1 . 设数组
,
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/238d69b84446b56240d2a8ea5de04837.png)
,数
称为数组
的元素.对于数组
,规定:
①数组
中所有元素的和为
;
②变换
,
将数组
变换成数组
,其中
表示不超过
的最大整数;
③若数组
,则当且仅当![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd7a172e3de92f315198a515eef6ebbe.png)
时,
.
如果对数组
中任意
个元素,存在一种分法,可将其分为两组,每组
个元素,使得两组所有元素的和相等,则称数组
具有性质
.
(Ⅰ)已知数组
,
,计算
,
,并写出数组
是否具有性质
;
(Ⅱ)已知数组
具有性质
,证明:
也具有性质
;
(Ⅲ)证明:数组
具有性质
的充要条件是
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad5b09d32784527e9d1201abb4ef099.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/238d69b84446b56240d2a8ea5de04837.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5c9fcb34bf3a74eae571ddad293b6cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
①数组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fb10fbfb7eb6d4d4ffd799cc5a363b4.png)
②变换
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/916765ac224780b2ab9ad42c16b279c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
③若数组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffd9a9a948c76032a62a91d11fbf5bae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd7a172e3de92f315198a515eef6ebbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5c9fcb34bf3a74eae571ddad293b6cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e7cbcf6e8ea569285cd05a6126d7190.png)
如果对数组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d51f9147b8265c0276c1f2c2659197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(Ⅰ)已知数组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d5994d78fc5debd0469e2a212561751.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a1a7b15221d91714fcb964379e1e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3292a7444b7e76515cee05ffe1eea50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9de0f6d2b676fb885e43df4db0ed66d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(Ⅱ)已知数组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b373da1d03d8fe9f96f75a1d310a24d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(Ⅲ)证明:数组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50513a655f357615a650b63daf00be2c.png)
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