1 . 20个巫师孤岛聚会.在此期间,任何三个巫师都曾在一起诅咒过别的某些巫师.证明:其中必存在某个巫师,他至少受到过其余九个巫师的诅咒.
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2 . 如图,
、
、
分别是
的边
、
、
上的点,且
与
交于点
、
与
交于点
,
与
交于点
.证明:
、
、
三线共点,当且仅当
、
、
三点共线.
![](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/0faed94a64b2dcfc6801b4fca0f16675.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://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b519e5794ef9932b64715619adf860db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09a2dedeb08d8d47252c3daec8308a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00bab17635a999236e8d2e35017a208d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9abaeba15f3abdd877bc701af52c5cd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b628d87cb667a0a31766a88c6c426324.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09a2dedeb08d8d47252c3daec8308a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b628d87cb667a0a31766a88c6c426324.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b519e5794ef9932b64715619adf860db.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/9/08e2e4a6-deb2-4de3-80ee-fbc2f77fc9bc.png?resizew=169)
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2010高三·山西·竞赛
3 . 如图,
是正
的中心,
是其所在平面上的任意一点,以
为直径的圆分别与直线
交于点
.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fbfc0ce508977423897abed5933856f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8785cef16dd00e01bbcd71146c47cab7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdc5c895153932c3e827a464664cef90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef91802a474d6fb95ca809844af5475a.png)
![](https://img.xkw.com/dksih/QBM/2018/12/19/2100142709342208/2104428801114112/STEM/0e5709a9-8cf0-4e5f-a21f-a83a5d47025d.png)
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2010高三·山西·竞赛
4 . 设
,约定
.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/122931583a58e218dc05475ce54fc231.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58a252d3f908b50ea6419b4f5207a543.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c5c4b189025e464f7154bceef1deb74.png)
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2013高三·山西·竞赛
5 . 如图,已知过
的三个顶点
各作其外接圆的切线,分别与相应顶点的对边所在直线交于点
证明:
三点共线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38335830b93ac4d99c28a8e209eecb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5137d6f8c78959a82f69a1c83f49bbb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c5b88ec996d2d117987e7303cefe4ad.png)
![](https://img.xkw.com/dksih/QBM/2018/12/13/2095808319389696/2096691277611008/STEM/d29d3d1aa9ec41188d4da4d2d7f14bbe.png?resizew=211)
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6 . 已知在正整数n的各位数字中,共含有
个1,
个2,⋯,
个n.证明:
并确定使等号成立的条件.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d04a8b7a7595251251b8e0b7e665e8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5453ec6a9e8b96357c888ea863ddcf7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/649fad26925be63140de835a44b78ed3.png)
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2018-12-07更新
|
370次组卷
|
3卷引用:2016年全国高中数学联赛山西赛区预赛试题
7 . 如图,
、
、
为
的外接圆
上的点,满足
,P为
上的任意一点.若直线
与BC、
与AC、
与AB分别交于点D、E、F,证明:D、E、F三点共线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50c64f2f03147277aad392390c1c483d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/800c5e266b4ad8462a46970f0a232d52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24006d28116bc097933cc90bcc0ea69f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ad3a1ea6790177130e16c2124984087.png)
![](https://img.xkw.com/dksih/QBM/2018/12/4/2089523208183808/2091037945249792/STEM/4a968c24c9ab408faa73f25dd09c3401.png?resizew=221)
您最近一年使用:0次
2014高三·山西·竞赛
8 . 数列
、
满足条件
,
,
.
证明:对每个正整数
,均有(1)
,
;(2)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4be2164a2c67d6163faee87a10942bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43b7e7cd571c8cd141cbbfe5d0890bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebaf2a2590bb84d646957f913d78f6dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ada18b47400ed40d1b9a1193b15413ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e194e7f25976502f399a81c56396ebd8.png)
证明:对每个正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0f9e68ae410c9fdd76b72406624f443.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acf1550ac7fc2b61e6b6bf02d3ae7581.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03ef91975d7e3fba8392f23add78c82d.png)
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2012高三·山西·竞赛
9 . 已知
内接于
是其内心,直线
分别与
交于点
,过点
作直线
,过点
作
的切线
,若
与
交于点
,证明:
三点共线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a05ddd5657aaf5371d869c85ed4373dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a7ef142e9e276a8220d5e19aa9d6572.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91e1e4115d78e625e9e0f47cdade3286.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5015bcf06e49e194bdda189204b2c04c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cc697381ccdea07ad097ebfb77b0598.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cc697381ccdea07ad097ebfb77b0598.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e9b0f5f44abbc6544a2f672b025b013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/927456b0989846a2f1573844bbaa2105.png)
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