名校
1 . 已知拋物线
过定点C(l,2),在抛物线上任取不同于点C的一点A,直线AC与直线y=x+3交于点P,过点P作x轴的平行线,与抛物线交于点B.
(1)证明:直线AB过定点;
(2)求△ABC面积的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fc58c62444bf42a25289c45425a00f2.png)
(1)证明:直线AB过定点;
(2)求△ABC面积的最小值.
您最近一年使用:0次
2 . 给定一个由
个小正方形拼成的棋盘形方格,这些小正方形的颜色黑白相间(如图).
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/8/691d10a0-cf8d-4eb2-9a06-a62e80c0bf56.png?resizew=192)
现定义一种运算A:把位于第i行的所有小正方形和位于第j列的所有小正方形都换成相反的颜色,即黑色的小正方形换成白色的,白色的小正方形换成黑色的,这里
.我们把A称为在位于第i行第j列上的小正方形上的一次运算.试问:能否经过若干次上述运算把棋盘上的所有小正方形全部换成同一种颜色?证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91c208d4f747f885000efa13c671779c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/8/691d10a0-cf8d-4eb2-9a06-a62e80c0bf56.png?resizew=192)
现定义一种运算A:把位于第i行的所有小正方形和位于第j列的所有小正方形都换成相反的颜色,即黑色的小正方形换成白色的,白色的小正方形换成黑色的,这里
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/385babde31579deec0dae3896c3a56a6.png)
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3 . 设正数数列
,
,
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
满足:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39310acb3c9b3cf6190f87b26364fd2a.png)
(
=3,4,
,
)且
,
.
(1)求
,
,
;
(2)证明
,
,
,
是自然数列,即
. (
=1,2,
,
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33fb97888c923a843a3b4806d3fd0733.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39310acb3c9b3cf6190f87b26364fd2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99ad6b84eabbce9f623343dca91ae7e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0748c346ed88f98e424de8edf278325.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36ba808c24aeae6a2f34b98ae5ec04ae.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c1ccc6c74b8754e9bcbb3f39a11b6f1.png)
(2)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea867f78d52b7382c38e8fc6e2e2a9c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37e5531913e2f170465d8df01795cd51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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4 . 设整数
是区间
中的不同整数.证明:集合
有这样的子集存在,它的所有元素之和能被
整除.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78f61798d78371d309f3ade451b77dad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e06ad75a82dbb501deb7e3db49b64a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b7d6f0860fa38e0b759277654378cd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d51f9147b8265c0276c1f2c2659197.png)
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5 . 已知a、b、c为正实数.证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d3f52b7bfe5b342f48c75c24d5f7eb8.png)
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