当前位置:文档之家› 高中数学竞赛均值不等式讲义

高中数学竞赛均值不等式讲义

均值不等式1.均值不等式知识点1: 二元均值不等式可以推广到n 元,即: 设,,,123a a a a nL 为n 个非负实数,则12n a a a n+++≥L 123a a a a n ====L ).如何证明?知识点2: 设,,,123a a a a nL 为n 个非负实数,n Q , 12nn a a a A n+++=L L,n G =, 12111n nnH a a a =++L L ,则n n n n Q A G H ≥≥≥(等号成立当且仅当123a a a a n ====L ) 更一般的平均值的定义: 设正数(1,2,3...)i a i n =,则α的幂平均值=11()ni i a nαα=∑,特别的,我们有:lim ()n f G αα→=,11()()ni i a f nααα==∑为关于α的增函数.知识点3:重要结论 (1)222,,,.a b c R a b c ab bc ac ∈++≥++(2) ()2,,,3().a b c R a b c ab bc ac ∈++≥++ (3) 2222,,,3()().a b c R a b c a b c ∈++≥++ (4) 2,,,()3().a b c R ab bc ca abc a b c ∈++≥++(5),,,()()()()().a b c R a b b c a c abc a b c ab cb ac ∈++++=++++(6) 222;2a a a b b a b b-≥-+≥(a,b,c>0)(7) 2222221()()3a b b c c a a b c a b c ++≤++++(a,b,c>0)(8)正实数(1,2,3...)i a i n =,则2111n ni i i ia n a ==⋅≥∑∑(当且仅当12...n a a a ===); (9) 222222222222()()()()()a b b c c a ab bc ca a b c a bc b ca c ab ++++=++++知识点4:加权平均值不等式已知12+...1(0,1,2.,,,)n i w w w w i n +=>=,则对任意正实数12112212........n w w w n n n w a w a w a a a a +++≥.均值不等式的使用前要注意两个方面,一个是观察题目中不等式证明方向,另外一个是取等条件,根据这些信息,相应去选择均值不等式的技巧、模型 ,不断尝试,最终解决问题 。

二、习题训练1. 设正实数,,a b c 满足2221a b c ++=,求证:22213a bc abc abc ++≤2.已知正实数,,a b c ,证明: 2222111()a b c ab c+++++≥ 3.已知正实数,,a b c ,求证: 1111000()()()27a b c a b c +++≥4.已知12,,...,0n x x x >,且123...1n x x x x =,证明:1(1)2nnii x =+≥∏改编: 已知12,,...,0n x x x >,且123...1n x x x x =,证明:1()(1)(0)nnii a x a a =+≥+>∏5.已知,,0x y z >,2221112111x y z ++=+++,求证: xyz ≤6. 已知,,0x y z >,3x y z ++=,求证: 33333312()888927x y z xy yz xz y z x ++≥++++++ 7.已知正实数,,a b c ,满足222,,a bc b ca c ab >>>,求222()()()abc a bc b ca c ab ---最大值 8. 已知正实数,,a b c ,证明:222a b c ≥+9.已知正数,,,a b c d 满足1abcd =,证明:211(1)a ≥+∑类似题: 已知正数,,,a b c d 满足1abcd =,证明:211(31)a ≥-∑10.已知正数,,a b c 满足1a b c ++=,求证: 2222a b b c c ab c c a a b+++++≥+++11.设大于等于2的正整数n,正实数12,,...,n x x x 满足:11ni i x ==∑,证明:11112n i j i i j n i n x x x =≤<≤⎛⎫⎛⎫≤ ⎪ ⎪-⎝⎭⎝⎭∑∑(16走向IMO57)12.设,1,2,...,,i x R i n n N ++∈=∈,求证: 111112(1)(1)1(1)22n n n n i n i i in x x n n i n n n n ix x x x --==≥+---++∑∑. 13.设正整数3,n ≥正实数23,,....,n a a a 满足234.....1n a a a a =,证明:23423423(1)(1)(1)(1).......(1)....nn n n n na a a a a a a ⎛⎫-++++> ⎪+++⎝⎭(52届IMO 预选题90)14.已知 0(1,2,..,,2)i a i n n >=≥,且11ni i a =<∑,证明:11111(1)1()(1)n niii i nn ni i i i a a n a a ==+==-≤-∑∏∑∏g 15. 非负实数12,,..,0,n a a a ≥12...1,n a a a +++=求01121.....n a a a a a a +++最大值16.对于正整数n,已知n 个12,,....,n x x x 的乘积为1,证明:1ni x =≥∑(2017年IMO71)17.设4n ≥,123....1n x x x x ++++=,12,,...,0n x x x ≥,求12323412.....n x x x x x x x x x +++的最大值(17IMO114)18.已知0(1,2,..,),2i x i n n ≥=≥,求最小的C ,使得42211()()ni i j i j i i j nC x x x x x =≤<≤⋅≥+∑∑19.设非负实数1236,,,..,x x x x 满足:611,i i x ==∑ 1352461540x x x x x x +≥, 若{}123234345456612max ,px x x x x x x x x x x x x x x q++++=(,)1p q =,(,)1,,p q p q N +=∈, 计算p q +。

20.设12,,..,n a a a R ∈,满足10nii a==∑,证明:{}{}{}{}1212231min ,,..,min ,min ,....min ,1n n na a a a a a a a a n ≥+++- 21. 已知123,,,.......,0n x x x x ≥,且123....1n x x x x ++++=,求1223341....n x x x x x x x x ++++的最大值.22.设非负实数123,,,..n x x x x 满足1,nii xn ==∑求122311....n n n x x x x x x x x -+++++的最大值。

23.给定集合(){},1,T i j i j n i j =≤<≤,对于任意满足123....1n x x x x ++++=的非负实数123,,,.......,,n x x x x ,求(,)i j i j Tx x ∈∑的最大值.24.已知n N +∈,2,n ≥22212....1n x x x +++=,求证:kmkm nx x≤≤∑25.设12,,..,0,n x x x ≥12....1n x x x +++=,求()i jij i jx x xx <+∑的最大值.26.已知n N +∈,2,n ≥22212....1n x x x +++=,证明: 12231...cos1n n x x x x x x n π-+++≤+27.已知12320172018........0,a a a a a ≥≥≥≥≥≥,且201811ii a==∑,求证:12345620172018135 (20174)a a a a a a a a ++++≤,并且指出等号成立的条件。

28. 已知非负实数22212100.....1a a a +++=,证明: 2221223100112 (25)a a a a a a +++≤29.设非负实数12,,....,n x x x 满足11ni i x ==∑,证明:222122314....27n x x x x x x +++≤29.设非负实数1212,,....,x x x 满足1211ii x==∑,求91231i i i i i S x x x x +++==∑最大值。

30. 设正实数12,,..,0n x x x >,求证:2121122123(..)1111...(1)(1)(1)21...n n n nx x x n x x x x x x x x x x ++++≥-++++ 31.求最大的正实数,使得下述不等式对一切正整数n 及正实数(1,2...,)i a i n =均成立:111+....nk kaλ=≥++∑32.求最大的正实数,使得下述不等式对一切正整数n 及正实数(1,2...,)i a i n =均成立:2111111+()(1)nnkk k k s s a a λ===≥+∑∑∑33.设正整数n(大于等于3),12,,..,0n t t t >,若212121111)(...)(...)n nn t t t t t t +>++++++(,证明: 对于满足1i j k n ≤<<≤所有的i,j,k ,正实数,,i j k t t t 总能构成三角形的三边长.34.定义12,,..,0n a a a >,用n g 表示几何平均,利用12,,...,n A A A 表示的算术平均为12...(1,2,3...,)kk a a a A k n k+++==,用n G 表示12,,...,n A A A 的几何平均,证明:1nng n G ≤+,并且确定等号成立的条件. 35.求满足以下条件的最小实数()m n :对于任意的0(1,2)i x i n n >≤≤≥,12...1n x x x =,不等式111nn ri i i i x x ==≤∑∑对于一切()r m n ≥成立. 36.已知123100,,,..,x x x x 为非负实数,且对于1,2,3.,,,,,100i =,有12101110221(,)i i i x x x x x x x ++++≤==,求和10021i i i S x x +==∑的最大值.37.对整数(2)n ≥,试确定最大实数n c ,使得任意正实数12,,..,n a a a 有:2222212121......()()n n n n a a a a a a c a a n n++++++≥+-38.设正整数2,n ≥求常数()C n 的最大值,使得对于所有满足()0,1(1,2,3,..,)i x i n ∈=;1(1)(1)(1)4i j x x i j n --≥≤<≤实数12,,..,n x x x均有11()(2ni i j i i j nx C n x x =≤<≤≥∑∑39.设,,0a b c >,44454690a b c ++=,求333523a b c ++的最大值. 40.求15sin sin 2sin34y x x x =++的最大值. 41.已知21x y +=,0,0x y >>,求11(,)f x y x y=++的最小值 42.求()f x =.43.设,,,0a b c d >44.设1212,,..,0,....1n n x x x x x x >+++=,求12111()ni x x =-∏的最小值. 45.确定最小的自然数k ,使得对于任意的[]0,1α∈及任意n N +∈,恒有31(1)(1)k n a a n -<+46. 0,(1,2,3,...,,2)2i i n n πα⎛⎫∈=≥ ⎪⎝⎭设,且1sin (n i i a a α==∏为正常数),求1cos ni i α=∏的最大值.47.设,,,1a b c d >,满足0a b c d abc abd acd bcd +++++++=,a,b,c,d 为实数,求证:101a >-∑48.设12,,..,0n x x x >,且11(3,,2)1k ni ki ix n n k N k n x +==-≥∈≤≤+∑,求证:1(1)n nki i x n =≥-∏ 49 设12..1n x x x =,且这n 个数都是正实数,记11n x x +=,求证:1111(1)()nnnnn ii ni i ii n xx x===+≥+∑∏∑50.已知123,,0a a a >,求证:12314()3(a a a a ++≥+ 51. 已知123,,,...,0n a a a a >,121 (1)n i ji j n a a a a a n n≤<≤+++-≥∑ 52.已知实数数列{}{}{},,,n n n a b c 满足10ni i i a b ==∑,证明:22222111114()(),n n nn niiii i i i i i i i i a b ca cbc =====≥∑∑∑∑∑53.已知n 是给定的正整数,12,,...,0n a a a >,对于任意的1k n ≤≤,均有123...!k a a a a k ≥; 求证:11212323!(1)!....31+(1)(2)(1)(2)(3)....()n n a a a a a a n a ++++<++++++! 54.已知n 是大于等于3的正整数,12,,..,0,n x x x ≥证明:224(1)(2)(22)n A B n B A n AC ++-≥+-,其中23111,,,n nni i i i i i A x B x C x ======∑∑∑55.对于两两不相等的实数12,,..,n x x x ,证明:221(1)n nk k j kj k x n x x =≠+≥-∑∏56.已知正实数12,,..,0,n x x x n N +>∈,求证:11211..)nnii i i i a a a e a ==<∑∑( 57.已知123,,0x x x ≥,证明:22231223311231234()27x x x x x x x x x x x x +++≤++。

相关主题