1 . 给定由正整数组成的数列![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80ef842458c67bd274549d90e916e5ac.png)
(1)求证:数列相邻项组成的无穷个整点
均在曲线
上.
(2)若设
,证明:
整除
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80ef842458c67bd274549d90e916e5ac.png)
(1)求证:数列相邻项组成的无穷个整点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4700f448e5e421847effc897502aba0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4db3ae18875992d69a104310d7b12a6d.png)
(2)若设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cbc2f2c80dfac0c350de25cd20e9c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
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2 . 已知复数
,
,…,
,
,
,…,
,满足
,
,…,
互不相同,
,
,…,
互不相同.已知对任意正整数
,均有
.求证:
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad7f66c97bfce4c00c53d86700c961b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54015ff5b49e3283901da1291b6b921d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f6872ffb1934339c53c2c2282d5889.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47233e81de2d9d728c80f02359f1af4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad7f66c97bfce4c00c53d86700c961b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54015ff5b49e3283901da1291b6b921d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46f6872ffb1934339c53c2c2282d5889.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47233e81de2d9d728c80f02359f1af4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58c8214c80f9e949b1ac8d6e41b597e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cddf38600954700caa8a227dae4e8ced.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8389928ac57b99f195e6bef80d2a28d3.png)
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3 . 已知多项式
.
(1)若
,且
有三个正实数根
,
,
,证明:
;
(2)对一般的正整数
,若
,
,
,
,证明:方程
的根不全是正实数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d310b3ca60508199bb95f15860232f4d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/291c25fc6a69d6d0ccfb8d839b9b4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8de1c943439d47ca9e9a02e558a1b2e.png)
(2)对一般的正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9772498c845b2043b375d1e8d8416b7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e506a31a62c9581edb62218fce59b51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f13c73c7894077a19b6c403587de96a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f4e01e8ef5adf43e1f21591adbc3851.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b92b70365c63607daecdc8deb73ecf.png)
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2023高三·全国·专题练习
4 . 设
,
是两个实系数非零多项式,且存在实数
使得
记
,证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14895b71b7bee6334172f84f0554931f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f65c792161a7882d13ae01b824453d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c98ff9ba195418273adfd773f78c8a5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37ed6dbdc5c0a115edac62a1eac695c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4720c811fe1d71070124bf972d9c5073.png)
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5 . 在
中,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e878d327fe80d3a1a722e856e3cd8ef.png)
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6 . 设多项式
,证明:
至少有一个根为虚根.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d7f6db9749ef0622795c4ba265a577f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
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2023高三·全国·专题练习
7 . 证明实系数奇数次多项式必有一实数根.
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8 . 设
是一个
次多项式,且
证明:
其中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14895b71b7bee6334172f84f0554931f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/257e61018107dd1618cb5c8e34722865.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aca54a3c1ca18b4f99c445f1a280ec9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bd82edc39d053d5ab04075e530763be.png)
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9 . 设
,满足
又设
满足
,证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6af3f54ea28a8d7df50ad784861176a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7850844ff8e0e33feadd667b69699e2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/850bc4fdf0ae1043eb3601b66bbfe1d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b8b13b8ed35ce32dd3899a0f6bdb418.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bef6887683a63bdae917280b410ec877.png)
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