解题方法
1 . 若
是棱长为
的正四面体
内一点,以
在四面体
的四个面上的射影为顶点的新四面体的体积的最大值为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94dbe9b1953ebfd11fc9c452d237321.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6c086edbf6e575465265662956df593.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82802218706a07a4482876b25d0f8705.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94dbe9b1953ebfd11fc9c452d237321.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82802218706a07a4482876b25d0f8705.png)
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2 . 设
.在
的方格表的每个小方格中填入区间
中的一个实数.设第
行的总和为
,第
列的总和为
,
.求
的最大值(答案用含
的式子表示).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f98da45d4de19a962cfa1d186e2755a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ace7d64e7ff100db25a07330654d5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8af908bca1b10f5de7e2d8979989c806.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ea8f47d8d8d9e1832d52b1c7425450.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4de122ae929b1acaff321dec137622ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2aba126d3f244b529547fa33b1dc5f48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a71c7129333f890292aa75bc1d080a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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3 . 正整数
称为“好数”,如果对任意不同于
的正整数
,均有
,这里,
表示实数
的小数部分.证明:存在无穷多个两两互素的合数均为好数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d66ee10f17600981c1756c08db4d8a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f344fd588492f8bcbb5f55b2946ea735.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
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4 . 求具有下述性质的最小正整数
:若将
中的每个数任意染为红色或者蓝色,则或者存在9个互不相同的红色的数
满足
,或者存在10个互不相同的蓝色的数
满足
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c3070b614e81435e29a6571bcf17271.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ceffa018e8a05505b1b275e97a5c9f7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c27ad2066ebe180072945da08f399e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd877ef4c893b656e492c8818822040c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e79213177565cc77ba4da1b49e743eb5.png)
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5 . 已知三棱柱
的9条棱长均相等.记底面
所在平面为
.若
的另外四个面(即面
,
,
,
)在
上投影的面积从小到大重排后依次为
,
,
,
,求
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dc551328e06a903cfb7fddd337317a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa35373ec4e4684107b42adb7a5161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b16cff607cdc2d69afc70dc778acbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d9a8181f7a7fe7f3fac872ce9534f15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38387ba1cadfd3dfc4dea4ca9f613cea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adbd3e8cf8325999cde03adf845d3dd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41322821ce31416fdac8dd6e0aa41c71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffa37aefb6d45efe4e20ba48c2e7dfa8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa35373ec4e4684107b42adb7a5161.png)
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6 . 已知函数
(
为常数)的图象上存在四个点
,过
的切线为
,其中
,且
围成的图形是正方形.
(1)求证:
;
(2)试求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/388c0990291dfcf9ce3060c06ddd810d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f70db67d96a5bf6d5c6b93ed64952d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e819b87f90651d89fcd258c276294e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb886661302d1bc974b0c4f2458fcea6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/693dd24614173c8295bc7cf97fd5725a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a7122f2ae84bff5b73095f78cafe04f.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f64dafa5de92d59009eda97f12ac5d71.png)
(2)试求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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名校
解题方法
7 . 已知动点
到点
的距离与到直线
的距离之比为
,记点
的轨迹为曲线
.
(1)求曲线
的方程;
(2)曲线
与
轴正半轴交于点
,过
的直线交曲线
于A,B两点(异于点
),连接
,
并延长分别交
于D,C,试问:以
为直径的圆是否恒过定点,若是,求出定点,若不是,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/092fd1b1d33979818300cd2e3699bff7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/495bb3e5a3a9d35f5c9f0cf1f5d51876.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(1)求曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(2)曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
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2023-03-02更新
|
730次组卷
|
3卷引用:安徽省太和中学2022-2023学年高二下学期数学竞赛试题
8 . 已知函数
(
),则函数
的最大值为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ac9a839b199426d5d7bf13386fd36e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f274f1247962a5b263183922fedfa086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
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9 . 已知数列
满足
,
.
(1)若
是递增数列,求实数
的取值范围;
(2)若
,且对任意大于
的正整数
,恒有
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31d0b399e3826e2721d683a357fe5dd4.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e144b442dc601367909266594699b10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaacfaef44a654c0a1c283ef03fc0550.png)
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10 . 我们称正有理数n为“友好数”,当且仅当
化为最简分数
时,a,b为奇数.则在集合
中优好数的个数为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3836223819fb20bad051b2f3fa5c9ef0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2122e3f1e76a635e58e4d54aa594c552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f5d6982b8a897a3e8174360d4abe973.png)
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