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 . (1)若实数x,y,z满足
,证明:
;
(2)若2023个实数
满足
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b77bd83db721327f2875fb9ce218f797.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1f921846cfd2491c21fbe42fd29fd80.png)
(2)若2023个实数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52011d2f4946b2ddd070bb20796a069d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca7909872111d29739539041cb524eaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4507435d75954f4991f50ad1376e5cb3.png)
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3 . 甲烷分子
的四个氢原子位于棱长为1的正四面体的四个顶点,碳原子C位于四面体的中心
,记四个氢原子分别为
,
,
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a13d906ff99fdc08d57f4de3785580c0.png)
_____ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c33fbdd535c70284a4e988fff972e4de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88e6fecc49564bf3341dc355e0ec4d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90dd27548b6259e9e7dced1a7e22cb42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a0c83fcca59db51cff4e3995c45081e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e54253784d559487fe67009c9223cb4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca9f017376a9c9c31829722d27857012.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a13d906ff99fdc08d57f4de3785580c0.png)
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4 . 如图,“爱心”是由曲线
和
所围成的封闭图形,在区域
内任取一点A,则A取自“爱心”内的概率![](https://staticzujuan.xkw.com/quesimg/Upload/formula/454a4be11ab1e6ab54f393c0b48cb6a6.png)
_____ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ea66cea52830fb855282de4266399cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ace4d72fbb3de77e5611f427583302b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bd7dac2b4af73fa517daa230c03b595.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/454a4be11ab1e6ab54f393c0b48cb6a6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/35896cc8-9c79-4686-ac95-4bc233a562da.png?resizew=158)
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5 . 如图,
,
是离心率都为
的椭圆,点A,B是分别是
的右顶点和上顶点,过A,B两点分别作
的切线
,
.若直线
,
的斜率分别为
,
,则
的值为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/bde56b8b-569f-4c5f-93d7-78f2e7d2acab.png?resizew=187)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/211b9e53e4677ae9e2b20d5f7ce0a4e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39a52e26180930ad5b56a8a45f28a0f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3da5d51cc23527feb04b419f83ec50c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39a52e26180930ad5b56a8a45f28a0f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/211b9e53e4677ae9e2b20d5f7ce0a4e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ce619c464f2aa04178ea0111ba60544.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9d9ccc4246c82939d8a659db9f28a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ce619c464f2aa04178ea0111ba60544.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9d9ccc4246c82939d8a659db9f28a87.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/8178a9bdd5f28b2eff1f886b5ab0fcb5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/bde56b8b-569f-4c5f-93d7-78f2e7d2acab.png?resizew=187)
A.![]() | B.![]() | C.![]() | D.![]() |
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名校
6 . 如图,以正方形一边为斜边向外作直角三角形,再以该直角三角形的两直角边分别向外作正方形,重复上述操作(其中
),得到四个小正方形
,记它们的面积分别为
,则以下结论正确的是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/275600c6-bdd1-40bb-a1b8-648a04b288ba.png?resizew=185)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aa9f89e08ecd078322a37521bd8e5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d21919c31c82e3bb78733605ef41255.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55eb79b9b81488b1ded67b3eb43491cb.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/275600c6-bdd1-40bb-a1b8-648a04b288ba.png?resizew=185)
A.![]() |
B.![]() |
C.![]() |
D.![]() |
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2022-10-19更新
|
659次组卷
|
5卷引用:2022年全国中学生数学奥林匹克(预赛)贵州省初赛试题
7 . 如图,M,N分别是
两直角边上的动点,P是线段MN的中点,则以下结论正确的是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/7b7d00f8-8d9f-46a1-96b9-879a897c94bc.png?resizew=118)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/7b7d00f8-8d9f-46a1-96b9-879a897c94bc.png?resizew=118)
A.当△AMN的面积为定值时,点P的轨迹为双曲线一支 |
B.当|MN|为定值时,点P的轨迹为一圆弧 |
C.当![]() |
D.当△AMN的周长为定值时,点P的轨迹为抛物线 |
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8 . 函数
的图像酷似教师批改作业时所画的“对勾”,所以我们常称
为“对勾函数”.其图像是双曲线,其渐近线方程为
(即
轴)与
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/c28cc4cf-c4ed-40e7-9c26-b0a8365caa55.png?resizew=165)
(1)求C顶点的坐标与离心率;
(2)求C焦点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a16803c866fdda79e4a0ebaef33b48f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cb788ae88e457017bb81120b6a2e5ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14fd2128777226fdda969aaaa541cea8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fcf9bfbf771cb6118f8e631724314e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7fdb3bf31cb105ddfe045d24f929ce6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/27/c28cc4cf-c4ed-40e7-9c26-b0a8365caa55.png?resizew=165)
(1)求C顶点的坐标与离心率;
(2)求C焦点坐标.
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9 . 已知
是曲线
上的点,C在
处的切线
交
轴于点
,过
作
轴的垂线交C于
,C在
处的切线
交
轴于
,过
作
轴的垂线交C于点
,C在
处的切线
交
轴于
,过
作
轴的垂线交C于
,重复上述操作,依次得到
,
,……,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b088cb297522e38dff96a4b5ea600e99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a098a0e65cc3a678c09ba4b90b97d91b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e367975cd9882f21334ffc8a92745161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ce619c464f2aa04178ea0111ba60544.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ffb694021b52653de5141ae27ba6d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9051c36cd615f33b0b998d098e49f0ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0235feb6050c98ad45cca7ceceaaf431.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ffb694021b52653de5141ae27ba6d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4f1b40221f57567c67c1ba5ceb2f57b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4f1b40221f57567c67c1ba5ceb2f57b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9d9ccc4246c82939d8a659db9f28a87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ffb694021b52653de5141ae27ba6d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32c57033e396170c7546b3f76e0cb5f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2d9d82c1949784452addc8047894384.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ffb694021b52653de5141ae27ba6d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/910ec9341909c0edd74ee755543c4c35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/910ec9341909c0edd74ee755543c4c35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92adfef6c3f4b8c041958fbf0b01f6a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ffb694021b52653de5141ae27ba6d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0269c74f81a17eca13334d53a3ae6176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f82cfd620516c5812e213ed4e0534a1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ffb694021b52653de5141ae27ba6d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5e056ec49e2b47361345a4db01ce788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7c643b0ffd4ca67a657e04a1337faac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d94f45a4cdb64e26f49c2490aaccd8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/297b6b3b19cde9602ca48acf1030e7d1.png)
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10 . 平面
与长方体的六个面所成的角分别为
,则
的值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07c9121cede0ee0562e23b8a26b34616.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2eec076403fb5f7e7ee03ba2231ed815.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1462beec3a6d100890d5295ddd3edee1.png)
A.2 | B.3 | C.4 | D.6 |
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