1 . 设双曲线Γ:
,
,B,C在Γ上且直线
经过A.设
分别为Γ在B,C处的切线,点D满足
,则D的轨迹方程是___________ ;若D的横纵坐标均为正整数,且二者之和大于2024,则D可以是_____ .(写出1个即可).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af3674d64026a371b001453280477ba5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40630a669f4eedf626bc24851df10c85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86de0e038b0304e34ca8c9b117fec917.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04e6aa92080d4e1f43a1ac2f9c2965fe.png)
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2 . 给定平面上的点集
,
中任三点均不共线.将
中所有的点任意分成83组,使得每组至少有3个点,且每点恰好属于一组,然后将在同一组的任两点用一条线段相连,不在同一组的两点不连线段,这样得到一个图案
.不同的分组方式得到不同的图案.将图案
中所含的以
中的点为顶点的三角形的个数记为
.
(1)求
的最小值
;
(2)设
是使
的一个图案,若将
中的线段(指以
的点为端点的线段)用4种颜色染色,每条线段恰好染一种颜色.证明存在一个染色方案,使
染色后不含以
的点为顶点的三边颜色相同的三角形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a91692e4495a039c3736a2aa090f91e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](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/23235a862992c52b040ba326ee263439.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23235a862992c52b040ba326ee263439.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26e93d8fb77f5bd2c0fc690752dfd771.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/350c5d8dc46eb245caa8e828597b30d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7acacd1de8f64feef7bd1dc61662d91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/350c5d8dc46eb245caa8e828597b30d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/350c5d8dc46eb245caa8e828597b30d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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3 . 设
均为大于1的整数,
为n个不超过m的互不相同的正整数,且
互素.证明:对任意实数x,均存在一个
,使得
,其中
表示实数r到与其最近的整数的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb6794910a17295ec8f9cbcad2895357.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8ed4bbcd82234606bc1b7a4e45f074f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8ed4bbcd82234606bc1b7a4e45f074f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90ed315453447a26d9d8bf99b938d7fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3b576e5a5aa36577e3d815767faab86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4fd4a969ee73c3c4801d2b65161a350.png)
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