sỞ giÁo dỤc vÀ ĐÀo tẠo khÁnh

8/14/2019 s Gio dc v o to Khnh

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S GIO DC V O TO KHNH HATRNG THPT CHUYN L QU N

------------------------

Chuyn :

BT NG THC C SI

TRONG CC K THI TUYN SINHI HC V CAO NG

Gio vin : Hunh Kim Linh

Nm hc 2008 - 2009

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Li ni u :Thc hin nhim v nm hc 2008 2009, Trng THPT Chuyn L

Qu n Khnh Ha c khuyn khch cc gio vin dy mn chuyn lmchuyn xy dng ti nguyn ca t chuyn mn

Chnh v vy ti thc hin v lm chuyn v

BT NG THC C SITRONG CC K THI TUYN SINH I HC V CAO NG

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BT NG THC C SITRONG CC K THI TUYN SINH I HC V CAO NG

Hunh Kim Linh

Trong cc k thi tuyn sinh i hc v cao ng, c mt hay hai cu kh phn loi th sinh v thng c mt cu v bt ng thc.

1) nh l ( Bt ng thc C si):Cho n s thc khng m :1 2 na ,a ,..., a .Ta c :

1 2 n n1 2 n

a a ... a a a ...an

+ + + .

ng thc xy ra khi v ch khi1 2 na a a= = =L .

2) Mt s bt ng thc lin quan n bt ng thc C si :2.1) Cc Bt ng thc dng phn thc

Vi x, y > 0. Ta c :

( )1 1 4 1x y x y+ +

( )( )2

1 4 2xy x y

+

ng thc xy ra khi v ch khi x = y.

Chng minh : (1) v( ) ( )22 x y 4xy + (ng)

Vi x, y, z > 0. Ta c :

( )1 1 1 9 3x y z x y z+ + + +ng thc xy ra khi v ch khi x = y = z.

2.2) Cc bt ng thc dng a thc :( )2 2 2x y z xy yz zx 4+ + + +

( ) ( ) ( )22 2 23 x y z x y z 5+ + + +

( ) ( ) ( )2x y z 3 xy yz zx 6+ + + +

ng thc xy ra khi v ch khi x = y = z.3


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3) MT S BI TON THI I HC :Bi ton 1 : thi tuyn sinh i hc khi A nm 2005

Cho x, y, z l cc s dng tha mn :1 1 1 4x y z

+ + = .

Chng minh rng :1 1 1 1.

2x y z x 2y z x y 2z+ + + + + + + +

Li gii :Cch 1 :

p dng bt ng thc :1 1 4x y x y

+ + vi x, y > 0, ta c :

( )1 1 1 1 1 1 1 1 1 1 1 18 2 4 1x y z x y y z z x x y y z z x

= + + = + + + + + + + + + + Tng t

( )

1 1 1 1 1 1 1 1 12x y y z z x x y x z x y y z y z z x

1 1 14 22x y z x 2y z x y 2z

+ + = + + + + + + + + + + + + + + + + + + + +

T (1) v (2) suy ra

1 1 18 82x y z x 2y z x y 2z + + + + + + + +

1 1 1 1.2x y z x 2y z x y 2z

+ + + + + + + +

ng thc xy ra khi 3x y z .4= = =

Cch 2 :

p dng bt ng thc :1 1 4x y x y

+ + vi x, y > 0, v bt ng thc Csi ta c :

( ) ( ) ( )2x y z x y x z 2 xy xz+ + = + + + +Do :

1 1 1 1 1 12x y z 2 8xy xz xy xz

+ + + + Tng t :

1 1 1 1x 2y z 8 xy yz

+ + +

1 1 1 1x y 2z 8 xz yz

+ + +

Cng v theo v 3 bt ng thc trn ta c :

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5/15

( )1 1 1 1 1 1 1 32x y z x 2y z x y 2z 4 xy yz zx

+ + + + + + + + + +

Mt khc theo bt ng thc Csi

( )1 1 1 1 1 1 1 1 1 1 1 14 42 x y 2 y z 2 z x xy yz zx

= + + + + + + +

T (3) v (4) suy ra :1 1 1 1.

2x y z x 2y z x y 2z+ + + + + + + +

Cch 3 :

p dng bt ng thc Csi cho 4 s dng

( ) 1 1 1 1x x y z 16x x y z + + + + + +

Suy ra1 1 2 1 1

2x y z 16 x y z + + + +

Tng t1 1 1 2 1

x 2y z 16 x y z + + + +

1 1 1 1 2x y 2z 16 x y z

+ + + + Cng v theo v 3 bt ng thc trn ta c :

1 1 1 1.2x y z x 2y z x y 2z

+ + + + + + + +

M rng bi ton 1 :Cho n s thc dng cho trc :1 2 na ,a , aK tha iu kin :

1 2 n

1 1 1ka a a+ + + =L . Vi n > 1 v k > 0 cho trc. Chng minh rng :

+ + + + + + + + + + + + + + +

LL L L L1 1 2 2 n n 2 1 n n 1 1 n n 1 1 2 n 1 n 1 2 n

1 1 1 kma m a m a m a m a ma m a ma m a m m m

Bi ton 2 : thi tuyn sinh i hc khi B nm 2005Chng minh rng : vi mi xR, ta c :

x x xx x x12 15 20

3 4 55 4 3

+ + + + Khi no ng thc xy ra.

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Li p dng bt ng thc Csi

( )2

1 1 1 13 3 2xy yz zx xyz

+ + =

T (1) v (2) suy ra iu cn chng minh.ng thc xy ra khi v ch khi x = y = z = 1.

M rng b i ton 3 :Cho cc s thc dng1 2 na ,a , aK tha mn : 1 2 na .a a 1=L

Chng minh rng :

( ) ( ) ( )

p p p p p p pm m m1 n 1 2 n n 1 n 2 m

q q q1 2 n 1 2 3 n n 1 n 2

1 a a 1 a a 1 a a an n

aa a a a a a a a

+ + + + + + ++ + + L L LLL L L

Trong m 2 l s nguyn dng, p, q l cc s thc ty

Hng dn :p dng bt ng thc Csi cho n s

( )pp p n

1 n 1 1 2 n 11 a a n a .a a + + L L

Bi ton 4 :D B 1 KHI A Nm 2005 :Cho x, y, z l ba s tha x + y + z = 0. Chng minh rng :

3 4 3 4 3 4 6 x y z + + + + +

Li gii :Ta c:

4x x x3 4 1 1 1 4 4 4+ = + + + + = 84x x x3 4 2 4 2. 4 .

Tng t

+ 8y x3 4 2 4 v 8z z3 4 2 4+

Vy+ + + + + + + + + =

38 8 8 8 24x y z x y z x y z x y z3 4 3 4 3 4 2 4 4 4 6 4 .4 .4 6 4 6

ng thc xy ra khi v ch khi x = y = z = 0.

Bi ton 5 :D B 2 KHI A Nm 2005 :

Chng minh rng : vi mi x, y > 0 ta c :( )2

91 1 1 256 + + +

y x

x y

.

ng thc xy ra khi no?Li gii :

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Ta c:3

43

x x x x1 x 1 4

3 3 3 3+ = + + + . ng thc xy ra khi v ch khi x = 3

34

3 3y y y y y

1 1 4x 3x 3x 3x 3 .x

+ = + + + . ng thc xy ra khi v ch khi y = 3x = 9

( )

3

4 39 3 3 3 3

1 1 4y y y y y

+ = + + + 2 6

43

9 31 16

y y

+

.

ng thc xy ra khi v ch khi y = 9

Vy ( ) + + + =

2 3 3 64

3 3 3 3y 9 x y 3

1 x 1 1 256 256x y 3 3 .x y

ng thc xy ra khi v ch khi x = 3 v y = 9.

Bi ton 6 :D B 1 KHI B Nm 2005 :

Cho a, b, c l ba s dng tha mn : a + b + c =34 . Chng minh rng :

3 3 33 3 3 3a b b c c a+ + + + + .Khi no ng thc xy ra ?

Li gii :Cch 1:

Ta c :

( ) ( )

( ) ( )

( ) ( )

3

3

3

a 3b 1 1 1a 3b 1.1 a 3b 2

3 3b 3c 1 1 1

b 3c 1.1 b 3c 23 3

c 3a 1 1 1c 3a 1.1 c 3a 23 3

+ + ++ = + +

+ + ++ = + +

+ + ++ = + +

Suy ra

( ) + + + + + + + + + = 3 3 3 1 1 3a 3b b 3c c 3a 4 a b c 6 4. 6 3

3 3 4

Du = xy ra3

a b c 1a b c4

4a 3b b 3c c 3a 1

+ + = = = =

+ = + = + =Cch 2:

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9/15

t33x a 3b x a 3b= + = +

= + = +33y b 3c y b 3c

= + = +33z c 3a z c 3a

( )3 3 3 3

x y z 4 a b c 4. 34+ + = + + = =.Bt ng thc cn chng minhx y z 3 + + .Ta c :

33 3x 1 1 3 x .1.1 3x+ + =3 33y 1 1 3 y .1.1 3y+ + =

33 3

z 1 1 3 z .1.1 3z+ + = ( )9 3 x y z + + (V 3 3 3x y z 3+ + = ).

Vyx y z 3+ +

Hay3 3 3a 3b b 3c c 3a 3+ + + + +

ng thc xy ra khi v ch khi= = =1a b c 4

Bi ton 7 :D B 2 KHI B Nm 2005 :Chng minh rng nu0 1 y x th

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x y y x .

ng thc xy ra khi no?Li gii :

Ta c20 x 1 x x

1 1x y y x x y y x

4 4 + (1)

Theo bt ng thc Cauchy :

+ + =2 21 1 1y x yx 2 yx . x y4 4 4 1x y y x 4

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Du = xy ra =

= = =

2

2

0 y x 1 x 1x x 1

y1 4yx4

Bi ton 8 :D B 1 KHI D Nm 2005 :Cho x, y, z l ba s dng v x yz = 1. Chng minh rng :

2 2 2 31 1 1 2 x y z

y z x+ + + + + .

Li gii : Ta c:

2 2x 1 y x 1 y2 . x1 y 4 1 y 4

+ ++ =+ +

2 2y 1 z y 1 z2 y

1 z 4 1 z 4+ ++ =+ +

2 2z 1 x z 1 x2 z

1 x 4 1 x 4+ ++ =+ +

Cng ba bt ng thc trn v theo v ta c:

( )2 2 2x 1 y y 1 z z 1 x x y z

1 y 4 1 z 4 1 x 4 + + ++ + + + + + + + + +

( ) ( )+ ++ +

+ + + + + + + +2 2 2 3 x y zx y z 3 x y z 3

x y z1 y 1 z 1 x 4 4 4 4

3 3 9 3 6 3.34 4 4 4 4 2

= = =( v 3x y z 3 xyz 3+ + =)

Vy2 2 2x y z 3

1 y 1 z 1 x 2+ + + + +

Bi ton 9 : thi tuyn sinh i hc khi A nm 2006

Cho hai s thcx 0, y 0 thay i v tha mn iu kin:( ) 2 2x y xy x y xy+ = + .

Tm gi tr ln nht ca biu thc 3 31 1A .

x y

= +

Li gii :

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T gi thit suy ra: 2 21 1 1 1 1 .x y x y xy+ = +

t1 1a, bx y

= = ta c: ( )2 2a b a b ab 1+ = +

Khi ( )( ) ( )23 3 2 2A a b a b a b ab a b .= + = + + = +

T (1) suy ra: ( )2

a b a b 3ab.+ = +

V2

a bab

2+

nn ( ) ( )2 23a b a b a b4+ + +

( ) ( )2a b 4 a b 0 0 a b 4 + + +

Suy ra: ( )2A a b 16.= +

Vi 1x y2

= = th A 16.= Vy gi tr ln nht caA l 16.

Bi ton 10 : D b 1 i hc khi A nm 2006Cho cc s thcx,y,z tha mn iu kin:x y z3 3 3 1. + + = Chng minh rng:

x y z x y z

x y z y z x z x y

9 9 9 3 3 3 .43 3 3 3 3 3+ + +

+ ++ + + + +

Li gii : t x y z3 a,3 b,3 c= = = . Ta c:a,b,c 0> v 1 1 1 1 ab bc ca abc.a b c+ + = + + =

Bt ng thc cn chng minh tng ng vi:2 2 2a b c a b c

a bc b ca c ab 4+ ++ + + + +

3 3 3

2 2 2

a b c a b ca abc b abc c abc 4

+ + + +

+ + +

( ) ( ) ( ) ( ) ( ) ( )3 3 3a b c a b c (1).

a b a c b c b a c a c b 4+ + + + + + + + + +

p dng bt ng thc C si ta c

( ) ( ) ( ) ( )

3 3

3a a b a c a a b a c 33 . . a (2)

a b a c 8 8 a b a c 8 8 4+ + + ++ + =+ + + +

( ) ( ) ( ) ( )

3 3

3 b b c b a b b c b a 3

3 . . b (3) b c b a 8 8 b c b a 8 8 4

+ + + ++ + =+ + + +

( ) ( ) ( ) ( )

3 3

3c c a c b c c a c b 33 . . c (4).

c a c b 8 8 c a c b 8 8 4+ + + ++ + =+ + + +

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Cng theo tng v cc bt ng thc (2), (3), (4) ta suy ra

( ) ( ) ( ) ( ) ( ) ( )

3 3 3a b c a b c.a b a c b c b a c a c b 4

+ ++ + + + + + + +

Vy (1) ng v ta c iu phi chng minh.

Bi ton 11 : D b 1 i hc khi B nm 2006Tm gi tr nh nht ca hm s:

211 7

y x 4 1 ,2x x

= + + + vi x 0> .

Li gii :

p dng bt ng thc :( ) ( )( )22 2 2 2a b c d ac bd+ + +

Ta c : ( )

2

2

7 79 7 1 3x x

+ + +

11 1 7y x 32x 2 x

+ + + 9 3 3 15x 6x 2 2 2

= + + + =

Khi x = 3 th y =152 nn gi tr nh nht ca y l152 .

Bi ton 12 : D b 2 i hc khi B nm 2006

Cho hai s dng x, y thay i tha mn iu kinx y 4.+ Tm gi tr nh nht ca biu thc :

2 3

2

3x 4 2 yA .4x y

+ += +

Li gii :

Ta c A =2 3

2 2

3x 4 2 y 3x 1 2 y4x 4 xy y

+ ++ = + + +

A 2x 1 1 y y x y24 x 8 8 2y

+= + + + + +

3 91 2 .2 2

+ + =

Vi x = y = 2 th A =92 .

Vy gi tr nh nht ca A l92

Bi ton 13 : D b i hc khi A nm 2007Cho x, y, z l cc bin s dng. Tm gi tr nh nht ca biu thc:

3 3 3 3 3 33 332 2 2

x y zP 4(x y ) 4(x z ) 4(z x ) 2

y z x

= + + + + + + + +

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Li gii : Vi x, y > 0 ta chng minh :

4(x3 + y3) (x + y)3 () Du = xy rax = yTht vy () 4(x + y)(x2 xy + y2) (x + y)3

4(x2 xy + y2) (x + y)2 do x, y > 0

3(x2

+ y2

2xy) 0 (x y)2

0 (ng)Tng t ta c

4(y3 + z3) (y + z)3 Du = xy ray = z4(z3 + x3) (z + x)3 Du = xy raz = x

Do

( ) ( ) ( )( )3 3 3 3 3 3 33 3 34 x y 4 y z 4 z x 2 x y z 6 xyz+ + + + + + +

Ta li c

3222 xyz6

xz

zy

yx2

++ Du = xy rax = y = z

Suy ra

12xyz1xyz6P

33

+

Du = xy ra

==

=

zyx

1x y z x = y = z = 1

Vy minP = 12 khi x = y = z = 1

Bi ton 14 : D b i hc khi D nm 2007Cho a, b > 0 tha mnab + a + b = 3.Chng minh :

23

baba

ab1a

b31b

a3 22 +++++++ .

Li gii : T gi thita, b > 0 v ab + a + b = 3. Suy ra:

ab 3 (a b)= + , (a+1)(b+1) = ab +a +b + 1 = 4Bt ng thc cho tng ng vi

2 2 3 3a(a 1) 3b(b 1) 3a b 12 (a 1)(b 1) a b

+ + ++ + + + + +

( )( ) 1

ba

3ba

4

3ba

4

3

2

3ba 2222

+++++++

( ) ( ) ( ) 4ba

12ba3ba36ba4 2222

+++++++

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( )2 2 12a b 3 a b 10a b + + + + (*)

t 2 2x a b 0 x (a b) 4ab 4(3 x)= + > = + = 2x 4x 12 0 x 6hay x 2 + x 2 ( v x > 0)

2 2 2x a b 2ab= + + 2 2 2 2a b x 2(3 x) x 2x 6 + = = +

Khi bt ng thc (*) thnh2 12x x 4 0

x + , vi x 2

3 2x x 4x 12 0 + , vi x 2 ( )( )2x 2 x x 6 0 + + , vi x 2 (hin nhin ng)Vy bt ng thc cho c chng minh.

4) Mt s bi ton cc bn t lm :Bi ton 1 5 : Cho x, y > 0 tha : x + y +z = 1 . Chng minh : xyz y x 16+Bi ton 16 : Chng minh rng vi a + b + c = 0 th cbacba 222888 ++++

Bi ton 17 : Cho a, b, c >0 : a + b + c = 1 . Chng minh(1 + a)(1 + b)(1 + c) 8(1 a)(1 b)(1 c)

Bi ton 18 : Cho a, b, c > 0. Chng minh :

bac

acb

cba

bac

acb

cba

+++++ 0 tha : 21 11 11 1 +++++ cba . Chng minh : 81abc

Bi ton 20 : Chng minh rng : vi a> b 0 th ( )( )3

14

2 ++

bbaa

Bi ton 21 : Chng minh : ( )333444

21

cbaba

cac

bcb

a +++++++vi a,b,c > 0

Bi ton 22 : Cho cc s thc dnga,b,c,d tha : 13333 =+++ d cba . Chng minh :

3443

333

2

333

2

333

2

333

2

++

+++

+++

+++ bca

d d ba

cd ca

bd cb

a

Bi ton 23 : Cho cc s thc dnga,b,c,d

tha :a + b + c + d = 1 . Chng minh :( ) ( ) ( ) ( )

2562333

12333

12333

12333

12222 ++

+++

+++

+++

=abcd d bacd acbd bca

P

Bi ton 24 : Cho a. b. c > 0. Chng minh : 3 abccba

ac

cb

ba ++++

Bi ton 25 : Cho a, b, c, d > 0. Chng minh :d cbad cba ++++++

6416411

Bi ton 26 : Cho a, b, c > 0 Chng minh :

cbacbacbacba 24

24

24111

++++++++++Bi ton 27 : Cho x, y, z l cc s thc dng tha tch xyz = 1. Chng minh:

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( ) ( ) ( ) ( ) ( ) ( )

3 3 3 31 1 1 1 1 1 4

x y z y z z x x y

+ + + + + + + +

Bi ton 28 : Cho a, b, c > 0 Chng minh :2111

34111

+++++++ accbbacabcab

Bi ton 29 : Cho a.b > 0 v 1+ba . Chng minh : 1432 22 ++

baabBi ton 30 : Cho a, b, c > 0. Chng minh :

( )( ) ( )( ) ( )( )( )23

221

221

221

cbaacabcacbbcba ++++++++++

Bi ton 31 : Cho a, b, c > 0 tha a + b + c = 1 . Chng minh

92

12

12

1222 +

++

++ abccabbca

Bi to n 32 : Cho x, y, z > 0 tha : x + y +z = 1. Chng minh :

43

111 +++++ z z

y y

x x

Bi ton 33 : Chng minh vi a,b,c > 0 th : ++++ cbacbacba 111333888

Bi ton 34 : Cho a, b, c > 0 tha :ab + cb + ca = 1. Chng minh :( )cbacba +++++++ 2111 222

Bi ton 35 : Cho a,b,c > 0 tha a+b+c =1. Chng minh :

3100111 222

++

++

+

cc

bb

aa

Bi ton 36 : Chng minh : vi mi x , y , z > 0 ta lun c :3 3 3 x y z

x y z yz zx xy

+ + + +

Bi ton 37 : Chng minh : vi a, b, c > 0 tha a + b + c = 3 ta c :

2 2 2

321 1 1

a b cb c a

+ + + + +

Bi ton 38 : Cho a, b, c 0. Chng minh :2 2 2

2 2 2 12 2 2a b c

a b b c c a+ +

+ + +

Bi ton 39 : Cho a, b, c, d dng vi a + b + c + d = 4. Hy Chng minh

a) 2 2 2 2 21 1 1 1a b c d b c d a

+ + + + + + +

b) 2 2 2 2 21 1 1 1a b c d b c c d d a a b

+ + + + + + +

c) 2 2 2 21 1 1 1 4

1 1 1 1a b c d

b c d a+ + + ++ + +

+ + + +

Bi ton 40 : Cho a, b, c > 0 v a + b + c = 3. Hy chng minh :a b c ab cb ca+ + + +

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