1 . 集合
,对于正整数m,集合S的任一m元子集中必有一个数为另外m-1个数乘积的约数.则m的最小可能值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4ccb195151d157a6abe2784592f52f0.png)
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2 . 求
的最大值,使得平面上有
个点,其中任意三点中必存在两点间距离为1.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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3 . 将
棋盘的每个方格都随意染黑白两色之一,每次操作是将其中同行、同列、同对角线的连续五个方格改变成相反的颜色.试问:能否经过有限次操作,使得所有方格的颜色都变成与原先相反的颜色?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be9649242affd44c4dbd7dee8352c895.png)
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4 . 在一次数学竞赛中,某些选手是朋友关系.记所有选手的集合为X,对集合X的子集Y,若可以将这些人两两分组,且每组中两名选手均是朋友关系,则称子集Y“可两两分组”.已知集合X不可两两分组,且对于任意选手
,若A、B不是朋友关系,则
可两两分组,且X中没有一个人与其他所有人均为朋友关系证明:对任意选手
,若a、b为朋友关系,b、c为朋友关系,则a、c也为朋友关系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2588f9e4d45702d31483b99bd45a79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0a7880f3e8368e4f6384660df97d366.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ce0e4d5674f7e5ab41cce7a7e8d8159.png)
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5 . 已知数列
满足
,
,
,
.
(1)是否存在正整数
,使得对任意的
,有
?
(2)设
,问:
是否为有理数?说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee7510780af5d70f33eeb66cb025b4b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e525160289dbe5aae1d3f07b5ee1687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/644abd575297a7271391f33aed9b9a4b.png)
(1)是否存在正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9e5270c151c0c9de54dd93e9687372c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b2b94cbf8f1acc77ed2618d9ba5756a.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c82361963bc08259b56d98b1eb66e8c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
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6 . 给定正整数数组
.若对任意的
,均有
,则集合
称为“
好的”.定义
为最大的正整数
,使得集合
可以分成两个集合
满足
,
,且
为
好的,
为
好的.若数组
满足
,且
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8845b362fa89e84beea37b11c61d3004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1796276e3bba753664e6e2e536a7248a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1f7c438c9d8857905fd4c4006d9c3ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26bdd1fc6840f36c86e57c02f8f0f93b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18a8ab9c2421408d202361aca2c944fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8f47f495b61b135d9ec764ddfd9d7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b790fb10e0391906cf406c4101814e17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2578b9b66df3cb080f572564c87ad60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07d037e9d6288bf61a1fceed25a419df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a71e896f2fa4f85dc29f7e71d7e0f6d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18a8ab9c2421408d202361aca2c944fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a67dad7337d3daaaae14dfb7ea8591fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ad4949bf2fd32d91c2d3ddbb7e66a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d492cf2f8b939b9f2f375b30195261e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df97a0b3b9e240a72ac26906a38ba35d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb7600263e0f3699ff394800ccdb4be0.png)
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7 . 空间中有5个点,任意4点不共面. 若连了若干条线段而图中不存在四面体,则图中至多有三角形个.
A.3 | B.4 | C.5 | D.6 |
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8 . 有10名乒乓球选手进行单循环赛.比赛结果显示,没有和局,且任意5人中既有1人胜其余4人,又有1人负其余4人.则恰好胜了两场的选手有______ 名.
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9 . 考虑
的方格表,其中每个方格内均填有数字0.每次操作可先选定三个实数
、
、
,然后选定一行,将这一行每个方格中的数都加上
(
为该方格所在的列数,
);或选定一列,将这一列每个方格中的数都加上
(
为该方格所在的行数,
),问:能否经过有限次操作,使该方格表中四个角的数字变成1,而其他格的数字仍为0?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd4be97adde17423d59d86d18318c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e7cd8891e6d25a9c180e7699598a9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cfdad5639ec97c97cb369ed6017eb54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/102d76767d5cc55728e95b3150dca280.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4f4bd2fab7acea4a223b4a5384f6bb1.png)
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