1 . 如图,已知
内接于抛物线
,且边
所在直线分别与抛物线
相切,F为抛物线M的焦点.求证:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/9363e490-d4f9-4149-82da-1203d0f10886.png?resizew=200)
(1)边
所在直线与抛物线M相切;
(2)A,C,B,F四点共圆.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/854f480c60b88b546cb15d3b5622e212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12bc980a6ff5fe01b746c65af072a05e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19409d6d120387b3bccd934c4911bb45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd646f98b097a1b09f0b8a76ec399340.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/9363e490-d4f9-4149-82da-1203d0f10886.png?resizew=200)
(1)边
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)A,C,B,F四点共圆.
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2 . 正数
,
满足
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4887473a8091e1ef53a169cc9f211e3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2fdeba282b028321696be7f90f2cbfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e2b49651d75b65c000f41a92c613508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a28a010582034341a048b08b77f6596.png)
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3 . 已知
,
,…,
是集合
的n个非空子集,如果对于任意的i,
,均有
,则n的最大值为___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9721059d158853671eaf19e39769b13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd8db4b168ddbcba90ac9b31d36a0432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64e11e77f3276edde5c0a9f2057beb44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a42daf2970136f21f3e5b3b2f96fe44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac30abcc83eecff9e2cb8dabc1d10ac8.png)
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4 . 如图,
、
分别是△ABC、△ACD的重心,
的外接圆与直线BD相交于点P,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc71c35d851ff74646c728756b24efb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/951d73cefe3fed1c5e70210ce71ae2fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5035afd29380c8912a02ac79a7db19b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7a69a73f2044f45086374c2e068b548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3269cf61861183004455d640aac8ccf0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/25/ada6f890-0c6f-474f-a6a8-d92f18e4566a.jpg?resizew=193)
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5 . 某校数学兴趣小组有14位同学,他们组成了n个不同的课题组.每个课题组有6位同学,每位同学至少参加2个课题组,且任意两个课题组至多有2位共同的同学,求n的最大值.
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6 . 已知椭圆C:
的离心率为
,
、
分别为椭圆C的左、右顶点,
、
分别为椭圆C的左、右焦点,B为椭圆C的上顶点,且
的外接圆半径为
.
(1)求椭圆C的方程;
(2)设与x不垂直的直线l交椭圆C于P、Q两点(P、Q在x轴的两侧),记直线
、
、
、
的斜率分别为
、
、
、
.已知
,求
面积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad7451b947d1adea22bc04070efdd2b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cfa1e7ffae662aefb49a44c52d4954d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9721059d158853671eaf19e39769b13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd8db4b168ddbcba90ac9b31d36a0432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e82fe25db889399bb3ca4ffd5dd5db84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a61d572ecf27dc02fcbd588f24647b1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/130c1bb7866e816804248f5eb8c59417.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c68316a6c48b33c53e2725274c8e1d65.png)
(1)求椭圆C的方程;
(2)设与x不垂直的直线l交椭圆C于P、Q两点(P、Q在x轴的两侧),记直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee1d228a6558961e7346d0326207dacf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c69286a11bc6a2641c73542167ce2c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cfdaad60640c4c023a339ffdc459d7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95dfc574d8ed4f6e828a9c5c40788d02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2562424a83aae7e4b3f8adf90307961a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03b58edb637c5dfa03794f3952de9d83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be76756ea33cd487013256cbe942a3cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9fa1f26b7d3a7a600ad370cfa176f39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7ee33f0fa8b278c793a691206b1fb83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05470dc94e4d9eea98d2c533ce576b07.png)
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7 . 工兵用信号探测器探测边长为2千米的等边三角形区域内的地雷,已知探测器的有效作业距离为
千米,从三角形的一个顶点出发,工兵至少需要行走多少距离才能完成探测任务?(要求说明理由)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62a711b601b52e4116fc5e8e7d2949e1.png)
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8 . 如图,已知
、
、
、
四点共圆,四边形
是平行四边形,
是
与
的交点,
.证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bcaadfa991b0edf9163751f42d39b54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60dc9f8e47e78b7f4ab2dd9cd559f601.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0914b68f106a912420705b2f3928ca42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce1132157a33c82610c2d5035493d024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/289d7a880379d6060065c829b45b0ed6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655b6a742dbacdab5aaa298007663dd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06e322e0c87479bba874db9ae9ba36b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9005a613eaecad3406f9818fb278357c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bcaadfa991b0edf9163751f42d39b54.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/25/434c7151-f90e-4ae0-ba47-2a6ccd55c11c.png?resizew=205)
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9 . 已知
,
,
,
,1,2,…,则满足
的最小正整数n为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8d22ac81dfffec9c7eabf4119eea7ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9675214dcec606b8b95fc40f3c1e07e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47b4c70672d5cdb3079f7d6ef0b8395b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fbde97699a18dfed26535425ad37fc7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65ea26b90e2f223146b6e88a12e99df8.png)
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10 . 设数列
满足
,
,则
的值为______ .(结果用
和
表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28ef6f1b2279af482557a8ea46f2e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0427551c379611f42d01e1fbc07c7c92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5fd2bff68aa7c5f33b0441756ef7530.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0927f25906574feaaba7c5b207a91e11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74a5771f2c592b64f54f35ba1e2df0dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d8ae104d924e78bb6fab0dc10d0f50c.png)
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