1 . 若
,求证:
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5be97cd1c7111b654d87d8fbb63b6a84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69b28c716d53e72489c55897f632f310.png)
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解题方法
2 . 证明下列结论.
(1)已知
,试用综合法证明:
;
(2)已知
,且
,试用分析法证明:
.
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf04fe8895c10624636a815d3d752975.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0da537e5284dc9786845fca39a9ca913.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce613eaa5df46a50174085ef5d1087fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e56f4504e0f80fd031c8b5f41832e03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8954db00a1de8263871cf3e26965eb4b.png)
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3 . 已知
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0a7e7ca61f9605b0d309923e3a64762.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4bf6cfdcec4b466f4d60ffa37527e05.png)
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2023高三·全国·专题练习
4 . 设
,给定数列
,其中
,
,
.证明:
(1)
.
(2)如果
,那么当
时,必有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6455e38ff53ede2508e4d9cb23f0b86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83cf38189d5cbf627d2b82ac0eb76006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7fab51121848ce166035ceab6f4e00b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75ab998a5e347af02ca52d29ced30940.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b87de3864d9d0ce93638a99b87590f3b.png)
(2)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06dfdac83e7aaac092e7e7a4e91a2e9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e29d198c2f468ff2309d420b89667e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc0e11a36f78b6ec85fa8ddb95c7beba.png)
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2023高三·全国·专题练习
5 . 设
是一个自然数,
是
的各位数字的平方和,定义数列
:
是自然数,
(
,
).
(1)求
,
;
(2)若
,求证:
;
(3)求证:存在
,使得
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff3bf2007903adc64d089a054c2284a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e13ee40e6cfa757f60396a5a93202c4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1505d56f0b35fe7f2de1fe1888036e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/621d38cb93dceb8487ff64a7c4d1c40f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f06aef3197f50d50505ab596dc31620.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/592871252c561ff04cf055064822eaa9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43ade3a1d01605706801e238726e55fb.png)
(3)求证:存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1741807e6cd730f285abd967f4bea5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/742a31e28de9309da5f9ef176a1c24fd.png)
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6 . 设
.
(1)
,证明:
;
(2)若
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2684b72f9f38f5046c8ecd4280b7b14b.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/114b1d85a3f52a2959c49ec25630dfaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ec307143b4bf45106369f256a796d61.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b140e221ddf537b8964fff8557cca0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2504fe7020ff898ca5955a423bed0096.png)
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2022高三·全国·专题练习
7 . 设不等式组
表示的平面区域为
,设
内整数坐标点的个数为
.设
, 当
时,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf7014de5cbe63dca175543775c19814.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66d4e8502106802f1485c3b0f28f2664.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66d4e8502106802f1485c3b0f28f2664.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ab5587f937dcd892b7cc0b69065b126.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65d52ce074304be72992e6695a44f71c.png)
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8 . 已知
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93b86eff6d1841d0bc77000eb4c7e180.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aa7584fcb7c6b8770d061c09158fcd1.png)
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9 . 已知
为锐角,
为钝角,且
,
,
成等差数列,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6581916f5a65edfea257c804efee007e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1565c50248de774cadd890f35dd2b690.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51d8dd18aa0b0d03c3c7702707f1daa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b196a81ab9eb8c98c0a2daac64ec839c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1974b4e41ca3a2ee154890b12397c490.png)
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解题方法
10 . 已知函数
,
,证明:
,其中,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d4433145e39bfbf2f69478d7d4aa9a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2efe8d2fdba8bfc8a40ef09c8c99b1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff5c26e5ef8a341f2f0f90fe639942d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
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