1 . 数列{an}满足
.
(1)证明:数列{an}是正整数数列;
(2)是否存在m∈Z+,使得2109|am,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b48158643c10dee10828f2a143f2c8d5.png)
(1)证明:数列{an}是正整数数列;
(2)是否存在m∈Z+,使得2109|am,并说明理由.
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2 . 设函数
(a≠0)满足
,
,
,求当
时
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1bd0587f5d6a3b5db9e4a93e0dbc0ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b501b3d9f1ed6143799e34b45e54742.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46006273eb4b084e8fbb46020fb908b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad150b78dc4b86bdcc650d7de2a5209a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac571d2a4938fc8d60b02d72c202fe39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fd713a9809d5df1de33c6f11b81eca7.png)
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2018-12-16更新
|
410次组卷
|
3卷引用:2018数学竞赛(重庆赛区)预赛试题
3 . 已知x,y≥0,x2019+y=1,求证:
.
注:可直接应用以下结论:(1)
;(2)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5dd3b0d007c29e09efee75e1da0f0ac.png)
注:可直接应用以下结论:(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73e59c70b8f0f7ebf84632d93e9018b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cd4c6a2cdb995618f6b83052ad718d9.png)
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4 . 设椭圆C的左、右顶点为A,
,过右焦点
作非水平直线
与椭圆C交于P,Q两点,记直线AP,BQ的斜率分别为
,
,试证:
为定值,并求此定值(用a的函数表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48d04015890783f6b8b0264b1d1c9127.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2ba2238d6afe0187534155dd9ac48c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6defc43285a40f7ccb74c1cc04265eba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b7ae39db552e60ee8b1d27312306f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67351fe10fcfc3f9072eec4c60bfaaa5.png)
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5 . 已知过点P(3,0)斜率为k的直线l交双曲线
右支于A、B两点,F为双曲线C的右焦点,
,求k的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2b30352c43707c4e54b94ce5b61f2e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4a15f197804d75d00d9252a9c3ce575.png)
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6 . 设
,
.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a564595e470d31c824b99575a53f9cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad63dc600cd9fcb1b69d360d909fc2d2.png)
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7 . 设f(m)是正整数m的各位数字的乘积,求方程
的正整数解.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcc425363985780a7f22c23257c0ab1e.png)
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