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Chuyn : BT NG THC Thy: L vn nh

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PPHHNN II:: LL TTHHUUYYTT CCNN BBNN

1. TNH CHT BT NG THC

TT iu kin Tnh cht c im1. a b

a cb c

> >> Bc cu

2. a b a c b c< + < + Cng 2 v BT vi mt s bt k3. c > 0 a b ac bc> > 4. c < 0 a b ac bc> <

Nhn hay chia 2 v BT vi mt s

5. a ba c b d

c d

> + > +

> Cng theo v 2 BT cng chiu

6. 0

0b

d

>>

a b ac bd c d

> >

> Nhn theo v 2 BT cng chiu

7.n nguyn dng

2 1 2 1

2 20

n n

n n

a b a b

a b a b

+ + < a a x a

12.0a >

x ax a

x a

13. a b a b a b + +

BT tr tuyt i

14. N> M A N < Suy t 2. v 4.15. 0

0N> >

>

A A

N< Suy t 9. v 3.


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2. CC BT NG THC THNG DNG BT NG THC CSI:

*Du hiu s dng: S dng khi cc s khng mL: Cho n s khng m a1 , a2 , . . . , an Ta c:

1 2

1 2 1 2 1 2

1 2 1 21 2 1 2

1 2

...

... ... ...

... ...... ...

ng thc xy ra ...

n

nn

n n n

n

n nnn n

n

a a a

a a a n a a a hay a a an

a a a a a aa a a hay a a a

n n

a a a

+ + +

+ + +

+ + + + + +

= = =

HQ: Cho n s dng a1 , a2 , . . . , an Ta c:

+ + + + + +

+ + + + + + = = =

2

2

1 2 1 2 1 2 1 2

1 2

1 1 1 1 1 1 1 1... ...

... ...ng thc xy ra ...

n n n n

n

nhay

a a a a a a a a a n a a aa a a

BT NG THC BUNHIAKPSKI:

*Du hiu s dng: Tng ca cc bnh phngL: Cho n cp s a1 , a2 , . . . , an v b1 , b2 , . . . , bn . Ta c:

( ) ( ) ( )

2 2 2 2 2 21 1 2 2 1 2 1 2

2 2 2 2 2 2 21 1 2 2 1 2 1 2

... ... ...

... ... ...

n n n n

n n n n

a b a b a b a a a b b b

hay a b a b a b a a a b b b

+ + + + + + + + +

+ + + + + + + + +

ng thc xy ra 1 21 2

... n

n

a a a

b b b = = =

*Ch : A & ( i k>0)A th A A k k A k V

HQ: Trong khng gian n chiu cho 2 vect 1 2 1 2( ; ;...; ) , ( ; ;...; )n nu a a a v b b b= =

: . .Tac u v u v

ng thc xy ra Hai vect trn cng phng

Gii thch: ( ): . . .cos ,Theonh ngha uv u v u v =

BT NG THC VECT ( Phng php hnh hc )

*Du hiu s dng: ng ca cc bnh phngT

1/ . ng thc xy ra Hai vect cng hnga b a b+ +

M rng: a b c a b c + + + +

2/ . . . ng thc xy ra Hai vect cng phngu v u v


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PPHHNN IIII:: LLUUYYNN TTPP CCNN BBNN

II.. CCHHNNGG MMIINNHH BBTT DDAA VVOO NNHH NNGGHHAA VV TTNNHH CCHHTT CCBBNN

1. Cho a, b > 0 chng minh: + +

33 3

a b a b2 2 (*)

Gii:

(*) + +

33 3a b a b0

2 2 ( )( )+

23a b a b 0

8. PCM.

2. Chng minh:+ +

2 2a b a b

2 2()

Gii: a + b 0 , () lun ng.

a + b > 0 , () + + +

2 2 2 2a b 2ab a b

04 2

( )

2

a b0

4, ng.

Vy:+ +

2 2a b a b

2 2.

3. Cho a + b 0 chng minh:+ +

3 3

3a b a b

2 2

Gii:+ +

3 3

3a b a b

2 2

( )+ +

3 3 3a b a b

8 2

( )( ) 2 23 b a a b 0 ( ) ( ) + 23 b a a b 0 , PCM.

4. Cho a, b > 0 . Chng minh: + +a b

a bb a

()

Gii:

() + +a a b b a b b a ( ) ( ) a b a a b b 0 ( )( ) a b a b 0 ( ) ( ) +

2a b a b 0 , PCM.


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5. Chng minh: Vi a b 1: + ++ +2 2

1 1 2

1 ab1 a 1 b

Gii:

+ ++ +2 2

1 1 2

1 ab1 a 1 b +

+ ++ +2 21 1 1 1

01 ab 1 ab1 a 1 b

( ) ( ) ( ) ( ) + + + + +

2 2

2 2ab a ab b 01 a 1 ab 1 b 1 ab ( )( )( ) ( )( )( ) + + + + +2 2

a b a b a b 01 a 1 ab 1 b 1 ab

+ + + 2 2b a a b

01 ab 1 a 1 b

( )( )

+ + + +

2 2

2 2

b a a ab b ba0

1 ab 1 a 1 b

( ) ( )

( )( ) ( )

+ + +

2

2 2

b a ab 10

1 ab 1 a 1 b, PCM.

V : a b 1 ab 1 ab 1 0.

6. Chng minh: ( )+ + + + +2 2 2a b c 3 2 a b c ; a , b , c RGii:

( )+ + + + +2 2 2a b c 3 2 a b c ( ) ( ) ( ) + + 2 2 2

a 1 b 1 c 1 0 . PCM.

7. Chng minh: ( )+ + + + + + +2 2 2 2 2a b c d e a b c d e Gii:

( )+ + + + + + +2 2 2 2 2a b c d e a b c d e

+ + + + + + + 2 2 2 22 2 2 2a a a aab b ac c ad d ae e 04 4 4 4

+ + +

2 2 2 2a a a a

b c d e 02 2 2 2

. PCM

8. Chng minh: + + + +2 2 2x y z xy yz zx Gii:

+ + + +2 2 2x y z xy yz zx + + 2 2 22x 2y 2z 2xy 2yz 2zx 0

( ) ( ) ( ) + + 2 22x y x z y z 0


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9. a. Chng minh:+ + + +

a b c ab bc ca

; a,b,c 03 3

b. Chng minh:+ + + +

22 2 2a b c a b c

3 3

Gii:

a. Chng minh: + + + + a b c ab bc ca ; a,b,c 03 3

+ + + +2 2 2a b c ab bc ca

+ + + + + + + + +

=

2 2 2 2a b c a b c 2ab 2bc 2ca ab bc ca

3 9 3

+ + + +

a b c ab bc ca

3 3

b. Chng minh:

+ + + +

22 2 2a b c a b c

3 3

( ) ( )+ + = + + + + +2 2 2 2 2 2 2 2 23 a b c a b c 2 a b c

( ) ( ) + + + + + = + + 22 2 2a b c 2 ab bc ca a b c

+ + + +

22 2 2a b c a b c

3 3

10. Chng minh: + + +2

2 2a b c ab ac 2bc4

Gii:

+ + +2

2 2a b c ab ac 2bc4

( ) + + 2

2 2a a b c b c 2bc 04

( )

2a

b c 02

.

11. Chng minh: + + + +2 2a b 1 ab a b Gii:

+ + + +2 2a b 1 ab a b + + 2 22a 2b 2 2ab 2a 2b 0

+ + + + + + + 2 2 2 2

a 2ab b a 2a 1 b 2b 1 0 ( ) ( ) ( ) + + 2 2 2

a b a 1 b 1 0 .

12. Chng minh: + + +2 2 2x y z 2xy 2xz 2yz Gii:

+ + +2 2 2x y z 2xy 2xz 2yz + + + 2 2 2x y z 2xy 2xz 2yz 0 (x y + z)2 0.


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13. Chng minh: + + + + +4 4 2 2x y z 1 2xy(xy x z 1) Gii:

+ + + + +4 4 2 2x y z 1 2x(xy x z 1) + + + + 4 4 2 2 2 2x y z 1 2x y 2x 2xz 2x 0

( ) ( ) ( ) + + 2 2 22 2x y x z x 1 0 .

14. Chng minh: Nu a + b 1 th: + 3 3 1a b

4

Gii: a + b 1 b 1 a b3 = (1 a)3 = 1 a + a2 a3

a3 + b3 = +

21 1 1

3 a2 4 4

.

15. Cho a, b, c l s o di 3 cnh ca 1 tam gic. Chng minh:a. ab + bc + ca a2 + b2 + c2 < 2(ab + bc + ca).b. abc (a + b c)(a + c b)(b + c a)c. 2a2b2 + 2b2c2 + 2c2a2 a4 b4 c4 > 0

Gii:a. ab + bc + ca a2 + b2 + c2 < 2(ab + bc + ca). ab + bc + ca a2 + b2 + c2 (a b)2 + (a c)2 + (b c)2 > > > a b c , b a c , c a b

> +2 2 2a b 2bc c , > +2 2 2b a 2ac c , > +2 2 2c a 2ab b a2 + b2 + c2 < 2(ab + bc + ca).

b. abc (a + b c)(a + c b)(b + c a)

( )> 22 2a a b c ( )( )> + + 2a a c b a b c

( )> 22 2b b a c ( )( )> + + 2b b c a a b c

( )> 22 2c c a b ( )( )> + + 2c b c a a c b

( ) ( ) ( )> + + + 2 2 22 2 2a b c a b c a c b b c a

( )( )( )> + + + abc a b c a c b b c a

c. 2a2b2 + 2b2c2 + 2c2a2 a4 b4 c4 > 0 4a2b2 + 2c2(b2 + a2) a4 b4 2a2b2 c4 > 0 4a2b2 + 2c2(b2 + a2) (a2 + b2)2 c4 > 0 (2ab)2 [(a2 + b2) c2]2 > 0 [c2 (a b)2][(a + b)2 c2] > 0 (c a + b)(c + a b)(a + b c)(a + b + c) > 0 . ng

V a , b , c l ba cnh ca tam gic c a + b > 0 , c + a b > 0 , a + b c > 0 , a + b + c > 0.


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IIII.. CCHHNNGG MMIINNHH BBTT DDAA VVOO BBTT CCSSII

1. Chng minh: + + + (a b)(b c)(c a) 8abc ; a,b,c 0 Gii: p dng bt ng thc Csi cho hai s khng m:

+ a b 2 ab , + b c 2 bc , + a c 2 ac

( )( )( )+ + + =2 2 2a b b c a c 8 a b c 8abc .

2. Chng minh: + + + + 2 2 2(a b c)(a b c ) 9abc ; a,b,c 0 Gii: p dng bt ng thc Csi cho ba s khng m:

+ + 3a b c 3 abc , + + 32 2 2 2 2 2a b c 3 a b c

( )( )+ + + + =32 2 2 3 3 3a b c a b c 9 a b c 9abc .

3. Chng minh: ( )( )( ) ( )+ + + +331 a 1 b 1 c 1 abc vi a , b , c 0

Gii:

( )( )( )+ + + = + + + + + + +1 a 1 b 1 c 1 a b c ab ac bc abc.

+ + 3a b c 3 abc , + + 3 2 2 2ab ac bc 3 a b c

( )( )( ) ( )+ + + + + + = +33 2 2 23 31 a 1 b 1 c 1 3 abc 3 a b c abc 1 abc

4. Cho a, b > 0. Chng minh: +

+ + +

m mm 1a b1 1 2

b a, vi m Z+

Gii:

+ + + + + + = + + =

m m m m mm m 1a b a b b a1 1 2 1 . 1 2 2 2 4 2

b a b a a b

5. Chng minh: + + + + bc ca ab

a b c ; a,b,c 0a b c

Gii: p dng BT Csi cho hai s khng m:

+ =2bc ca abc

2 2ca b ab

, + =2bc ba b ac

2 2ba c ac

, + =2ca ab a bc

2 2ab c bc

+ + + +bc ca ab

a b ca b c

.


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6. Chng minh:+

6 9

2 3x y 3x y 16 ; x,y 04

Gii:+

6 9

2 3x y 3x y 16 ; x,y 04

+ + 6 9 2 3x y 64 12x y ( ) ( )+ + 3 32 3 3 2 3x y 4 12x y

p dng BT Csi cho ba s khng m:

( ) ( )+ + =3 32 3 3 2 3 2 3x y 4 3x y 4 12x y .

7. Chng minh: + +

4 22

12a 3a 1

1 a.

Gii:

+ +

4 22

12a 3a 1

1 a + + + +

+4 4 2 2

2

1a a a 1 4a

1 a.

p dng BT Csi cho 4 s khng m: +

+

4 4 22

1a , a , a 1,

1 a

( )+ + + + + =+ +

4 4 2 4 4 2 242 2

1 1a a a 1 4 a a a 1 4a

1 a 1 a

8. Chng minh: ( )> 1995a 1995 a 1 , a > 0Gii:

( )> 1995a 1995 a 1

> + >1995 1995a 1995a 1995 a 1995 1995a

+ > + = + + + + =

19951995 1995 1995 1995

1994 soa 1995 a 1994 a 1 1 ... 1 1995 a 1995a

9. Chng minh: ( ) ( ) ( )+ + + + + 2 2 2 2 2 2a 1 b b 1 c c 1 a 6abc .Gii:

( ) ( ) ( )+ + + + + = + + + + +2 2 2 2 2 2 2 2 2 2 2 2 2 2 2a 1 b b 1 c c 1 a a a b b b c c c a p dng bt ng thc Csi cho 6 s khng m:

+ + + + + =62 2 2 2 2 2 2 2 2 6 6 6a a b b b c c c a 6 a b c 6abc

10. Cho a , b > 0. Chng minh: + + + + + + +2 2 2 2 2 2

a b c 1 1 1 1

2 a b ca b b c a c

Gii:

=+2 2a a 1

2ab 2ba b, =

+2 2b b 1

2bc 2cb c, =

+2 2c c 1

2ac 2aa c

Vy:

+ + + + + + +2 2 2 2 2 2

a b c 1 1 1 1

2 a b ca b b c a c


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11. Cho a , b 1 , chng minh: + ab a b 1 b a 1.Gii:

( ) ( )= + = + a a 1 1 2 a 1, b b 1 1 2 b 1

ab 2b a 1 , ab 2a b 1

+ ab a b 1 b a 1

12. Cho x, y, z > 1 v x + y + z = 4. Chng minh: xyz 64(x 1)(y 1)(z 1)Gii:

( ) ( )= + = + + + x x 1 1 x 1 x y z 3

( ) ( ) ( ) ( ) ( ) ( )( )= + + + 24x 1 x 1 y 1 z 1 4 x 1 y 1 z 1

Tng t: ( )( ) ( ) 24y 4 x 1 y 1 z 1 ; ( )( )( )

24z 4 x 1 y 1 z 1

xyz 64(x 1)(y 1)(z 1).

13. Cho a > b > c, Chng minh: ( )( ) 3

a 3 a b b c c .Gii:

( ) ( ) ( )( )= + + 3a a b b c c 3 a b b c c

14. Cho: a , b , c > 0 v a + b + c = 1. Chng minh:a) b + c 16abc.b) (1 a)(1 b)(1 c) 8abc

c)

+ + +

1 1 11 1 1 64

a b c

Gii:

a) b + c 16abc.

+

2b c

bc2

( )+

= =

2 22b c 1 a

16abc 16a 16a 4a 1 a2 2

( ) ( )( ) ( ) ( ) = = = + 2 224a 1 a 1 a 4a 4a 1 a 1 1 2a 1 a b c

b) (1 a)(1 b)(1 c) 8abc

(1 a)(1 b)(1 c) = (b + c)(a + c)(a + b) =2 bc.2 ac.2 ab 8abc

c)

+ + +

1 1 11 1 1 64

a b c

+ + +

+ =

4 21 a a b c 4 a bc1

a a a

+ 4 21 4 ab c

1b b

+ 4 21 4 abc

1c c

+ + +

1 1 11 1 1 64

a b c


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15. Cho x > y > 0 . Chng minh:( )

+

1x 3

x y y

Gii:

( )( )

( )

( )

= + + =

3

x y y1VT x y y 3 3

x y y x y y

16. Chng minh:

a)+

+

2

2

x 22

x 1,x R b)

+

x 86

x 1, x > 1 c)

+

+

2

2

a 54

a 1

Gii:

a)+

+

2

2

x 22

x 1 + +2 2x 2 2 x 1 + + +2 2x 1 1 2 x 1

b) +

x 8x 1

= + = + = x 1 9 9 9x 1 2 x 1 6

x 1 x 1 x 1

c. ( ) ( )+ + + = +2 2 2a 1 4 2 4 a 1 4 a 1+

+

2

2

a 54

a 1

17. Chng minh:+ +

+ + >+ + +

ab bc ca a b c; a, b, c 0

a b b c c a 2

Gii: V : + a b 2 ab

=+

ab ab ab

a b 22 ab, =

+bc bc bc

b c 22 bc, =+

ac ac ac

a c 22 ac

+ + + +a b c ab bc ca , da vo: + + + +2 2 2a b c ab bc ca .

+ + + +

+ + + + +

ab bc ca ab bc ac a b c

a b b c c a 2 2

18. Chng minh: + + +

2 2

4 4

x y 1

41 16x 1 16y, x , y R

Gii:

( )

= =+ +

2 2 2

4 2 2

x x x 1

81 16x 2.4x1 4x

( )

= =+ +

2 2 2

4 2 2

y y y 1

81 16y 2.4y1 4y

+ + +

2 2

4 4

x y 1

41 16x 1 16y


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19. Chng minh: + + + + +a b c 3

b c a c a b 2; a , b , c > 0

Gii: t X = b + c , Y = c + a , Z = a + b.

a + b + c = 12

(X + Y + Z)

+ + +

= = =Y Z X Z X Y X Y Z

a , b , c2 2 2

+ + = + + + + + + + +

a b c 1 Y X Z X Z Y3

b c a c a b 2 X Y X Z Y Z [ ] + + =

1 32 2 2 3

2 2.

Cch khc:

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

a b c a b c1 1 1 3

b c a c a b b c a c a b

( ) ( ) ( )[ ] = + + + + + + + + + +

1 1 1 1a b b c c a 3

2 b c a c a b

p dng bt ng thc Csi cho ba s khng m:

( ) ( ) ( )[ ]

+ + + + + + + = + + +

1 1 1 1 9 3a b b c c a 32 b c a c a b 2 2

20. Cho a , b , c > 0. C/m: + + + + + + + +3 3 3 3 3 3

1 1 1 1

abca b abc b c abc c a abc

Gii:

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

( ) ( )+ + + + = + +3 3a b abc a b ab abc ab a b c , tng t

( ) ( )+ + + + = + +3 3b c abc b c bc abc bc a b c

( ) ( )+ + + + = + +3 3c a abc c a ca abc ca a b c

( ) ( ) ( ) + + + + = + + + + + + + + 1 1 1 1 a b cVT

ab a b c bc a b c ca a b c a b c abc

21. p dng BT Csi cho hai s chng minh:

a. + + + 4a b c d 4 abcd vi a , b , c , d 0 (Csi 4 s)

b. + + 3a b c 3 abc vi a , b , c 0 , (Csi 3 s )Gii:

a. + + + 4a b c d 4 abcd vi a , b , c , d 0 (Csi 4 s)

+ + a b 2 ab , c d 2 cd

( ) ( )+ + + 4a b cd 2 ab cd 2 2 ab. cd 4 abcd b. + + 3a b c 3 abc vi a , b , c 0 , (Csi 3 s )

+ + + +

+ + + 4a b c a b c

a b c 4. abc3 3

+ + + +

4a b c a b c

abc3 3

+ + + +

4a b c a b c

abc3 3

+ +

3a b c

abc3

+ + 3a b c 3 abc .


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22. Chng minh: + + + +3 3 3 2 2 2a b c a bc b ac c ab ; a , b , c > 0Gii:

+ 3 2a abc 2a bc , + 3 2b abc 2b ac , + 3 2c abc 2c ab

( )+ + + + +3 3 3 2 2 2a b c 3abc 2 a bc b ac c ab

( ) ( )+ + + +3 3 3 2 2 22 a b c 2 a bc b ac c ab , v : + + 3 3 3a b c 3abc Vy: + + + +3 3 3 2 2 2a b c a bc b ac c ab

23. Chng minh: + + 3 942 a 3 b 4 c 9 abc Gii: p dng bt ng thc Csi cho 9 s khng m:

= + + + + + + + + 3 3 3 94 4 4 4VT a a b b b c c c c 9 abc

T BI 24 N BI 38 C TH DNG PHNG PHP O HM

24. Cho = +x 18

y2 x

, x > 0. nh x y t GTNN.

Gii:

p dng BT Csi cho hai s khng m: = + =x 18 x 18

y 2 . 62 x 2 x

Du = xy ra = = = 2x 18

x 36 x 62 x

, chn x = 6.

Vy: Khi x = 6 th y t GTNN bng 6

25. Cho = + >

x 2y ,x 1

2 x 1. nh x y t GTNN.

Gii:

= + +

x 1 2 1y

2 x 1 2

p dng bt ng thc Csi cho hai s khng m

x 1 2

,2 x 1

:

= + + + =

x 1 2 1 x 1 2 1 5y 2 .

2 x 1 2 2 x 1 2 2

Du = xy ra ( )=

= = =

2 x 3x 1 2x 1 4

x 1(loai)2 x 1

Vy: Khi x = 3 th y t GTNN bng 52


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26. Cho = + > +

3x 1y , x 1

2 x 1. nh x y t GTNN.

Gii:

+

= + +

3(x 1) 1 3y

2 x 1 2

p dng bt ng thc Csi cho hai s khng m( )+

+3 x 1 1

,2 x 1 :

( ) ( )+ += + =

+ +3 x 1 1 3 3 x 1 1 3 3

y 2 . 62 x 1 2 2 x 1 2 2

Du = xy ra ( )

( )

= + = + =

+=

2

6x 1

3 x 1 1 2 3x 12 x 1 3 6

x 1(loai)3

Vy: Khi = 6x 1

3

th y t GTNN bng 3

6

2

27. Cho = + >

x 5 1y ,x

3 2x 1 2. nh x y t GTNN.

Gii:

= + +

2x 1 5 1y

6 2x 1 3

p dng bt ng thc Csi cho hai s khng m

2x 1 5

,6 2x 1

:

+= + + + =

2x 1 5 1 2x 1 5 1 30 1

y 2 .6 2x 1 3 6 2x 1 3 3

Du = xy ra ( )

+= = =

+=

2

30 1x

2x 1 5 22x 1 306 2x 1 30 1

x (loai)2

Vy: Khi+

=30 1

x2

th y t GTNN bng+30 1

3

28. Cho = +x 5

y1 x x

, 0 < x < 1 . nh x y t GTNN.

Gii:

( ) +

= + = + + + = + x 5 1 x 5x x x 1 x 1 x

f(x) 5 5 2 5 5 2 5 51 x x 1 x x 1 x x

Du = xy ra

= = =

2x 1 x x 5 5

5 5 x1 x x 1 x 4

(0 < x < 1)

Vy: GTNN ca y l +2 5 5 khi

=5 5

x4


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29. Cho+

=3

2

x 1y

x, x > 0 . nh x y t GTNN.

Gii:

+

= + = + + =3

32 2 2 2 3

x 1 1 x x 1 x x 1 3x 3

2 2 2 2 4x x x x

Du = xy ra = = 2x x 12 2 x

= 3x 2 .

Vy: GTNN ca y l3

3

4khi = 3x 2

30. Tm GTNN ca+ +

=2x 4x 4

f(x)x

, x > 0.

Gii:

+ +

= + + + =

2x 4x 4 4 4x 4 2 x. 4 8x x x

Du = xy ra =4

xx

x = 2 (x > 0).

Vy: GTNN ca y l 8 khi x = 2.

31. Tm GTNN ca = +2 32

f(x) xx

, x > 0.

Gii:

+ = + + + + =

322 2 2 22 5

3 3 3 3 52 x x x 1 1 x 1 5x 5

3 3 3 3 27x x x x

Du = xy ra = =2

53

x 1x 3

3 x x = 2 (x > 0).

Vy: GTNN ca y l5

5

2 7khi = 5x 3 .

32. Tm GTLN ca f(x) = (2x 1)(3 5x)

Gii: f(x) = 10x2 + 11x 3 =

= +

22 11x 11 1 110 x 3 10 x

10 20 40 40

Du = xy ra =1 1

x2 0

Vy: Khi =1 1

x2 0

th y t GTLN bng 14 0

.


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33. Cho y = x(6 x) , 0 x 6 . nh x y t GTLN.Gii: p dng BT Csi cho 2 s khng m x v 6 x (v 0 x 6):

( ) ( )= + 6 x 6 x 2 x 6 x x(6 x) 9 Du = xy ra x = 6 x x = 3 Vy: Khi x = 3 th y t GTLN bng 9.

34. Cho y = (x + 3)(5 2x) , 3 x 52

. nh x y t GTLN

Gii:

y = (x + 3)(5 2x) = 12

(2x + 6)(5 2x)

p dng BT Csi cho 2 s khng m 2x + 6 v 5 2x ,

53 x

2:

( ) ( ) ( )( )= + + + 11 2x 6 5 2x 2 2x 6 5 2x 12

(2x + 6)(5 2x) 1218

Du = xy ra 2x + 6 = 5 2x = 1x 4

Vy: Khi = 1

x4

th y t GTLN bng121

8.

35. Cho y = (2x + 5)(5 x) , 5

x 52

. nh x y t GTLN

Gii:

y = (2x + 5)(5 x) = 12

(2x + 5)(10 2x)

p dng BT Csi cho 2 s khng m 2x + 5 , 10 2x , 5 x 52 :

( ) ( ) ( )( )+ + + 2x 5 10 2x 2 2x 5 10 2x 12

(2x + 5)(10 2x) 6 2 5

8

Du = xy ra 2x + 5 = 10 2x = 5x4

Vy: Khi =5

x4

th y t GTLN bng625

8

36. Cho y = (6x + 3)(5 2x) , 1

2

x 5

2

. nh x y t GTLN

Gii: y = 3(2x + 1)(5 2x)

p dng BT Csi cho 2 s khng m 2x + 1 , 5 2x ,

1 5x

2 2:

( ) ( ) ( )( )+ + + 2x 1 5 2x 2 2x 1 5 2x (2x + 1)(5 2x) 9 Du = xy ra 2x + 1 = 5 2x x = 1 Vy: Khi x = 1 th y t GTLN bng 9.


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37. Cho =+2x

yx 2

. nh x y t GTLN

Gii:

+ =2 22 x 2 2x 2x 2 + 2

1 x

2 2 2 x

1y

2 2

Du = xy ra = 2

x 2 & x > 0 x= 2 Vy: Khi =x 2 th y t GTLN bng

1

2 2.

38. Cho( )

=+

2

32

xy

x 2. nh x y t GTLN

Gii:

+ = + + 32 2 2x 2 x 1 1 3 x .1.1 ( )

( )+

+

232 232

x 1x 2 27x

27x 2

Du = xy ra = =

2

x 1 x 1 Vy: Khi = x 1 th y t GTLN bng

1

27.

IIIIII.. CCHHNNGG MMIINNHH BBTT DDAA VVOO BBTT BBUUNNHHIIAACCPPXXKKII

1. Chng minh: (ab + cd)2 (a2 + c2)(b2 + d2) BT BunhiacopxkiGii :

(ab + cd)2 (a2 + c2)(b2 + d2)

+ + + + +2 2 2 2 2 2 2 2 2 2 2 2a b 2abcd c d a b a d c b c d

+ 2 2 2 2a d c b 2abcd 0 ( ) 2

ad cb 0 .

2. Chng minh: + sin x cos x 2 Gii : p dng BT Bunhiacopski cho 4 s 1 , sinx , 1 , cosx :

+ =sinx cosx ( ) ( )+ + + =2 2 2 21. sinx 1. cosx 1 1 sin x cos x 2

3. Cho 3a 4b = 7. Chng minh: 3a2 + 4b2 7.Gii :

p dng BT Bunhiacopski cho 4 s 3 , 3 a , 4 , 4 b :

( )( )+ = + + +2 23a 4b 3. 3a 4. 4b 3 4 3a 4b 3a2 + 4b2 7.


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4. Cho 2a 3b = 7. Chng minh: 3a2 + 5b2725

47.

Gii :

= 2 3

2a 3b 3a 5b3 5

p dng BT Bunhiacopski cho 4 s

2 3

, 3 a , , 5b3 5 :

( )

+ +

2 22 3 4 93 a 5b 3a 5b3 53 5

3a2 + 5b27 3 5

4 7.

5. Cho 3a 5b = 8. Chng minh: 7a2 + 11b22464

13 7.

Gii :

= 3 5

3a 5b 7 a 11b7 11

p dng BT Bunhiacopski cho 4 s 3 5, 7 a , , 11b7 11

:

( )

+ +

2 23 5 9 257 a 11b 7a 11b7 117 11

7a2 + 11b22 4 6 4

13 7.

6. Cho a + b = 2. Chng minh: a4 + b4 2.Gii : p dng BT Bunhiacopski:

( )( )

= + + +

2 2

2 a b 1 1 a b a2

+ b

2

2 ( ) ( )( ) + + +2 2 4 42 a b 1 1 a b a4 + b4 2

7. Cho a + b 1 Chng minh: + 2 21

a b2

Gii :

( ) ( ) + + + + 2 2 2 2 2 21

1 a b 1 1 a b a b2


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PPHHNN IIIIII.. TTHHII II HHCCNM 1999

1. (H Y Dc TP HCM 1999)

Cho 3 s a, b, c khc 0. Chng minh: + + + +2 2 2

2 2 2

a b c a b c

b c ab c a

Gii:p dng BT Csi ta c:

* + + =2 2 2 2 2 2

32 2 2 2 2 2

a b c a b c3 . . 3

b c a b c a(1)

* + 2

2

a a1 2

bb; +

2

2

b b1 2

cc; +

2

2

c c1 2

aa

+ + + +

2 2 2

2 2 2

a b c a b c2 3

b c ab c a(2)

Kt hp (1) v (2) ta c: + + + +

2 2 2

2 2 2

a b c a b c2 2

b c ab c a

+ + + +2 2 2

2 2 2

a b c a b c

b c ab c a

2. (H Hng hi 1999)Cho x, y, z 0 v x + y + z 3. Chng minh rng:

+ + + ++ + ++ + +2 2 2

x y z 3 1 1 1

2 1 x 1 y 1 z1 x 1 y 1 z

Gii:

Do (x 1)2 0 nn x2 + 1 2x + 22x

1 x 1

Tng t ta cng c:+ 22y

1 y 1;

+ 22 z

1 z 1

Do :+ 22 x

1 x+

+ 22 y

1 y+

+ 22 z

1 z 3

Hay: + + + + +2 2 2x y z 3

21 x 1 y 1 z(1)

p dng BT Csi cho 3 s khng m ta c:

+ ++ + +

=+ + + + + +3 3

1 1 1

1 11 x 1 y 1 z

3 (1 x)(1 y)(1 z) (1 x)(1 y)(1 z)

+ + ++ +

+ + +

33 (1 x)(1 y)(1 z)1 1 1

1 x 1 y 1 z

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

3 2

+ ++ + +3 1 1 1

2 1 x 1 y 1 z(2)

Kt hp (1) v (2) ta c BT cn chng minh.


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3. (H An ninh HN khi D 1999)Cho 3 s x, y, z thay i, nhn gi tr thuc on [0;1]. Chng minh rng:

2(x3 + y3 + z3) (x2y + y2z + z2x) 3 (*)

Gii:V 0 x, y, z 1 nn x2 x3; y2 y3; z2 z3.Suy ra: 2(x3 + y3 + z3) (x2y + y2z + z2x) 2(x2 + y2 + z2) (x2y + y2z + z2x)Do nu ta chng minh c:

2(x2 + y2 + z2) (x2y + y2z + z2x) 3 (1)th (*) ng.Ta c: (1 y)(1 + y x2) 0 x2 + y2 x2y 1 0 (2)

Du = (2) xy ra

=

= =

y 1

x 1

y 0

Tng t ta cng c: x2 + z2 z2x 1 0 (3)y2 + z2 y2z 1 0 (4)

Cng (2), (3), (4) v theo v ta c:2(x2 + y2 + z2) (x2y + y2z + z2x) 3

Vy (1) ng (*) ng

Nhn xt: Du = (*) xy ra (x; y; z) { }(1;1;1),(1;1;0),(1;0;1),(0;1;1)

NM 2000

4. (HQG HN khi A 2000)Vi a, b, c l 3 s thc bt k tho iu kin a + b + c = 0.Chng minh rng: 8a + 8b + 8c 2a + 2b + 2c

Gii:t x = 2a, y = 2b, z = 2c th x, y, z > 0..kin a + b + c = 0 xyz = 2a+b+c = 1, do theo BT Csi: x + y + z 3Mt khc: x3 + 1 + 1 3x x3 3x 2Tng t: y3 3y 2; z3 3z 2 x3 + y3 + z3 3(x + y + z) 6 = (x + y + z) + 2(x + y + z 3) x + y + z

8a

+ 8b

+ 8c

2a

+ 2b

+ 2c


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5. (HQG HN khi D 2000)Vi a, b, c l 3 s thc dng tho iu kin: ab + bc + ca = abc. Chng minh rng:

+ + ++ +

2 2 2 2 2 2b 2a c 2b a 2c3

ab bc ca

Gii:

Ta c: + += = +2 2 2 2

2 2 2 2b 2a b 2a 1 12.ab a b a b

t x = 1a

; y = 1b

; z = 1c

th gi thit>

+ + =

a,b,c 0

ab bc ca abc

>

+ + =

x,y,z 0

x y z 1

v pcm + + + + + 2 2 2 2 2 2x 2y y 2z z 2x 3 Theo BT Bunhiacopxki ta c:

3(x2 + 2y2) = 3(x2 + y2 + y2) (x + y + y)2 + +2 2 1x 2y (x 2y)

3

Vit 2 BT tng t, ri cng li, ta c:

+ + + + + + + =2 2 2 2 2 2 1x 2y y 2z z 2x (3x 3y 3z) 33

ng thc xy ra x = y = z = 13

a = b = c = 3

6. (H Bch khoa HN khi A 2000)

Cho 2 s a, b tho iu kin a + b 0. Ch. minh rng:+ +

33 3a b a b

2 2

Gii:

Ta c:+ +

33 3a b a b

2 2 4(a3 + b3) (a + b)3

(a + b) [4(a2 + b2 ab) (a2 + b2 + 2ab)] 0 (a + b)(3a2 + 3b2 6ab) 0 (a + b)(a b)2 0BT cui cng ny ng, nn BT cn chng minh l ng.ng thc xy ra a = b.

7. (HSP TP HCM khi DE 2000)Cho 3 s a, b, c bt k. Chng minh cc BT:

a) a2

+ b2

+ c2

ab + bc + ca b) (ab + bc + ca)2

3abc(a + b + c)Gii:a) a2 + b2 2ab; b2 + c2 2bc; c2 + a2 2ca

a2 + b2 + c2 ab + bc + ca.ng thc xy ra a = b = cb) (ab + bc + ca)2 = (ab)2 + (bc)2 + (ca)2 + 2(abbc + bcca + caab)

abbc + bcca + caab + 2abc(a + b + c) = 3abc(a + b + c)


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8. (H Nng nghip I khi A 2000)Cho 3 s dng a, b, c tho iu kin abc = 1.

Tm gi tr nh nht ca biu thc: P= + ++ + +2 2 2 2 2 2bc ca ab

a b a c b c b a c a c b

Gii: Ta c: = = = + + ++

2

2 2 22

1bc bc 1 a

1 11 1a b a c a (b c) ab cb c

t x = 1a

; y = 1b

; z = 1c

th gi thit

a, b, c > 0

abc = 1

>

x, y, z 0

xyz=1v P = + +

+ + +

2 2 2x y z

y z z x x y

Theo BT Bunhiacopxki ta c:

(y + z + z + x + x + y).P

+ + + + +

+ + +

2x y z

y z. z x. x y.

y z z x x y

2(x + y + z).P (x + y + z)2 P 12

(x + y + z) =31 1

.3 xyz .32 2

P 32

Nu P = 32

th x = y = z = 1 a = b = c = 1. o li, nu a = b = c = 1 th P = 32

.

Vy minP = 32

9. (H Thu li II 2000)Chng minh rng vi mi s dng a, b, c ta u c:

(a + 1).(b + 1).(c + 1) ( )+331 abc

Gii:(a + 1).(b + 1).(c + 1) = 1 + a + b + c + ab + bc + ca + abc

1 + 3 + 3 2 2 23 abc 3 a b c + abc = ( )+331 abc

10. (H Y HN 2000)

Gi s x, y l hai s dng tho iu kin + =2 3

6x y

.

Tm gi tr nh nht ca tng x + y.

Gii: ( )

+ = + + +

22 2 3 2 3

2 3 . x . y (x y)x y x y = 6(x + y) x + y

( )+2

2 3

6

Gi tr( )+

22 3

6t c

( )

=

+ + =

2

2 3: x : y

x y

2 3x y

6

+=

+ =

2( 2 3)x

6

3( 2 3)y

6

Vy min(x + y) =+5 2 6

6


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11. (H An Giang khi D 2000)Cho cc s a, b, c 0. Chng minh: ac + 1 + bc + 1 ab(ac 1 + bc 1)

Gii:Gi s a b 0 ac(a b) bc(a b) ac + 1 + bc + 1 ab(ac 1 + bc 1)

12. (H Ty Nguyn khi AB 2000)

CMR vi mi x, y, z dng v x + y + z = 1 th xy + yz + zx > +18xyz

2 xyz

Gii:p dng BT Csi cho 6 s dng ta c:

2 = x + y + z + x + y + z 6 3 xyz (1) v xy + yz + zx 32 2 23 x y z (2)

Nhn cc BT (1) v (2) v theo v ta c:2(xy + yz + zx) 18xyz (3)Mt khc ta c: xyz(xy + yz + zx) > 0 (4)Cng cc BT (3) v (4) v theo v ta c:

(xy + yz + zx)(2 + xyz) > 18xyz xy + yz + zx >+

18xyz

2 xyz(v 2 +xyz > 0)

13. (H An Ninh khi A 2000)Chng minh rng vi mi s nguyn n 3 ta u c: nn + 1 > (n + 1)n

Gii:

Ta c: 34 = 81, 43 = 64 34 > 43 BT cn chng minh ng vi n = 3.

Vi n > 3, pcm n >+

nn 1

n

+

n1

1n

< n (1)

Ta c: +

n1

1n

==n

kn k

k 0

1C

n= 1 +

++ + +

2 n

n n(n 1) 1 n(n 1)...(n n 1) 1. ... .

n 2! n!n n

= 1 + 1 +

+ +

1 1 1 1 2 n 11 ... 1 1 ... 1

2! n n! n n n 0 th:

+ +

1 1 4x y x y

(1). Du = xy ra x = y.

p dng (1) ta c: + = +

1 1 4 4

p a p b p a p b c

+ = + 1 1 4 4

p b p c p b p c a, + =

+

1 1 4 4

p c p a p c p a b

Cng 3 BT trn v theo v, ta c:

+ + + +

1 1 1 1 1 12 4

p a p b p c a b c pcm

Du = xy ra a = b = c.

20. (H Nng nghip I HN khi A 2001)Cho 3 s x, y, z > 0. Chng minh rng:

+ + + ++ + +3 2 3 2 3 2 2 2 2

2 y2 x 2 z 1 1 1

x y y z z x x y z

Gii:p dng BT Csi cho 2 s dng x3, y2 ta c:

x3 + y2 2 =3 2x y 2xy x =+3 2

2 x 2 x 1xy2xy xx y

p dng BT Csi cho 2 s dng2 2

1 1,

x yta c:

+

2 2

1 1 1 1

xy 2 x y

+ +

3 2 2 2

2 x 1 1 1

2x y x y

Tng t ta cng c:

+

+ 3 2 2 2

2 y 1 1 1

2y z y z

;

+

+ 3 2 2 2

2 z 1 1 1

2z x z x

Suy ra: + + + ++ + +3 2 3 2 3 2 2 2 2

2 y2 x 2 z 1 1 1

x y y z z x x y z

Du = xy ra = = =

= = =

3 2 3 2 3 2x y y z z xva va

x y y z z x x = y = z = 1


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21. (H PCCC khi A 2001)Ch. minh rng vi a 2, b 2, c 2 th: + + ++ + >b c c a a blog a log b log c 1

Gii:Trc ht ch rng nu a > 1, x > 1 th hm s y = alo g x l ng bin v dng.

Do hm s y = logxa =a

1

log xl nghch bin.

V vai tr ca a, b, c l nh nhau, nn ta c th gi thit a b c. Ta c:VT= + + + + + + ++ + + + =b c c a a b a b a b a b a blog a log b log c log a log b log c log abc

V a, b, c 2 nn abc 2ab = ab + ab > a + bDo VT loga+babc > loga+b(a + b) = 1.

22. (H Quc gia HN khi D 2001)Ch. minh rng vi mi x 0 v vi mi > 1 ta lun c: x + 1 x.T chng minh rng vi 3 s dng a, b, c bt k th:

+ + + +3 3 3

3 3 3

a b c a b c

b c ab c a Gii:

Xt f(x) = x x + 1 (x 0)f(x) = (x 1 1); f(x) = 0 x = 1

Vy vi x 0 v > 1 th f(x) 0 hay x + 1 x.

BT cn chng minh: + + + +

3 3 32 2 2a b c a b c

b c a b c a

p dng BT chng minh vi = 32

, ta c:

+

3

2a 1 3 a.

b 2 2 b;

+

3

2b 1 3 b.

c 2 2 c;

+

3

2c 1 3 c.

a 2 2 a

Mt khc, theo BT Csi ta c:

+ +

3 3 3

2 2 21 a b c 3

2 b c a 2

Cng 4 BT trn, v theo v, ta c:

+ + + + + +

3 3 3

2 2 23 a b c 3 3 a b c 3

2 b c a 2 2 b c a 2

Suy ra: + + + +

3 3 3

2 2 2a b c a b c

b c a b c a


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23. (H Thi Nguyn khi D 2001)

Cho a 1, b 1. Chng minh rng: + a b 1 b a 1 ab (*)Gii:

BT (*)

+ a b 1 b a 1

1ab ab

+

1 1 1 11 1 1

b b a a(1)

Theo BT Csi ta c:

+ =

1 111 1 1b b

1b b 2 2

+ =

1 11

1 1 1a a1

a a 2 2

Cng 2 BT li ta c BT cn chng minh.

Du = xy ra

= =

= =

1 1 11

b b 2

1 1 11a a 2

a = b = 2.

24. (H Vinh khi A, B 2001)Chng minh rng nu a, b, c l di ba cnh ca mt tam gic c chu vi bng 3 th:

3a2 + 3b2 + 3c2 + 4abc 13Gii:

Ta c: 3 2a = a + b + c 2a = b + c a > 0.Do theo BT Csi ta c:

(3 2a)(3 2b)(3 2c)

+ +

33 2a 3 2b 3 2c

3 = 1 27 9(2a + 2b + 2c) + 3(4ab + 4bc + 4ca) 8abc 1 27 54 + 12(ab + bc + ca) 8abc 1 4abc 6(ab + bc + ca) 14 3(a2 + b2 + c2) + 4abc 3(a2 + b2 + c2) + 6(ab + bc + ca) 14 = 3(a + b +c)2 14 =13ng thc xy ra 3 2a = 3 2b = 3 2c a = b = c = 1.

25. (H Y Thi Bnh khi A 2001)

Cho a, b, c l nhng s dng v a + b = c. Ch. minh rng: + >2 2 2

3 3 3a b c Gii:

T gi thit ta c: +a b

c c= 1 0 +

2 2

3 3a b a b

c c c c= 1

T suy ra: + >2 2 2

3 3 3a b c


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NM 200226. (i hc 2002 d b 1)

Gi x, y, z l khong cch t im M thuc min trong ca ABC c 3 gc nhn ncc cnh BC, CA, AB. Chng minh rng:

+ ++ +

2 2 2a b cx y z

2R(a, b, c l cc cnh ca ABC, R l bn knh

ng trn ngoi tip). Du = xy ra khi no?

Gii: + + = + +1 1 1

x y z . ax . by . cza b c

+ +

1 1 1(ax+by+cz)

a b c

+ +

1 1 1.2S

a b c=

+ +

1 1 1 abc

a b c 2R=

+ +ab bc ca2R

+ +2 2 2a b c

2R

Du = xy ra = =

= =

a b c

x y z

ABC eu

M trung vi trong tam G cua ABC

27. (i hc 2002 d b 3)

Gi s x, y l hai s dng thay i tho mn iu kin x + y = 54 . Tm gi tr nh nht

ca biu thc: S = +4 1

x 4y

Gii:

Cch 1: S = + + + + 5

1 1 1 1 1 5

x x x x 4y x.x.x.x.4y

+ + + +5.5

x x x x 4y= 5

minS = 5

=

=

+ =

1 1

x 4y

x 4y

5x y4

=

=

x 1

1y

4

Cch 2: S = +

4 1

x 5 4x= f(x), 0 < x < 5

4

f(x) = +2 2

4 4

x (5 4x); f(x) = 0

=

<


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28. (i hc 2002 d b 5)Gi s a, b, c, d l 4 s nguyn thay i tho mn 1 a < b < c < d 50.

Chng minh bt ng thc:+ +

+ 2a c b b 50

b d 50b

v tm gi tr nh nht ca biu thc: S = +a c

b d.

Gii:V a 1, d 50 v c > b (c, b N) nn c b + 1 thnh th:

S = +a c

b d

++

1 b 1

b 50=

+ +2b b 5050b

Vy BT ca ra c chng minh.

Du = xy ra

=

= = +

a 1

d 50

c b 1

tm minS, ta t+ +2b b 5050b = + +

b 1 1

50 b 50 v xt hm s c bin s lin tc x:

f(x) = + +x 1 1

50 x 50(2 x 48)

f(x) =

=2

2 2

1 1 x 50

50 x 50x; f(x) = 0

=

2x 50

2 x 48 =x 5 2

Bng bin thin:

2

Chuyn v biu thc f(b) =+ +2b b 5050b

(2 b 48, b N)

T BBT suy ra khi b bin thin t 2 n 7, f(b) gim ri chuyn sang tng khi b binthin t 8 n 48. Suy ra minf(b) = min[f(7); f(8)].

Ta c f(7) =+

=49 57 53

350 175; f(8) =

+= >

64 58 61 53

400 200 175

Vy minS = 5 31 7 5 khi

= = = =

a 1

b 7

c 8

d 5 0


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29. (i hc 2002 d b 6)

Cho tam gic ABC c din tch bng 32

. Gi a, b, c ln lt l di cc cnh BC, CA,

AB v ha, hb, hc tng ng l di cc ng cao k t cc nh A, B, C. Chngminh rng:

+ + + +

a b c

1 1 1 1 1 13

a b c h h h

Gii:

Ta c din tch tam gic: S = = =a b c1 1 1

ah bh ch2 2 2

ha =2S

a; hb =

2S

b; hc =

2S

c + + = + +

a b c

1 1 1 1(a b c)

h h h 2S

+ + + + = + + + +

a b c

1 1 1 1 1 1 1 1 1 1(a b c)

a b c h h h 2S a b c

p dng BT Csi ta c: (a + b + c)

+ +

1 1 1

a b c 9

v v S =3

2, nn ta c:

+ + + + =

a b c

1 1 1 1 1 1 93

a b c h h h 3

NM 2003

30. (CGT II 2003 d b)

Cho 3 s bt k x, y, z. CMR: + + + + +2 2 2 2 2 2x xy y x xz+z y yz+z

Gii:Trong mt phng to Oxy, xt cc im:

A

+

y 3x ; z

2 2 , B

+

3 30; y z

2 2, C

y z;0

2 2

Ta c: AB =

+ + = + +

222 2y 3x y x xy y

2 2

AC = + + = + +

222 2z 3x z x xz z

2 2

BC =

+ + = +

222 2y z 3 (y z) y yz+z

2 2 2

Vi 3 im A, B, C ta lun c: AB + AC BC

+ + + + +2 2 2 2 2 2x xy y x xz+z y yz+z


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31. (i hc khi A 2003)Cho x, y, z l 3 s dng v x + y + z 1. Chng minh rng:

+ + + + + 2 2 22 2 2

1 1 1x y z 82

x y z

Gii:

Vi mi

u,v ta c: + +

u v u v (*)

t = = =

1 1 1a x; ; b y; ; c z;

x y z

p dng bt ng thc (*), ta c: + + + + + +

a b c a b c a b c

Vy P = + + + + +2 2 2

2 2 2

1 1 1x y z

x y z

+ + + + +

22 1 1 1(x y z)

x y z

Cch 1:

Ta c: P

+ + + + +

22 1 1 1

(x y z) x y z ( )

+

22

33

1

3 xyz 3 xyz = +9

9t t

vi t = 23( xyz) 0 < t + +

2x y z 1

3 9

t Q(t) = 9t +9

tQ(t) = 9

2

9

t< 0, t

10;

9Q(t) gim trn

10;

9

Q(t) Q

1

9= 82. Vy P Q(t) 82

Du "=" xy ra x = y = z = 13

.

Cch 2: Ta c:

(x + y + z)2 +

+ +

21 1 1

x y z= 81(x + y + z)2 +

+ +

21 1 1

x y z 80(x + y + z)2

18(x + y + z).

+ +

1 1 1

x y z 80(x + y + z)2 162 80 = 82

Vy P 82

Du "=" xy ra x = y = z = 13

.


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32. (i hc khi A 2003 d b 1)

Tm gi tr ln nht v gi tr nh nht ca hm s:y = sin5x + 3 cosxGii:

Tm max: y = sin5x + 3 cosx sin4x + 3 cosx (1)

Ta chng minh: sin4x + 3 cosx 3 , x R (2)

3 (1 cosx) sin4x 0 3 (1 cosx) (1 cos2x)2 0 (1 cosx).[ 3 (1 cosx)(1 + cosx)2] 0 (3)Theo BT Csi ta c:

(1 cosx)(1 + cosx)(1 + cosx) = 12

(2 2cosx)(1 + cosx)(1 + cosx)

=


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NM 2005

34. (i hc khi A 2005)

Cho x, y, z l cc s dng tho mn : + + =1 1 1

4x y z

.

Chng minh rng:+ +

+ + + +

1 1 11

2x+y+z x 2y z x y 2z Gii:

Vi a, b > 0 ta c:

4ab (a + b)2+

+1 a b

a b 4ab

+ +

1 1 1 1

a b 4 a b

Du "=" xy ra khi v ch khi a = b.p dng kt qu trn ta c:

+ +

1 1 1 1

2x+y+z 4 2x y z

+ +

1 1 1 1 1

4 2x 4 y z=

+ +

1 1 1 1

8 x 2y 2z(1)

Tng t:

+ + + +

1 1 1 1

x 2y z 4 2y x z

+ +

1 1 1 1 1

4 2y 4 x z=

+ +

1 1 1 1

8 y 2z 2x(2)

+ + + +

1 1 1 1

x y 2z 4 2z x y

+ +

1 1 1 1 1

4 2z 4 x y=

+ +

1 1 1 1

8 z 2x 2y(3)

Vy:

+ + + + + + + +

1 1 1 1 1 11

2x+y+z x 2y z x y 2z 4 x yz= 1

Ta thy trong cc bt ng thc (1), (2), (3) th du "=" xy ra khi v ch khi

x = y = z. Vy ng thc xy ra khi v ch khi x = y = z = 34

.

35. (i hc khi B 2005)Chng minh rng vi mi x R, ta c:

+ + + +

x x xx x x12 15 20 3 4 5

5 4 3

Khi no ng thc xy ra?Gii:

p dng bt ng thc Csi cho 2 s dng ta c:

+

x x x x12 15 12 15

2 .

5 4 5 4

+

x x12 15

5 4

2.3x (1)

Tng t ta c:

+

x x12 20

5 3 2.4x (2)

+

x x15 20

4 3 2.5x (3)

Cng cc bt ng thc (1), (2), (3), chia 2 v ca bt ng thc nhn c cho 2 tac pcm.ng thc xy ra (1), (2), (3) l cc ng thc x = 0.


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36. (i hc khi D 2005)Cho cc s dng x, y, z tho mn xyz = 1. Chng minh rng:

+ + + + + ++ +

3 3 3 3 3 31 x y 1 y z 1 z x3 3

xy yz zx

Khi no ng thc xy ra?Gii:

p dng bt ng thc Csi cho 3 s dng ta c:

1 + x3 + y3 3 3 33 1.x .y = 3xy + +

3 31 x y 3

xy xy(1)

Tng t:+ +

3 31 y z 3

yz yz(2);

+ +

3 31 z x 3

zx zx(3)

Mt khc + + 33 3 3 3 3 3

3xy yz zx xy yz zx

+ + 3 3 3 3 3xy yz zx

(4)

Cng cc bt ng thc (1), (2), (3), (4) ta c pcm.ng thc xy ra (1), (2), (3), (4) l cc ng thc x = y = z = 1.

37. (i hc khi A 2005 d b 1)

Cho 3 s x, y, z tho x + y + z = 0. CMR: + + + + +x y z3 4 3 4 3 4 6Gii:

Ta c: 3 + 4x = 1 + 1 + 1 + 4x 4 4 x4 + = 84x x x3 4 2 4 2 4

Tng t: + 8y y3 4 2 4 ; + 8z z3 4 2 4

Vy + + + + +x y z3 4 3 4 3 4 2 + + 8 8 8x y z4 4 4

3 8 x y z6 4 .4 .4 6 + +24 x y z4 = 6

38. (i hc khi A 2005 d b 2)

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

+ + +

2y 9

1 x 1 1x y

256

ng thc xy ra khi no?Gii:

Ta c: 1 + x = 1 + + + 3

43

x x x x43 3 3 3

v 1 +yx

= 1 + + + 34

3 3y y y y4

3x 3x 3x 3 x

1 +9

y= 1 + + +

3

43

3 3 3 34

y y y y

+

2 6

43

9 31 16

y y

Vy: ( )

+ + +

2y 9

1 x 1 1x y

2563 3 6

43 3 3 3

x y 3. .

3 3 x y= 256


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van

39. (i hc khi B 2005 d b 1)

Cho 3 s dng a, b, c tho mn: a + b + c = 34

. Chng minh rng:

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

Gii: Cch 1:

Ta c: + + ++ = + +3 a 3b 1 1 1(a 3b).1.1 (a 3b 2)3 3

+ + ++ = + +3

b 3c 1 1 1(b 3c).1.1 (b 3c 2)

3 3;

+ + ++ = + +3

c 3a 1 1 1(c 3a).1.1 (c 3a 2)

3 3

Suy ra: [ ]+ + + + + + + +3 3 31

a 3b b 3c c 3a 4(a b c) 63

+

1 34. 6

3 4= 3

Du "=" xy ra

+ + = + = + = +

3a b c

4a 3b b 3c c 3a=1

a = b = c = 14

Cch 2:

t x = +3 a 3b x3 = a + 3b; y = +3 b 3c y3 = b + 3c;

z = +3 c 3a z3 = c + 3a

x3 + y3 + z3 = 4(a + b + c) = 4. 34

= 3. BT cn ch. minh x + y + z 3

Ta c: x3 + 1 + 1 3 3 3x .1.1 = 3x ; y3 + 1 + 1 3 33 y .1.1 = 3y ;

z3 + 1 + 1 3 3 3z .1.1 = 3z 9 3(x + y + z) (v x3 + y3 + z3 = 3)Vy x + y + z 3

Du "=" xy ra

= = =

+ + =

3 3 3x y z 1

3a b c

4

+ = + = +

a 3b b 3c c 3a=1

3a+b+c=

4 a = b = c = 1

4

40. (i hc khi B 2005 d b 2)

Chng minh rng nu 0 y x 1 th 1

x y y x4

.

ng thc xy ra khi no?Gii:

Ta c: 0 x 1 x x2

1

x y y x4

+1

x y y x4 (1)

Theo BT Csi ta c: + + =2 21 1 1

y x yx 2 yx . x y4 4 4

1

x y y x4

Du "=" xy ra

=

=

= =

2

2

0 y x 1 x 1x x 1

y1 4yx4


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41. (i hc khi D 2005 d b 2)

Cho x, y, z l 3 s dng v xyz = 1. CMR: + + + + +

2 2 2x y z 3

1 y 1 z 1 x 2

Gii:

Ta c:+ +

+ =

+ +

2 2x 1 y x 1 y2 . x

1 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 3 bt ng thc trn, v theo v, ta c: + + +

+ + + + + + + + + +

2 2 2x 1 y y 1 z z 1 xx y z

1 y 4 1 z 4 1 x 4

+ +

+ + + + ++ + +

2 2 2x y z 3 x y z

x y z1 y 1 z 1 x 4 4

+ +

3(x y z) 3

4 4

= =3 3 9 3 3

.34 4 4 4 2

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

Vy: + + + + +

2 2 2x y z 3

1 y 1 z 1 x 2.

NM 2006

42. (CBC Hoa Sen khi A 2006)Cho x, y, z > 0 v xyz = 1. Chng minh rng: x3 + y3 + z3 x + y + z.

Gii:

x3 + y3 + z3 3 3 3 33 x y z 2(x3 + y3 + z3) 6

x3 + 1 + 1 3 3 3x x3 + 2 3x (1)

Tng t: y3 + 1 + 1 3 33 y y3 + 2 3y (2)

z3 + 1 + 1 3 3 3z z3 + 2 3z (3)Cng (1), (2), (3) v theo v suy ra bt ng thc cn chng minh.


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43. (CKTKT Cn Th khi A 2006)Cho 3 s dng x, y, z tho x + y + z 1.

Tm gi tr nh nht ca biu thc: A = x + y + z + + +1 1 1

x y z

Gii: Cch 1:

Theo BT Csi: 1 x + y + z 3 3 xyz > 0 , + + 31 1 1 3

x y z xyz

T : A 3 3 xyz +3

3

xyz

t: t = 3 xyz , iu kin: 0 < t 13

Xt hm s f(t) = 3t + 3

tvi 0 < t 1

3, f(t) = 3

2

3

t=

2

2

3(t 1)

t< 0, t

10;

3

Bng bin thin:1

3

T bng bin thin ta suy ra: A 10. Du "=" xy ra khi x = y = z = 13

Vy Amin = 10 t c khi x = y = z =1

3

.

Cch 2:

Theo BT Csi: 1 x + y + z 3 3 xyz > 0 3

1

xyz 3

x + 1 2

9x 3, y +

1 2

9y 3, z +

1 2

9z 3

T : A= + + + + + + + +

1 1 1 8 1 1 1x y z

9x 9y 9z 9 x y z 2 +

3

8 3

9 xyz 10

Du "=" xy ra khi x = y = z = 13

.Vy Amin = 10 t c khi x = y = z =1

3


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44. (CSPHCM khi ABTDM 2006)

Cho x, y l hai s thc dng v tho x + y = 54

.

Tm gi tr nh nht ca biu thc: A = +4 1

x 4y.

Gii:

Ta c: x + y =5

4 4x + 4y 5 = 0

A = +4 1

x 4y= + +

4 14x+ 4y 5

x 4y A 2

4.4x

x+ 2

1.4y

4y 5 A 5

Du "=" xy ra

=

= + = >

44x

x1

4y4y

5x y

4x,y 0

=

=

x 1

1y

4

. Vy Amin = 5.

45. (CKTKT Cn Th khi B 2006)Cho 4 s dng a, b, c, d. Chng minh bt ng thc:

+ + ++ + + + + + + +

a b c d

a b c b c d c d a d a b< 2

Gii:V a, b, c, d > 0 nn ta lun c:

+ < + =+ + + + + +

a c a c

1a b c c d a a c a c v + < + =+ + + + + +

b d b d

1b c d d a b b d b d Cng v theo v cc BT trn ta c pcm.

46. (CKT Cao Thng khi A 2006)

Chng minh rng nu x > 0 th (x + 1)2

+ + 2

1 21

xx 16.

Gii:

Ta c: (x + 1)2

+ +

2

1 21

xx 16 (1) (x + 1)2

+

21

1

x 16

(x + 1)

+

11

x 4 (do x > 0) (x + 1)2 4x (x 1)2 0 (2)

(2) lun ng nn (1) c chng minh.


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47. (CKTKTCN1 khi A 2006)

Cho 3 s dng a, b, c. Ch. minh rng:+ + + + + +

+ + a b c a b c a b c

9a b c

Gii:

Xt v tri ca BT cho: VT =+ + + + + + + +

b c a c a b1 1 1

a a b b c c

= 3 +

+ + + + +

b a c a c b

a b a c b c

Do a, b, c > 0 nn theo BT Csi ta c:

+ =b a b a

2 . 2a b a b

; + =b c b c

2 . 2c b c b

; + =c a c a

2 . 2a c a c

Khi : VT 3 + 2 + 2 + 2 = 9 (pcm).

48. (CKTYT1 2006)Cho cc s thc x, y thay i tho mn iu kin: y 0; x2 + x = y + 12.Tm gi tr ln nht, nh nht ca biu thc: A = xy + x + 2y + 17

Gii:y 0, x2 + x = y + 12 x2 + x 12 0 4 x 3y = x2 + x 12 A = x3 + 3x2 9x 7t f(x) = A = x3 + 3x2 9x 7 vi 4 x 3f(x) = 3x2 + 6x 9 ; f(x) = 0 x = 1 hoc x = 3

f(4) = 13, f(3) = 20, f(1) = 12, f(3) = 20Vy maxA = 20 (x = 3, y = 0), minA = 12 (x = 1, y = 10).

49. (CBC Hoa Sen khi D 2006)Cho x, y, z > 0; x + y + z = xyz. Tm gi tr nh nht ca biu thc A = xyz.

Gii:

Ta c: x + y + z 3 3 xyz xyz 3 3 xyz (xyz)2 27 xyz 3 3

Du "=" xy ra x = y = z = 3 .Vy minA = 3 3 .


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50. (i hc khi A 2006)Cho 2 s thc x 0, y 0 thay i v tho mn iu kin:

(x + y)xy = x2 + y2 xy.

Tm gi tr ln nht ca biu thc: A = +3 3

1 1

x y.

Gii:

Cch 1:T gi thit suy ra: + = + 2 2

1 1 1 1 1

x y xyx y.

t 1x

= a, 1y

= b, ta c: a + b = a2 + b2 ab (1)

A = a3 + b3 = (a + b)(a2 ab + b2) = (a + b)2T (1) suy ra: a + b = (a + b)2 3ab.

V ab +

2a b

2nn a + b (a + b)2 + 2

3(a b)

4

(a + b)2 4(a + b) 0 0 a + b 4 Suy ra: A = (a + b)2 16

Vi x = y = 12

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

Cch 2:t S = x + y, P = xy vi S2 4P 0. T gi thit S, P 0.

Ta c: SP = S2 3P P =+

2S

S 3

A = +3 3

1 1

x y=

+3 3

3 3

x y

x y=

+ + 2 2

3 3

(x y)(x y xy)

x y=

+ 2

3 3

(x y) xy

x y=

+ 2

2 2

(x y)

x y

A =+

=

2

2

S S 3

SP

k: S2 4P 0 S2 +

24S

S 3 0 S2

+

S 1

S 3 0 +

S 1

S 3 0 (v S0)

<

S 3

S 1(*)

t h = f(S) =+S 3S

h = 2

3

S< 0, S tho (*)

T bng bin thin, ta c: 0 < h 4 v h 1, S tho (*).

M A = h MaxA = 16 khi x = y = 12

(S = 1, P = 14

).


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Cch 3:

(x + y)xy =

+

2 2y 3yx

2 4> 0

++ =

1 1 x y

x y xy> 0

A = +3 3

1 1

x y=

+3 3

3 3

x y

x y=

+

21 1

x y = +

1 1A

x y

D chng minh c: + +

3 3 3a b a b2 2

(vi a + b > 0)

du "=" xy ra khi a = b.

p dng vi a =1

x, b =

1

y, ta c:

++

333 1 11 1x yx y

2 2

3A A

2 2 A 16.

Du "=" xy ra khi = =1 1

2x y

. Vy Max A = 16.

Cch 4:A =

2

2

S

P, suy ra = =

2S 3S

AP S SP

S2 4P 0 S2 4 2S SP

3 0

P1

S1 43

0 P 1

S 4(chia cho S2)

Nn: A =2

2

S

P 16. Vy Max A = 16 (khi x = y = 1

2).

51. (i hc khi B 2006)Cho x, y l cc s thc thay i. Tm gi tr nh nht ca biu thc:

A = ( ) ( ) + + + + + 2 22 2x 1 y x 1 y y 2 Gii:Trong mpOxy, xt M(x 1; y), N(x + 1; y).

Do OM + ON MN nn:

( ) ( ) + + + + + = +2 22 2 2 2x 1 y x 1 y 4 4y 2 1 y

Do : A 2 + + 21 y y 2 = f(y)

Vi y 2 f(y) = 2 + 21 y + 2 y f(y) =+2

2y

y 1 1

f(y) = 0 2y = + 21 y = +

2 2y 04y 1 y

y = 13

Do ta c bng bin thin nh trn

Vi y 2 f(y) 2 + 21 y 2 5 > 2 + 3 .Vy A 2 + 3 vi mi s thc x, y.

Khi x = 0 v y = 13

th A = 2 + 3

Nn gi tr nh nht ca A l 2 + 3 .


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NM 2007

52. (i hc khi A 2007)

Cho x, y, z l cc s thc dng thay i v tha mn iu kin xyz =1.

Tm gi tr nh nht ca biu thc:2 2 2( ) ( ) ( )

2 2 2

x y z y z x z x yP

y z z z z x x x x y y

+ + += + +

+ + +

Gii:

Ta c: 2 2 2( ) 2 . : ( ) 2 , ( ) 2y z x x y z x y y z x y z z + + + Tngt

22 2

2 2 2

y y x x z z P

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

+ + +

t: 2 , 2 , 2a x x y y b y y z z c z z x x= + = + = +

Suy ra4 2 4 2 4 2

, ,9 9 9

c a b a b c b c a x x y y z z

+ + + = = =

Do : 2 4 2 4 2 4 2 2 4 69 9 9 9 9

c a b a b c b c a c a b a b cPb c a b c a

+ + + + + = + + + + +

2(4.3 3 6) 2

9 + =

1 1 2 2 1 4 1 3c a b c a b a b

Dob c a b c a b a

+ + = + + + + =

Du "=" xy ra x = y = z = 1. Vy gi tr nh nht ca P l 2.

53. (i hc khi B 2007)Cho x, y, z l ba s thc dng thay i. Tm gi tr nh nht ca biu thc:

1 1 12 2 2

x y z P x y z

yz zx xy

= + + + + +

Gii:

Ta c :2 2 2 2 2 2

2 2 2

x y z x y z P

xyz

+ += + + + .

Do2 2 2 2 2 2

2 2 2

2 2 2

x y y z z x x y z xy yz zx

+ + ++ + = + + + +

2 2 2

1 1 12 2 2 x y z P

x y z + + + + +

Xt hm s :2 1

( ) , 02

t f t t

t= + >

Lp bng bin thin ca hm s ny ta c3

( ) , 0.2

f t t >

Suy ra :9

2P . Du bng xy ra x = y = z = 1. Vy gi tr nh nht ca P l 9 .


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54. (i hc khi D 2007)

Cho1 1

0. : 2 22 2

b a

a b

a ba b CMR

> + +

Gii:

( ) ( ) ( ) ( )ln 1 4 ln 1 41 12 2 1 4 1 42 2

a bb a

b aa b a ba b a b

+ + + + + +

Xt hm s :( ) ( ) ( )

( )/

2

ln 1 4 4 ln 4 1 4 ln 1 4( ) , 0. ( ) 0

1 4

x x x x x

x f x x f x

x x

+ + += > = 0 nn f(a) f(b)Vy ta c iu phi chng minh.

NM 2009

55. (i hc khi A 2009)Chng minh rng vi mi s thc dng x,y,z tha mn x( x + y + z )= 3xy, ta c:

3 3 3( ) ( ) 3( )( )( ) 5( )x y x z x y x z y z y z + + + + + + + +

Gii:t a = x + y , b = z + c , c = x + y , ,

2 2 2

b c a c a b a b c x y z

+ + + = = =

T iu kin x(x + y + z) = 3yz 2 2 2 2 2 24 ( ) 3( ) (*)a b c b c a b bc c = + + = + Ta c bt ng thc cho tng ng:

3 3 35 3a b c abc + + 25 ( ) 3 (**)a a b c bc + +

T (*)

2

22 2 2 2 2 ( )2 2( ) 2 2

2

a bc

b ca b c bc b c a b c

+

= + + +

2

2

2 ( )3 3

a a b c

a bc

+

(**) ng pcm

Du = xy ra khi a= b= c tc l x = y = z.


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56. (i hc khi B 2009)

Cho cc s thcx, y thay i v tho mn ( ) 4 2 x y xy+

+ +

Tm gi tr nh nht ca biu thc ( ) ( )4 4 2 2 2 23 2 1A x y x y x y= + + + +

Gii: Kt hp ( ) ( ) ( ) ( )3 2 3 2

4 2 & 4 2 1 y xy x y xy suy ra x y x y x y+ + + + + + +

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

24 4 2 2 2 2 2 2 4 4 2 2

2 2 22 2 2 2 2 2 2 2 2 2

3 33 2 1 2 12 23 3 9

2 1 2 12 2 4

A x y x y x y x y x y x y

x y x y x y A x y x y

= + + + + = + + + + +

+ + + + + + + +

t 2 2t x y= + , ta c( )

2

2 2 1 1

2 2 2

x y x y t

++

Do :29 2 1

4t t +

Xt 2 /

1;2

9 9 1 1 9( ) 2 1 ; ( ) 2 0, min ( )

4 2 2 2 16

t t t t f t t t f t f +

= + = > = =

9

16A . ng thc xy ra

1

2x y = = . Vy gi tr nh nht ca A l

9

16

57. (i hc khi D 2009)Cho cc s thc khng mx; y thay i v tho mn 1x y+ =

Tm gi tr ln nht v gi tr nh nht ca biu thc ( ) ( )2 24 3 4 3 25S x y y x xy= + + +

Gii: Do x + y = 1 nn ( )2 2 3 316 12 9 25S x y x y xy xy= + + + +

( ) ( )32 2 2 216 12 3 34 16 2 12 x y x y xy x y xy x y xy = + + + + = +

t t = xy ta c2

2 1 116 2 12; 0 0;2 4 4

x yS t t xy t

+ = + =

Xt hm s 21

( ) 16 2 12 0;4

f t t t = +

lien tuc tren oan

/ / 1 1 191 1 25( ) 32 2 ; ( ) 0 ; (0) 12 , ,16 16 16 4 2

f t t f t t f f f = = = = = =

110;0;

44

1 25 1 191max ( ) ; min ( )

4 2 16 16 f t f f t f

= = = =

Gi tr ln nht ca S bng ( )125 1 1

; ;42 2 2

x ykhi x y

xy

+ = = =

Gi tr nh nht ca S bng191

16

( ) ( )1

2 3 2 3 2 3 2 3; ; ; ;1

4 4 4 416

x y

khi x y hay x yxy

+ = + + = = =


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anh

le

van

58. (Cao ng ABD 2009)Cho a v b l hai s thc tha mn 0 1a b< < < .Chng minh rng: 2 2ln ln ln lna b b a a b >

Gii:2 2

2 2

ln lnln ln ln ln

1 1

a ba b b a a b

a b >

+ +

Do hm s f(t) ng bin trn khong (0;1)

M 0


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anh

le

van

NM 2010

60. (Cao ng ABD 2010)Cho hai s thc dng thay ix, ytha mn iu kin 3 1y+

Tm gi tr nh nht ca biu thc1 1

Ax xy

= +

Gii:

1 1 1 2 1 2 4 8 82 8

2 ( ) 32 ( )A

x x x y x x y x x y x y xy x x y= + + = =

+ + + + ++

ng thc xy ra1

4x y = = . Vy gi tr nh nht ca A bng 8.

61. (i hc B 2010)Cho cc s thc khng m a, b, c tha mn: a + b + c = 1.Tm gi tr nh nht ca biu thc

M = 3(a2b2+b2c2+c2a2) + 3(ab + bc + ca) + 2 2 22 a b c+ + .Gii:

t t = ab + bc + ca, ta c: a2 + b2 + c2 ab + bc + ca 1 = (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) 3(ab + bc + ca)

a2 + b2 + c2 = 1 2t v1

03

t

Theo B.C.S ta c : t2 = (ab + bc + ca)2 3(a2b2 + b2c2 + c2a2)

M 2 3 2 1 2 ( )t t t f t + + =

f(t) =2

2 31 2

tt

+

f (t) =3

22

(1 2 )t

< 0, t

10,

3

f(t) l hm gim

/ / 1 11( ) 2 33 3

f t f =

> 0 f tng f(t) f(0) = 2, t 1

0,3

M 2, a, b, c khng m tha a + b + c = 1Khi a = b = 0 v c = 1 th M = 2. Vy min M = 2.

KKKKKKKKhhhhhhhhoooooooonnnnnnnngggggggg ccccccccoooooooo vvvvvvvviiiiiiiieeeeeeeecccccccc gggggggg kkkkkkkkhhhhhhhhoooooooo ,,,,,,,, cccccccchhhhhhhh ssssssss lllllllloooooooonnnnnnnngggggggg kkkkkkkkhhhhhhhhoooooooonnnnnnnngggggggg bbbbbbbbeeeeeeeennnnnnnnaaaaaaaa oooooooo nnnnnnnnuuuuuuuu iiiiiiii vvvvvvvvaaaaaaaa llllllllaaaaaaaapppppppp bbbbbbbbiiiiiiiieeeeeeeennnnnnnn ,,,,,,,, qqqqqqqquuuuuuuuyyyyyyyyeeeeeeeetttttttt cccccccchhhhhhhh aaaaaaaatttttttt llllllllaaaaaaaammmmmmmm nnnnnnnneeeeeeeennnnnnnn

Bất đẳng thức LTDH _ Giải chi tiết

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