名校
1 . (1)已知集合
,任意从中取出k个四元子集
,均满足
的元素个数不超过2个,求k的最大值.(举出一个例子即可,无需证明)
(2)已知集合
,任意从中取出k个三元子集
,均满足
的元素个数不超过一个,求k的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89f4949a8f15f43c0adbd5b86d935a79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afdee66380bcaf6444095a37e6dc2052.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/738aa208e410305523fa64b8518ba6b2.png)
(2)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1402be16d42c77f5eb8e12f1d5723690.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afdee66380bcaf6444095a37e6dc2052.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/738aa208e410305523fa64b8518ba6b2.png)
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2 . 给定整数
,记
为集合
的满足如下两个条件的子集
的元素个数的最小值:ⅰ.
,
;ⅱ.子集
中的元素(除1外)均为
中的另两个(可以相同)元素的和.
(1)求
的值;
(2)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8715a3f984d2627afd7c40c61347b7cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab6a1d0b6255d2fe84bfad2e4ad232d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35a2410ce34b36954ed4923e600d42f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/006862363f0414fa9eaa25411bfbc1f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9347bb4ffedcbea2f4c16d047a138d75.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecd61ca77c54e98408bd4bfe5898671b.png)
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2018-12-14更新
|
216次组卷
|
3卷引用:2012年新知杯上海市高中数学竞赛试题
3 . 对集合
(
),若对于任何的
不能整除
的任何一个非空子集的元素和,则称
是“好集”(
表示集合
除去元素
后的集合).
(1)若
是好集,求
的最小值
;
(2)证明:集合
不是好集,其中,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2344e0bdfa0fc62283a93d4fe67ea89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b58b34e73f045f54f9b2ee7daad5fc98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5805e7b270434d58308046727e963ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5805e7b270434d58308046727e963ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3bb14e6163867fe0acaffbb05ad2f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7e9f86738335a22298559db41037a4.png)
(2)证明:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ff7ed34e0562a4542fe4b860753995f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4806dc988e47163bc2178983196b9bff.png)
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