2f15c4.png)
(2)设函数F(x)=|f(x)|(x∈[1,e])有极小值,求a的取值范围.
您最近一年使用:0次
2020-05-11更新
|
544次组卷
|
2卷引用:2019年全国高中数学联赛四川省预赛
2006高三·四川·竞赛
3 . 如图,在
内取一点P,使
,作
于点D,
于点E.求证:DE的垂直平分线必过BC的中点M.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7b1f10e660e2f364038eb8aab17bc5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/545e18836bc7fee22f8f813a6f525d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bc56fdf70e65bd88980c64af96b83da.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/7/28e313ee-08ff-418f-baee-493400a5fd0d.png?resizew=163)
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4 . 设
是
的
边外的旁切圆,
,
,
分别是
与
,
和
的切点(如图).
![](https://img.xkw.com/dksih/QBM/2018/12/5/2090236281135104/2091849065775104/STEM/8fc982e59fe64cf3903456244c2d56c9.png?resizew=110)
(1)若
与
相交于
,求证:
平分
.
(2)已知
,
,
,且
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9abaeba15f3abdd877bc701af52c5cd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/2018/12/5/2090236281135104/2091849065775104/STEM/8fc982e59fe64cf3903456244c2d56c9.png?resizew=110)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e4c6b0869d7248b6c2f964718a67702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e781a2489271bfd1597cba1bb6f5887.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df81cda12d7601d58b1d9c7c180c4d66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c884a45b56bc34d79273b067c1520b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46196aec06c25d5c8f9b1d3a8f50a889.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/404805afe92b11e5e6125f88ec1de580.png)
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5 . 已知:
.求证:
(1)若
,且
,则 ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b8f6bf5c2f04c84550957271c5cb115.png)
(2)当
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d17e3231e8d07e6f17cca273949d2bbc.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb7961cbe98aac6a5fdee94582c341b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f364bd19eda02fa19a278441b2aa401b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b8f6bf5c2f04c84550957271c5cb115.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e468312d09c6563c9094b710a35a65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eadd42e1f5910bcf2080d46d60db4f91.png)
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6 . 设函数
.
(1)若
在其定义域内为单调递增函数,求实数
的取值范围;
(2)设
,且
,若在
上至少存在一点
,使得
成立,求实数
的取值范围;
(3)求证:对任意的正整数
,都有
成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9e46c8aabf3375ef2fc95aa00b93455.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9413be6f2821909223cae6cfac464e91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5abd313d4e92a762fb7fb0c1cb65263d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/579e2c39e6c0a640357e3b0ccd6f954a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae17887ccf795c630b89ff5c9f72813a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
(3)求证:对任意的正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9b22b70a93c0a81b1d2bd0c3bd0932.png)
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2007高三·四川·竞赛
7 . 已知:
. 求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77e6c9383cbb460fbc307f78b7c70d21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a8142944ef3d840b0eb7d0bdf0fd93c.png)
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2005高三·四川·竞赛
8 . 如图,
、
是
的两条高,
、
分别是
、
的中点,
是
的外心.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af9aef21635a4eb796ea2f9db486de9d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/9/1d9a86a6-fb1b-4bd5-9b02-6d180a95233a.png?resizew=179)
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9 . 已知:实数
满足
,且
.求证
.提示:先证
,再证
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5983c77c76205ec7e864a0be1ef346f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c148bfb45d171c5d2a5b6e1148dbbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/830cf43e7bd98da31c6f0121d21de04c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf7474ff31b72b184cabf9396b0e1bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/489b32cf016aa026eee9d4df246299a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f85c5784b5d690046a1b7b901e851799.png)
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