1 . 数列
满足:
是大于1的正整数,试证明:在数列
中存在相邻的两项,它们除以
余数相同.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57c4d9c843ed628701f262f3e80ccb62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58e51235780886a13ff7ab8918e97d64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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解题方法
2 . 一个顶点为
、底面中心为
的圆锥体积为1,若正四棱锥
内接于该圆锥,平面
与该圆锥底面平行,
这4个点都在圆锥的侧面上,则正四棱锥
的体积的最大值是__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06a5faf3cbb633fc4294c8ce703c64c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06a5faf3cbb633fc4294c8ce703c64c3.png)
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3 . 若3个整数
满足
,则这样的有序整数组
共有__________ 组.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82a74f84c202f34850d0ea31e034c06f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10a57d1215099fab4a97db12b2fa8f14.png)
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解题方法
4 . 已知
是同一平面上的3个向量,满足
,且向量
与
的夹角为
,则
的最大值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae951e0bb5a2a406f1572fc1e4964265.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c71ddc7c848789bd46e341bb5fa6316.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/015804533449e04f3e4f9f2d54326f3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/222fd0837d18068c2471dac135e79f7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15615de1a6df206dbd081251f676578e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83634e314df24df73ae37d25a44d20e3.png)
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5 . 若关于
的方程
存在一个模为1的虚根,则正整数
的最小值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ceadc4dc689abd18738645c901316b44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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6 . 求所有正整数
,满足正
边形能内接于平面直角坐标系
中椭圆
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8715a3f984d2627afd7c40c61347b7cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48da128547c4cf9745e8e4b99988a3db.png)
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7 . 将正整数
填入
方格表中,每个小方格恰好填1个数,要求每行从左到右10个数依次递减,记第
行的10个数之和为
. 设
满足:存在一种填法,使得
均大于第
列上的10个数之和,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deacb2d14b3b685334af74c9eb08e708.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f23f3b42cd6069f0e461035e76459ee4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aba753aa5e77c45b0d328c036a954a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70e72518ba0d330df05786f6c48db9b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77efd8c62dacd2212c3ff5db6b02a5dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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8 . 计算![](https://staticzujuan.xkw.com/quesimg/Upload/formula/375672a5135e75e4783ad9628165b3aa.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/375672a5135e75e4783ad9628165b3aa.png)
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9 . 已知函数
有两个零点,则实数
的取值范围是__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10e93aa746ec436ef3f412fb65f14401.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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