1 . 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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2 . 设
为正整数,
,
,令![](https://staticzujuan.xkw.com/quesimg/Upload/formula/649309617a29d1b9d4b2373d3c1c2946.png)
.求证:存在
使得
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4887473a8091e1ef53a169cc9f211e3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e65369cec2aac72a87c314f4c7ba143.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0be022d2478bb5f67d3e98f1319ea821.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/649309617a29d1b9d4b2373d3c1c2946.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/264d2ca5912fe1d6c354d7b0fd5efe4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00a28be4d5a16cf245f6fa7c4088fee4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0f7741f4975d9df3daa0c1cdd23c9df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/264d2ca5912fe1d6c354d7b0fd5efe4f.png)
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3 . 设
,
,
,
,直线
与双曲线
相交于
,
两点,若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/289bcfd18f773ebabcfe06683b4ebc79.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eec6a840369a7f6d3af9ec20dc066aee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd653739edd1793d5e2f5bc2a1da6f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5e5f44f92c1622e4cd89b5f40b48437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df0e05a41bd2560573dd2533f76a92b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed16a0a363ba8da5a730897dfd0658d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0914b68f106a912420705b2f3928ca42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1da9dcf6c319174c9ea2b1ceaed1649a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/587d5d656482d60ae672024af4e37e8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/289bcfd18f773ebabcfe06683b4ebc79.png)
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4 . 设
是严格单调递增的函数,其反函数为
.设
,
分别是方程
和
的解,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75b04a597429e99f3406acf168857b10.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89f4eff3125c5e63a994ba1ad5be58e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f749e2471684db63546a9ce817a6415.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b12a4eecd249473a831d0ee472470240.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9565876bc50bceb63e5793c8c67a9032.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac8506490bdfeef3771ca828e6ceac5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0e124011e4d140a8ea89685b41b76f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75b04a597429e99f3406acf168857b10.png)
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5 . 设
,
且
,
.若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30800bcb030cf5ee97d0e21fbbe8c368.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb559ff1fd340bf86f4f857eea672ae1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9467b82088c5ebcab031c03ac63fc798.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b250172ed268999cdb7d38980855e6ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/264d2ca5912fe1d6c354d7b0fd5efe4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84a2f8008f8af0bbec60ddaada7ce3f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30800bcb030cf5ee97d0e21fbbe8c368.png)
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6 . 设
为正整数,若
对所有正整数
成立,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d22d7c214d9685da7be80319ed80d8df.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a425978da20cebf8c4c63953579e7b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/533674a39b4b03b2b25432840172aa22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab4717e4827480f0f6f4ded85e52eab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d22d7c214d9685da7be80319ed80d8df.png)
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7 . 已知
,
.设
,则
的整数部分为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7360cd8a3b80d6b572255042528f0740.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3625fa11729e51745eee9ed2cfa29be6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53ee923e5c38b4cef1df8fd9f0f30e9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ffb694021b52653de5141ae27ba6d0.png)
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8 . 给定正整数n,
,记
从
的一一映射f称为是可
划分的:若X可划分为k个非空子集
,
,…,
,且
(
,2,…,k)(即
,且
,
,…,
两两的交集为空集,
).已知f是一个X的
划分的一一映射,
,
,…,
是1,2,…,n的一个排列,则
的最小值为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7863f720c810be5158d84a0e9b5e5be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b056db510d3e75cc12adc855015ad448.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31556cfdfe2ebbf0dacfbb9e28f968ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307e85746fa628650484d89e3b9809ba.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/1d7f2c72ab559a0615db4c51327b78d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd2781fa12b524b3b01f02d7b9d8f68a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a88501f72db91e5cf1a42114b4300ad6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb67fb9b5a8e475087ecc2a0aaca869a.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/1d7f2c72ab559a0615db4c51327b78d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7187f7b32e990b39cf9df84428660a42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307e85746fa628650484d89e3b9809ba.png)
![](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/2a10902a745902e85100175a0d7788fe.png)
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9 . 设锐角
,O,H分别为其外心和垂心,直线
分别交
,
于L,K.已知
与
的面积相等,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8db9f09c5f5d6cdbf2c48d1ab600488.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ff214d99784c6b23b7784bdaf3ed37a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4cd68cc82e90a5e2049a7ea3171b84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54435ea54b0214eb58ce6f4a4c53317d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc1eb72c3df18cdaa2ff9510b4742dfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8db9f09c5f5d6cdbf2c48d1ab600488.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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