1 . 四面体,又叫三棱锥,是一种简单多面体.指空间两两不相交且不共线的四个平面在空间割出的封闭多面体.它有
个面、
个点、
条棱、
个二面角.若一个四面体的四个顶点
,
,
,
.则可记为四面体
.对下列特殊的四面体,请选择正确得选项( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
A.若四面体![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
B.若四面体![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
C.各面均为直接三角形且有至少三条棱长为![]() ![]() |
D.若一个平面与正四面体相交得到一个钝角三角形,则该钝角总小于![]() |
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2 . 如图,给定两个相交的圆
与
,A、B为
、
的交点,一动直线经过B与
交于点C,与
交于点D,且B在线段
内,过C的
的切线与过D的
的切线相交于点M,连结
交
于点E,过点E作
的平行线交
于点K,求点K的轨迹.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0385df6c7f8ee7ab503b6ed35933695b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cd73875650e1538c4c61d5e16d3db29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0385df6c7f8ee7ab503b6ed35933695b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cd73875650e1538c4c61d5e16d3db29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0385df6c7f8ee7ab503b6ed35933695b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cd73875650e1538c4c61d5e16d3db29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0385df6c7f8ee7ab503b6ed35933695b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cd73875650e1538c4c61d5e16d3db29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a424b50eaeafa6f302ffd95476cb86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
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3 . 已知
为椭圆
上的点,对椭圆
上的任意两点P、Q,用如下办法定义它们的“和”
:过点S作一条平行于
(若点P与Q重合,则直线
表示椭圆
在P处的切线)的直线l与椭圆
交于不同于S的另一点,记作
(若l与椭圆
相切,则规定S为
).并规定
.
(1)若点
,求
、
以及
的坐标.
(2)在椭圆
上是否存在不同于S的点P,满足
?若存在,求出所有满足条件的点P的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cacd4c2119a2381d627a45de68e2d263.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/459208d3062060f65fa347e6368e1457.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f460a3d908ee5647b7ba52c19740ef08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f460a3d908ee5647b7ba52c19740ef08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f460a3d908ee5647b7ba52c19740ef08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9eb1745e3e06d9a875107480c658b54.png)
(1)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abc301000af55006ce42a131e6d56097.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f460a3d908ee5647b7ba52c19740ef08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1624d5e6bace3bfe3f7b69ddbd774a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ded53338ed7b2c20fd5e96522debbe4.png)
(2)在椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75cb0ff741c5a50624b7fa1a0ddfc3b1.png)
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4 . 给定
.若
共取有限个不同值,证明:x,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5ab4b75fa22deba7fcbcdcb31dd45b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/596e69bb701c3cabbfe11848b349366c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d69160111209ee77c2d6449a73227438.png)
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5 . 定义数列
,满足
,且
.
为
前n项的平方和.求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c38f0e2ee53fad0810444913431ad92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dcd2bf39cd4f7ddd412827758a7df2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1cb96b923f8e64642a00dbde61adba39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c38f0e2ee53fad0810444913431ad92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e331ed850756b7366c34094f4d386919.png)
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6 . 实数x、y满足
则x、y的大小关系是___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/356ffd3ab48054894785cc839c71e370.png)
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7 . 设
是数
的一个排列,满足
是2的幂
.求
最大可能值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6ba62a51e634eded7d06af92f2f6494.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c20aa1f4136a8fdd2ad61397ba825244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c89d03c39db535b7da8ce5883db5a583.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7746dc2215c6b8db658c63597a11527b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3d7f8a99686ee40a31fb0909aacb7ab.png)
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8 . 如图,锐角
的外接圆为
,D是A在
上的射影,假设
,点M为
中点,
的角平分线与
交于点N,
上一点P满足
.直线
与
交于点F,直线
与圆
再交于点Q.直线
与
的外接圆再交于点E.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d2530e7023b2345c651e8f53629ff1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3981e7286d41960daf4e110c1c84e03a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745cd464c272fc1e45254682591b0405.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3533837e3d08c461dea031a44e5424d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fb26d84907c923278ac4626a9d58947.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bcd8ee2d8367c167d6ae0abc741b6b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/797e43acfc591c6493bb06de927aeffd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c6ef3581369b6b5b374c1301e4d2ed3.png)
![](https://img.xkw.com/dksih/QBM/2021/7/20/2768310052290560/2809301599690752/STEM/c505ba87bb994159b998eafd242dff71.png?resizew=267)
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9 . 将圆周
等分于点
,在以其中每三点为顶点的三角形中,含有圆心的三角形个数为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2def5aa62f497709e1bd8258583d62fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58aec728e0915e6bc12efa884a310a1f.png)
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10 . 证明:对任意正整数
,都存在正整数
和
个互不相同的正整数
,使
是完全平方数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1563da7b0f046a469476668a3686e8f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fe1c31a81f198c443e71b83ca662939.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cdc0ee2a438b747bf35b7cf4541b0f7.png)
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