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2021年放缩法证明不等式例题.pdf

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1、放缩法证明不等式一、放缩法原理为了证明不等式BA，我们可以找一个或多个中间变量C 作比较，即若能判定BC,CA同时成立，那么BA显然正确。所谓“放”即把A 放大到 C,再把 C 放大到B；反之，由B 缩小经过C 而变到 A,则称为“缩”，统称为放缩法。放缩是一种技巧性较强的不等变形，必须时刻注意放缩的跨度，做到“放不能过头，缩不能不及”。二、常见的放缩法技巧、基本不等式、柯西不等式、排序不等式放缩2、糖水不等式放缩：)ba,0m(mambab.3、添（减）项放缩4、先放缩，后裂项（或先裂项再放缩）5、逐项放大或缩小：)1n(n1n1)1n(n1221n2)1n(nn)12)(32(1)12(1

2、2nnn)12)(12(1)12(12nnn)22(21)12(12nnn三、例题讲解例 1：设a、b、c是三角形的边长，求证cbacbacbacba3 例 2：设a、b、c0，且3cba，求证abccba2322229精品w o r d 学习资料可编辑资料-精心整理-欢迎下载-第 1 页，共 7 页例 3：已知*21().nnanN求证：*122311.().23nnaaannNaaa例：函数f（x）=xx414，求证：f（1）+f（2）+f（n）n+)(2121*1Nnn.例 5：已知 an=n，求证：nk=1ka2k3精品w o r d 学习资料可编辑资料-精心整理-欢迎下载-第 2

3、页，共 7 页文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2

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5、:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L

6、9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P

7、6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S

8、8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z

9、4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5例 6：已知数列na，132a，113(2,*)21nnnnaannNan（1）求数列na的通项公式；（2）对一切正整数n，不等式123!naaaan恒

10、成立，试求正整数的最小值。例 7：已知数列na，212221111,123(1)naann，(2,*)nnN求证：(1)2211(*)(1)nnannNan(2)12111(1)(1)(1)4()nnNaaa精品w o r d 学习资料可编辑资料-精心整理-欢迎下载-第 3 页，共 7 页文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z

11、4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G

12、3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2

13、Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码

14、:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L

15、9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P

16、6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S

17、8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5例 8：（1）已知54nan，证明：不等式51mnmnaa a对任何正整数m n，都成立.（2）证明：对于任意正整数R，有.)111()11(1nnnn例 9：在xoy平面上有一系列点),(,),(),(222111nnnyxPyxPyxP，对每个自然数n，点nP位于函数)0(2xxy的图象上.以点nP为圆心的nP与x轴都相切，且nP与1nP又彼此外切，若11x，且nxxnn(1N*).(1)求证：数列nx1是等差数列；(2)设nP的面积为nS，nnSSST21，求证：2

18、3nT.例 10：已知数列、nnnabx满111112,4,()nnnnnnnnnaababb babxnNb.（1）填空当 2n时，nx1（填“、”不必说明理由）；（2）试用nx表示nxn(1N*)；（3）求证：1nx与nx中一个比5大，另一个比5小.并说明1nx与nx中哪一个更接近于5？（4）求证：1551nkkx.精品w o r d 学习资料可编辑资料-精心整理-欢迎下载-第 4 页，共 7 页文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8

19、S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 H

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21、A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 Z

22、J5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y1

23、0V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文

24、档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF

25、10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5针对性练习1、求证：2222111171234n2、设)1(433221nnan求证：2)1(2)1(2nannn3、已知函数),0(,12)(xxxxf，数列nx满足),2,1)(1nxfxnn，且11x.(1)设2nnxa，证明：nnaa1；(2)设(1)中的数列na的前n项和为nS，证明22nS.精品w o r d 学习资料可编辑资料-精心整理-欢迎下

26、载-第 5 页，共 7 页文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3

27、Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y

28、5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:

29、CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9

30、P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6

31、 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8

32、I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y54、已知数列na满足2111,0,2nnaaa求证：1211().32nkkkkaaa5、设a、b、c是三角形的边长，求证222)()()(b

33、abacacacbcbcba)()()(31222accbba6、设 0abc1，求证：)1)(1)(1(111cbabacacbcba1 7.数列01,naaa 满足20111,(0,1,)2kkkaaaaknn，求证：111kan。8.（2008 浙江高考）：已知数列na，0na，01a，*)(12121Nnaaannn记：nnaaaS21，)1()1)(1(1)1)(1(11121211nnaaaaaaT求证：当Nn时，（）1nnaa；（）2nSn；（）3nT。精品w o r d 学习资料可编辑资料-精心整理-欢迎下载-第 6 页，共 7 页文档编码:CF10L9P8S7P6 HV1S

34、8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z

35、4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G

36、3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2

37、Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码

38、:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L

39、9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P

40、6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5精品w o r d 学习资料可编辑资料-精心整理-欢迎下载-第 7 页，共 7 页文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S

41、8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z

42、4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G

43、3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2

44、Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码

45、:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L

46、9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5文档编码:CF10L9P8S7P6 HV1S8I5A9Z4 ZJ5G3Y10V2Y5

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