Bất đẳng thức Bunhiacopxki và ứng dụng

Trng THPT Chuyen Tien Giang GV o Kim Sn

Chuyen e boi dng hoc sinh gioi K10 Page1

I.Bt ng thc Bunhiacpxki ( BCS ) : Cho 2 b s thc ( )1 2; ;...; na a a v ( )1 2; ;...; nb b b , mi b gm n s. Khi ta c: ( ) ( )( )2 2 2 2 2 2 21 1 2 2 1 2 1 2... ... ...n n n na b a b a b a a a b b b+ + + + + + + + + Du ng thc xy ra khi v ch khi:

1 21 2

... nn

aa ab b b= = = vi quy c nu mu bng 0 th t phi bng 0.

II. Cc h qu : H qu 1:

Nu 1 1 ... n na x a x C+ + = (khng i) th ( )2 21 2 21

min ......n n

Cx xa a

+ + = + +

t c khi 11

... nn

xxa a= =

H qu 2: Nu 2 2 21 ... nx x C+ + = (khng i) th ( ) 2 21 1 1max ... ...n n na x a x C a a+ + = + +

t c khi 11

... 0nn

xxa a= =

( ) 2 21 1 1min ... ...n n na x a x C a a+ + = + + Du = xy ra 1

1

... 0nn

xxa a

= = III.Bt ng thc Bunhiacpxki m rng: M rng bt ng thc Bunhiacpxki cho 3 dy s thc khng m

( )1 2; ;...; na a a ; ( )1 2; ;...; nb b b ; ( )1 2; ;...; nc c c ta lun c : ( ) ( )( )( )2 3 3 3 3 3 3 3 3 31 1 1 2 2 2 1 2 1 2 1 2... ... ... ...n n n n n na b c a b c a b c a a a b b b c c c+ + + + + + + + + + + +

Chng minh: t 3 3 3 3 3 3 3 3 33 3 31 2 1 2 1 2... , ... , ...n n nA a a a B b b b C c c c= + + + = + + + = + + +

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Nu 0A = hoc 0B = hoc 0C = th bt ng thc hin nhin ng v khi c hai v ca bt ng thc u bng 0. Vy ta ch xt trng hp 0; 0; 0A B C> > > t ; ;i i ii i i

a b cx y z

A B C= = = vi 1;2;3i =

Khi ta c:

3 3 31 2 3

3 3 31 2 3

3 3 31 2 3

1

1

1

x x xy y yz z z

+ + = + + = + + = v bt ng thc cn chng minh tr thnh: 1 1 1 2 2 2 3 3 3 1x y z x y z x y z+ +

p dng bt ng thc Cauchy cho 3 s khng m: ( )3 3 3; ; 1;2;3i i ix y z i = ta c:

3 3 31 1 1

1 1 1

3 3 32 2 2

2 2 2

3 3 33 3 3

3 3 3

3

3

3

x x xx y z

x x xx y z

x x xx y z

+ + + + + +

Cng cc bt ng thc trn li ta c: 1 1 1 2 2 2 3 3 3 1x y z x y z x y z+ + (pcm)

ng thc xy ra

1 1 1

1 1 12 2 2

2 2 2

3 3 33 3 3

a b cA B Cx y za b cx y zA B C

x y z a b cA B C

= == = = = = = = = = =

Hay ( ): : : : 1;2;3i i ia b c A B C i= = tc l: 1 1 1 2 2 2 3 3 3: : : : : :a b c a b c a b c= = Tng qut : bt ng thc Bunhiacpxki m rng cho rng cho m dy s thc khng m: Cho m dy s thc khng m: ( )1 2; ;...; na a a , ( )1 2; ;...; nb b b , , ( )1 2; ;...; nK K K Ta c: ( ) ( )( ) ( )1 1 1 2 2 2 1 2 1 2 1 2... ... ... ... ... ... ... ...m m m m m m m m m mn n n n n na b K a b K a b K a a a b b b K K K+ + + + + + + + + + + + Du = xy ra khi v ch khi:

1 1 1 2 2 2: : ... : : : ... : : : ... :n n na b K a b K a b K= = ( chng minh tng t nh trn) I- MT S V D : Bi 1: Cho , ,x y z l ba s dng tha 4 9 16 49x y z+ + = . Chng minh rng:

1 25 64 49Tx y z

= + +

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ng thc xy ra khi no? Hng dn gii

p dng BT Bunhiacpxki cho su s 2 ;3 ;4x y z v 1 5 8; ;x y z

ta c:

( ) ( ) ( ) ( ) 22 22 2 21 25 84 1 5 849. 4 9 16 2 3 4T x y z x y zx y z x y z = + + + + = + + + +

2

21 5 82 . 3 . 4 . 49x y zx y z

+ + =

1 25 64 49Tx y z

= + +

ng thc xy ra khi

121 5 852 3 43

4 9 16 49 2

x

x y z yx y z z

= = = = + + = =

Bi 2 : Cho 0; 0x y> > v 2 2x y x y+ + .Chng minh:

3 2 5x y+ + Hng dn gii

Gi thit: 2 2

2 2 1 1 12 2 2

x y x y x y + + +

p dng BT Bunhiacpxki cho 2 b s: ( ) 1 11;3 ; ;2 2

x y ta c: 2 2 21 1 1 11. 1 3. 10 5

2 2 2 2y x y

+ +

( )23 2 5x y + 3 2 5x y + 3 2 5x y + +

ng thc xy ra khi

1 52 101 3 52 10

x

y

= + = +

Bi 3 : Cho , , 0a b c ; 1a b c+ + = .Chng minh:

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2 2 2

1 1 1 1 30ab bc aca b c

+ + + + + Hng dn gii

Gi 2 2 21 1 1 1A

ab bc aca b c= + + ++ +

p dng BT Bunhiacpxki cho 2 b s:

( )2 2 2

2 2 2

1 1 1 1; ; ;

;3 ;3 ;3

ab bc caa b c

a b c ab bc ca

+ + + +

Ta c: ( ) ( )2 2 2 21 3 3 3 9 9 9a b c ab bc ca A+ + + + + + + + ( ) ( )2100 7a b c ab bc ca A + + + + + (*)

M ( )21 1 (do 1)3 3

ab bc ca a b c a b c+ + + + = + + = Do : (*) 30.A ng thc xy ra khi 1

3a b c= = =

Bi 4 : Cho ; ; 0x y z > v tho 1x y z+ + .Chng minh : 2 2 22 2 21 1 1 82x y zx y z+ + + + + Hng dn gii

Gi 2 2 22 2 21 1 1S x y zx y z

= + + + + +

p dng BT Bunhiacpxki cho 2 b s: ( ) 11;9 ; ;xx

Ta c: 2 22 29 1 11 81. 82.x x xx x x

+ + + = + (1)

Tng t: 2

2

9 182.y yy y

+ + ` (2)

2 29 182.z zz z

+ + (3)

Cng (1),(2) v (3) theo v ta c: 1 1 1. 82 9S x y zx y z

+ + + + +

Hay ( ) ( )1 1 1. 82 81 9 80S x y z x y zx y z + + + + + + +

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( ) 1 1 12.9.3. 80 162 80 82x y zx y z

+ + + + =

Vy 2 2 22 2 21 1 1 82x y zx y z

+ + + + + Bi 5 : Cho ba s thc dng , ,a b c tho ab bc ca abc+ + = .Chng minh rng:

2 2 2 2 2 22 2 2 3b a c b a cab bc ca+ + ++ +

Hng dn gii

Ta c: 2 2 2 2

2 2 2 2

2 2 1 12b a b aab a b a b+ += = + (do ,a b dng)

t 1 1 1; ;x y za b c

= = = th

gi thit , , 0 ; ; 0

1a b c x y zab bc ca abc x y z

> > + + = + + =

v (pcm) 2 2 2 2 2 22 2 2 3x y y z z x + + + + + p dng BT Bunhiacpxki ta c:

( ) ( ) ( )22 2 2 2 23 2 3x y x y y x y y+ = + + + + ( )2 2 12 2

3x y x y + +

Tng t ( )2 2 12 23

y z y z+ +

( )2 2 12 23

z x z x+ +

Vy ( )2 2 2 2 2 2 12 2 2 3 3 3 33

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

ng thc xy ra khi 13

x y z= = =

Vi 13

x y z= = = th 3a b c= = = Bi 6 : Chng minh: ( )1 1 1 1a b c c ab + + + vi mi s thc dng ; ; 1a b c

Hng dn gii t 2 2 21 ; 1 ; 1a x b y c z = = = Vi ; ; 0.x y z > Bt ng thc cn chng minh tr thnh:

( ) ( )( )2 2 21 1 1 1x y z z x y + + + + + + p dng BT Bunhiacpxki ta c:

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( )( ) ( )( )2 2 2 21 1 1 1x y x y x y z x y z+ + + + + + + + (1) ( )( ) ( )( )2 2 2 2 21 1 1 1 1. 1x y z x y z+ + + + + + + (2)

Kt hp (1) v (2) ta c ( ) ( )( )2 2 21 1 1 1x y z z x y + + + + + + Vy ( )1 1 1 1a b c c ab + + + (pcm) Bi 7 : Cho ; ; 0a b c > v tho 1abc = .Chng minh: ( ) ( ) ( )3 3 3

1 1 1 32a b c b c a c a b


t 1 1 1; ;x y za b c

= = = 1; 0; 0; 0xyz x y z = > > >

Ta cn chng minh bt ng thc sau : A=2 2 2 3

2x y z

y z z x x y+ + + + +

p dng BT Bunhiacpxki cho 2 b s : ( ); ; ; ; ;x y zy z z x x yy z z x x y

+ + + + + +

Ta c: ( ) ( )2x y z y z z x x y A+ + + + + + + 33 3.

2 2 2x y zA xyz+ + = (do 1xyz = ) 3

2A

ng thc xy ra khi 1x y z= = = Vi 1x y z= = = th 1.a b c= = = Bi 8 : Cho ; ; 0a b c > .Chng minh:

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

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

Hng dn gii

p dng BT Bunhiacpxki cho 2 b s: ( ) ( ); ; ;a b c a Ta c:

( ) ( )( ) ( )( )2ac ab a b c a ac ab a b c a+ + + + + + ( )( )a ac ab a a b c a + + + + +

( )( )a a a

a ac ab a b ca a b a c =+ + + ++ + + (1)

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Tng t: ( )( )b b

a b cb b c b a + ++ + + (2)

( )( )c c

a b cc c a c b + ++ + + (3)

Cng (1),(2) v (3) theo v ta c:

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

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

ng thc xy ra khi a b c= = .

Bi 9 : Cho ; 0a b > v tho 2 2 9a b+ = .Chng minh : 3 2 33 2

aba b

+ + Hng dn gii

Ta c: 2 2 9a b+ = ( )( )( )

22 9

2 3 3

ab a b

ab a b a b

= + = + + +

2 33

33 2 2

ab a ba b

ab a ba b

= + + ++ = + +

M theo BT Bunhiacpxki th 2 22. 3 2a b a b+ + = Nn 3 2 3

3 2ab

a b+ +

ng thc xy ra khi 2 2

; 0392

a b

a b a b

a b

> + = = = =

Bi 10: Cho ; ; ;a b c d dng tu .Chng minh : 1 1 1 p q p q p qa b c pa qb pb qc pc qa

+ + ++ + + ++ + + Hng dn gii

p dng BT Bunhiacpxki ta c

( ) ( )2

2 . .p q p qp q pa qb pa qba b a b

+ = + + +

Tng t ta chng minh c

( ) ( ) ( ) ( )2 2 ; p q p qp q pb qc p q pc qab c c a

+ + + + + + Cng cc v tng ng ca ba bt ng thc ta c :

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( ) ( )2 1 1 1 1 1 1p q p qpa qb pb qc pc qa a b c

+ + + + + + + + +

Hay ( ) 1 1 1 1 1 1p qpa qb pb qc pc qa a b c

+ + + + + + + +

Vy 1 1 1 p q p q p qa b c pa qb pb qc pc qa

+ + ++ + + ++ + + Bi 11 : Cho 4 s dng ; ; ;a b c d .Chng minh:

3 3 3 3 2 2 2 2

3a b c d a b c d

b c d c d a b d a a b c+ + ++ + + + + + + + + + +

Hng dn gii

t 3 3 3 3a b c dP

b c d c d a b d a a b c= + + ++ + + + + + + +

p dng BT Bunhiacpxki cho 2 b s:

( ) ( ) ( ) ( )( )3 3 3 3; ; ; ; ; ; ;a b c d a b c d b c d a c d b a d a b cb c d c d a b d a a b c + + + + + + + + + + + + + + + + Ta c:

( ) ( ) ( ) ( ) ( )22 2 2 2a b c d P a b c d b c d a c d a b d a b c+ + + + + + + + + + + + + + ( ) ( ) ( )2 22 2 2 2 2 2 2 2a b c d P a b c d a b c d + + + + + + + + + (1)

p dng BT Bunhiacpxki cho 2 b s: ( ) ( ); ; ; ; 1;1;1;1a b c d ta c: ( ) ( )2 2 2 2 24a b c d a b c d+ + + + + + (2)

T (1) v (2) ta c ( ) ( )22 2 2 2 2 2 2 22 2 2 2

3

3

a b c d P a b c d

a b c d P

+ + + + + + + + +

Vy 3 3 3 3 2 2 2 2

3a b c d a b c d

b c d c d a b d a a b c+ + ++ + + + + + + + + + +

Bi 12 : Cho cc s dng ; ;a b c tha a + b + c = 1 . Chng minh : 11 1 1

a b cb a c b a c


t 1 1 1 2 2 2

a b c a b cAb a c b a c b c c a a b

= + + = + ++ + + + + + p dng BT Bunhiacpxki ta c:

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

22 2 2 2

2 2 2

2 2 22 2 2

a b ca b c a b c b c a c a bb c c a a b

a b c a b c b c a c a bb c c a a b

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

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( )( )

2

3a b c

Aab bc ca+ + + +

Ta li c:

( ) ( )2 3a b c ab bc ca+ + + + . Suy ra ( )( )3

13

ab bc caA

ab bc ca+ + =+ +

Vy 11 1 1

a b cb a c b a c

+ + + + +

Du ng thc xy ra khi 2 2 2

13

1

b c c a a ba b c a b ca b c

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

Bi 13 : Gi s cc s thc ; ; ;x y z t tho mn iu kin: ( ) ( )2 2 2 2 1a x y b z t+ + + = vi ;a b l hai s dng cho trc. Chng minh: ( )( ) a bx z y t

ab++ +

Hng dn gii Do ; 0a b > nn t gi thit ta c: ( ) ( ) 2 2 2 22 2 2 2

2 2 2 2

11

1

x y z ta x y b z tb a ab

x z y tb a b a ab

+ ++ + + = + =

+ + + =

p dng BT Bunhiacpxki ta c:

( ) ( )2 2 2

2 . .x z x zx z b a b ab ab a

+ = + + + (1)

Tng t : ( ) ( ) 2 22 y ty t b ab a

+ + + (2)

Cng tng v (1) v (2) ta c:

( ) ( ) ( ) 2 2 2 22 2 x z y t a bx z y t b ab a b a ab

++ + + + + + + = (3)

Mt khc ( ) ( ) ( )( )2 2 2x z y t x z y t+ + + + + (4) Do t (3) v (4) suy ra: ( )( ) a bx z y t

ab++ +

Du ng thc xy ra

x zb a x yy t

axb a z tbx z y t

= = = = = + = +

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Bi 14 : Cho cc s thc dng ; ; ;x y z t tho mn 1.xyzt = Chng minh:

( ) ( ) ( ) ( )3 3 3 31 1 1 1 4

3x yz zt ty y xz zt tx z xt ty yx t xy yz zx+ + + + + + + + + + +

Hng dn gii

Vi ; ; ;x y z t dt 1 1 1 1; ; ; ( ; ; ; 0)a b c d a b c dx y z t

= = = = > v 1abcd = 1 1 1 1; ; ;x y z ta b c d

= = = = Bt ng thc cn chng minh tng vi:

3 3 3 3

1 1 1 1 41 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3

bc cd bd ac cd ad ad bd ab ab bc aca b c d

+ + + + + + + + + + +

3 3 3 3 43

a b c db c d c d a d a b a b c

bcd adc abd abc

+ + + + + + + + + + +

( ) ( ) ( ) ( )3 3 3 3 4

3a b c d

a b c d b c d a c d a b d a b c + + + + + + + + + + + (v 1abcd = )

2 2 2 2 43


+ + + + + + + + + + +

t 2 2 2 2a b c dS

b c d c d a d a b a b c= + + ++ + + + + + + +

p dng BT Bunhiacpxki ta c: ( ) ( ) ( ) ( ) ( )2.S b c d c d a d a b a b c a b c d+ + + + + + + + + + + + + +

( )( ) ( )

21

3 3a b c d

S a b c da b c d+ + + = + + ++ + + (1)

p dng BT Cauchy vi 2 s dng: 2 ; 2a b ab c d cd+ +

Suy ra ( )2a b c d ab cd+ + + + Li p dng BT Cauchy cho 2 s dng ;ab cd ta c:

42 2 2ab cd abcd abcd+ = = (v 1abcd = ) (2) T (1) v (2) suy ra 4

3S

Vy

3 3 3 3

1 1 1 1 41 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3

bc cd bd ac cd ad ad bd ab ab bc aca b c d

+ + + + + + + + + + +

Du ng thc xy ra khi 1 1a b c d x y z t= = = = = = = = .

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4 4 4 41 2 3 43 3 3 31 2 3 4

14

x x x xx x x x+ + + + + +

Bi 15 : Cho 1 2 3 4; ; ;x x x x dng tho iu kin 1 2 3 4 1x x x x+ + + = .Chng minh :

Hng dn gii p dng BT Bunhiacpxki ta c:

( ) ( )2 2 2 2 21 2 3 4 1 2 3 41 4x x x x x x x x= + + + + + + 2 2 2 21 2 3 4

14

x x x x + + + (1) ( ) ( )22 2 2 2 3 3 3 31 2 3 4 1 1 2 2 3 3 4 4. . . .x x x x x x x x x x x x+ + + = + + + ( )( )3 3 3 31 2 3 4 1 2 3 4x x x x x x x x + + + + + + 3 3 3 31 2 3 4x x x x= + + + (v 1 2 3 4 1x x x x+ + + = )

3 3 3 32 2 2 21 2 3 41 2 3 42 2 2 2

1 2 3 4

x x x xx x x x

x x x x+ + + + + ++ + + (2)

( )23 3 3 31 2 3 4x x x x+ + + ( )2 2 2 21 1 2 2 3 3 4 4. . . .x x x x x x x x= + + + ( )( )2 2 2 2 4 4 4 41 2 3 4 1 2 3 4x x x x x x x x + + + + + +

4 4 4 4 3 3 3 31 2 3 4 1 2 3 43 3 3 3 2 2 2 21 2 3 4 1 2 3 4

x x x x x x x xx x x x x x x x+ + + + + + + + + + + + (3)

T (1);(2) v (3) suy ra:

Bi 16 : Cho bn s dng ; ; ;a b c d .Chng minh:

( )( ) ( )( ) ( )( ) ( )( )4 4 4 4

2 2 2 2 2 2 2 2 4a b c d a b c d

a b a b b c b c c d c d d a d a+ + ++ + + + + + + + + + +

Hng dn gii p dng BT Bunhiacpxki ta c: ( ) ( ) ( )( ) ( ) ( )2 22 2 2 2 2 2 2 22 2 4a b a b a b a b a b a b+ + + + + + (1) ( )( ) ( )

4 4

2 2

14

a b a ba b a b

+ ++ +

Mt khc: ( )( )4 4

2 2

a b a ba b a b

= + +

t ( )( ) ( )( ) ( )( ) ( )( )4 4 4 4

2 2 2 2 2 2 2 2

a b c dNa b a b b c b c c d c d d a d a

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

4 4 4 41 2 3 43 3 3 31 2 3 4

14

x x x xx x x x+ + + + + +

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Ta c: ( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )( )( )

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

2 2 2 2 2 2 2 22

a b a b b c b c c d c d d a d aN

a b a b b c b c c d c d d a d a

+ + + + + + + += + + ++ + + + + + + + (1)

( ) ( ) ( ) ( )1 1 1 124 4 4 4

N a b a b b c b c c d c d d a d a + + + + + + + + + + +

( ) ( )1 124 4

N a b b c c d d a N a b c d + + + + + + + + + + ( pcm ) Bi 17 : Cho ; ;a b c l cc s thc dng.Chng minh:

2 2 21

8 8 8

a b ca bc b ac c ab

+ + + + +

(Trch thi Olympic Ton Quc T ln th 42, nm 2001) Hng dn gii

t 2 2 28 8 8

a b cAa bc b ac c ab

= + ++ + +

p dng BT Bunhiacpxki hai ln ta c:

( )2

2 2 2 24 4 4

2 2 24 4 4

2 2 2

2 2 2

3 3 3

. . 8 . . 8 . . 88 8 8

. 8 8 88 8 8

. . 8 . 8 . 8

a b ca b c a a bc b b ac c c aba bc b ac c ab

a b c a a bc b b ac c c aba bc b ac c ab

A a a abc b b abc c c abc

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

( )( )3 3 3. 24A a b c a b c abc + + + + + (1) Mt khc

( ) ( )( )( )3 3 3 3 3a b c a b c a b b c a c+ + = + + + + + + p dng BT Cauchy vi hai s dng ta c:

2 ; 2 ; 2a b ab b c bc a c ac+ + + Suy ra:

( )( )( ) 8a b b c a c abc+ + + ( ) ( )( )( )3 3 3 3 3 3 33 24a b c a b c a b b c a c a b c abc + + = + + + + + + + + + (2) T (1) v (2) suy ra:

( ) ( )( ) ( )2 3 2. .a b c A a b c a b c A a b c+ + + + + + = + + Do 1A , ngha l

2 2 21

8 8 8

a b ca bc b ac c ab

+ + + + +

Du ng thc xy ra khi a b c= = . Bi 18 : Cho ; ;x y z + tho 1xy yz zt tx+ + + = .Chng minh:

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3 3 3 3 13

x y z ty z t x z t x y t x y z

+ + + + + + + + + + + Hng dn gii

p dng BT Bunhiacpxki ta c: ( ) ( )( )2 2 2 2 2 2 2 2 2xy yz zt tx x y z t y z t x+ + + + + + + + +

2 2 2 21 x y z t + + + (1) t: ; ; ;X y z t Y x z t Z x y t T x y z= + + = + + = + + = + + Khng mt tnh tng qut gi s: x y z t

2 2 2 2x y z t v 3 3 3 3x y z t v y z t x z t x y t x y z X Y Z T+ + + + + + + + 1 1 1 1

X Y Z T

p dng BT Tr-b-sp cho hai dy s sau: 3 3 3 3

1 1 1 1x y z t

X Y Z T

( )3 3 3 3 3 3 3 31 1 1 1 14x y z t x y z tX Y Z T X Y Z T + + + + + + + + + (2) p dng BT Tr-b-sp cho hai dy

2 2 2 2

x y z tx y z t

( ) ( )( )3 3 3 3 2 2 2 214x y z t x y z t x y z t+ + + + + + + + + Mt khc:

( ) ( )1 13 3

x y z t x y z x y t x z t y z t X Y Z T+ + + = + + + + + + + + + + + = + + +

( ) ( ) ( )3 3 3 3 2 2 2 21 1.4 3x y z t x y z t X Y Z T + + + + + + + + + (3) T (2) v (3) rt ra:

( )( )3 3 3 3 2 2 2 21 1 1 1 148x y z t x y z t X Y Z TX Y Z T X Y Z T + + + + + + + + + + + + Theo (1) ta li c: 2 2 2 21 x y z t + + + p dng BT Cauchy cho ; ; ; 0X Y Z T > ta c:

( )

4

4

4 . . .

1 1 1 1 14. . .

1 1 1 1. 16

X Y Z T X Y Z T

X Y Z T X Y Z T

X Y Z TX Y Z T

+ + + + + +

+ + + + + +

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Vy 3 3 3 3 1 1.1.16

48 3x y z tX Y Z T+ + + =

Thay ; ; ;X Y Z T ta c kt qu: 3 3 3 3 1

3x y z t

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

Du ng thc xy ra khi 12

x y z t= = = = Bi 19 : Cho n l s t nhin.Chng minh rng:

( )1 2 ... 2 1n nn n nC C C n+ + + Hng dn gii

Chn hai dy ( ) ( )1 21 2 1 2; ;...; ; ... 1nn n n n na C a C a C b b b= = = = = = = p dng BT Bunhiacpxki ta c: ( ) ( )( )21 2 1 2... ... 1 1 ... 1n nn n n n n nC C C C C C+ + + + + + + + + (1) Theo nh thc Newton ta c: ( )

1

nn k k n k

nk

a b C a b =

+ = Cho 1a b= = .Ta c:

0 1 12 ... 2 1 ...n n n nn n n n nC C C C C= + + + = + + Vy t (1) ta c:

( )1 2 ... 2 1n nn n nC C C n+ + + Du ng thc xy ra khi 1 2 ... 1nn n nC C C n= = = = .

Bi 20 : Cho ; ; ; 0a b c d > .Chng minh : 22 3 2 3 2 3 2 3 3


+ + + + + + + + + + + (Trch d b Quc T Ton M nm 1993)

Hng dn gii

p dng BT Bunhiacpxki ta c:2

1 1 1

n n ni

i i ii i ii

xx y x

y= = =

vi ( ) ( ) ( ) ( )1 2 3 4 1 2 3 44; ; ; ; ; ; ; ; ; ; ; 2 3 ; 2 3 ; 2 3 ; 2 3n x x x x a b c d y y y y b c d c d a d a b a b c= = = + + + + + + + + VT ( )( )

2

4a b c d

ab ac ad bc bd cd+ + + + + + + + (1)

Mt khc ( ) ( )238

ab ac ad bc bd cd a b c d+ + + + + + + + (2)

T (1) v (2) VT 23

( pcm )

Bi 21 : Cho 0; 0; 0a b c> > > .Chng minh : 4 4 4 3 3 3

2a b c a b c

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

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Hng dn gii

t 4 4 4

2 2 21 2 3; ;

a b cx x xb c c a a b

= = =+ + + v ( ) ( ) ( )2 2 2 2 2 2

1 2 3; ;a b c y b c a y c a b y+ = + = + = p dng BT Bunhiacpxki ta c cho cc s 1 2 3; ;x x x v 1 2 3; ;y y y ta c:

( ) ( ) ( ) ( )4 4 4 2 22 2 2 3 3 3a b c a b c b c a c a b a b cb c c a a b + + + + + + + + + + + + Nn

( )( ) ( ) ( )

23 3 34 4 4

2 2 2

a b ca b cb c c a a b a b c b c a c a b

+ ++ + + + + + + + + + chng minh c bi ton ta cn chng minh: ( ) ( ) ( ) ( )23 3 3 2 22 a b c a b c b c a c a b+ + + + + + + (**)

(**) 3 3 2 2 3 3 2 2 3 3 2 2 0a b a b b a b c b c bc c a c a ca + + + + + ( ) ( ) ( ) ( ) ( ) ( )2 2 2 0a b a b b c b c c a c a + + + + + (***) Bt ng thc (***) l ng (**) l ng Bi ton ng. Vy

4 4 4 3 3 3

2a b c a b c

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

Bi 22 : Cho 0; 1;2;...;ix i n> = c 1 2 ... 1nx x x+ + + = .Cho 1 2; ;...; ni i ix x x l hon v ca 1 2; ;...; nx x x .Chng minh:

( )22 21

11

k

n

kk i

nx

x n=

++ Hng dn gii

Theo Bunhiacpxki: 22 2

1 1 1 1

1 1 1.k k k

n n n n

k k kk k k ki i i

n x x xx x x= = = =

+ + = +

M 1

1n

kk

x=

= 22 21 1 1

1

1 1k

k kk

n n n

i nk k ki i

ik

nx n nx x x= = =

=

=

Vy ( )22 2

1

11

k

n

kk i

nx

x n=

++

BI TP :

Bi 1: Cho ; ; ; 0a b c d > v tha ( )32 2 2 2c d a b+ = + .Chng minh: 3 3 1a bc d+

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Bi 2: Cho ; ; ; 0a b c d > .Chng minh: 1 1 4 16 64a b c d a b c d+ + + + + +

Bi 3: Cho ; ;a b c l 3 s dng v 2 2 2 1a b c+ + .Chng minh:3 3 3 1

2a b c

b c c a a b+ + + + +

Bi 4: Cho 2 2 2 1a b c+ + = .Chng minh: 1 3a b c ab ac bc+ + + + + + Bi 5: Cho ; ;a b c l cc s dng.Chng minh:

4 4 4 2 2 2

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

a ba b b bc c c ac a+ ++ + + + + + + +

Bi 6: Cho 3 s ; ;x y z tho ( ) ( ) ( ) 41 1 13

x x y y z z + + .Chng minh: 4x y z+ +

Bi 6: Cho ; ;a b c l 3 s khng m.Chng minh: 2 2 23 3 3

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

Bi 7: Cho 3 s dng ; ;a b c c 1abc = .Chng minh: 2 2 2 2 2 2 32bc ca ab

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

Bi 8: Cho 3 s dng ; ;x y z c 1x y z+ + = .Chng minh: 11 1 9 3 32

yx zy z z x x y

++ + ++ + + + +

Bi 9: Chng minh:( )2a b ca b c

x y z x y z

+ ++ + + +

Bi 10: Cho 0x y z > .Chng minh: ( )2 2 2 22 2 2x y y z z x x y zz x y+ + + + Bi 11: Cho 1; 1a b .Chng minh: 2 2 2log log 2 log 2

a ba b + +

Bi 12: Cho ; ; 0a b c > .Chng minh: ( ) ( )23 3 3 1 1 1a b c a b ca b c + + + + + + Bi 13: Cho ; ;a b c .Chng minh: ( ) ( ) ( )2 2 22 2 2 3 21 1 1

2a b b c c a+ + + + +

Bi 14: Cho ; ; 0x y z > v 32

x y z+ + .Chng minh: 2 2 22 2 21 1 1 3 172x y zx y z+ + + + + Bi 15: Cho trc 2 s dng ;a b v 2 s dng ;c d thay i sao cho a b c d+ < + .Chng minh:

( )22 2a cc ac d a b c d a b

+ + + + . Du = xy ra khi no?

Bi 16: Cho 1 2; ;...; na a a l cc s thc tho mn2 2 21 2 ... 3na a a+ + + = .Chng minh: 1 2 ... 22 3 1

naa an

+ + + .Chng minh: 3a b cpb qc pc qa pa qb p q

+ + + + + + Bi 18: Chng minh rng vi mi ( )1;2;...;ia i n = ta c:

( ) ( ) ( )2 2 22 2 21 2 2 3 11 1 ... 1 2nna a a a a a+ + + + + +

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Bi 1: Cho ABC tho mn h thc:3 3 3 22( )

9a b c a b c

br cR cr aR ar bR R+ ++ + =+ + + (1).CM ABC u

Hng dn gii

n gin ta t: 000

x br cRy cr aRz ar bR

= + >= + >= + >

(2)

vy (1)3 3 3 22( )

9a b c a b cx y z R

+ + + + = T (2) ta c:

( )( )ax by cz ab bc ca r R+ + = + + + (3)3 3 3

4 4 4 2 2 2 2 2 2( )( ) ( ) ( ) ( )a b c y x z y x zax by cz a b c ab a b bc b c ca c ax y z x y y z z x

+ + + + = + + + + + + + + Theo BTCauchy,ta c:

3 3 34 4 4 2 2 2 2( )( ) 2 . .2 .2 ( )a b cax by cz a b c ab ab bc bc ca ca a b c

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

Suy ra : ( )3 3 3 2 2 2( )( )

( )a b c a b cx y z ab bc ca r R

+ ++ + + + + (theo 3) (4) mt khc ta lun c (Cauchy): 2 2 2a b c ab bc ca+ + + + nn (4):

3 3 3 2 2 2 2 2 2 2

2 2 2

( )( )( )

a b c a b c a b cx y z a b c r R r R

+ + + ++ + =+ + + +

2( )

3( )a b c

r R+ + + (theo BT BCS)

M 92 3( ) 3( )2 2R RR r r R R + + =

t :3 3 3 22( )

9a b c a b cx y z R

+ ++ + 3 3 3 22( )

9a b c a b c

br cR cr aR ar bR R+ + + + + + +

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du = xy ra khi

2 2 2 2 2 2, ,

a b cR r

y y z y x za b b c c ax z y z y x

= = = = = =

ABC u

Bi 2 : CM: 1 cos cos cos 3 sin sin sinA B C A B C+ vi A, B,C nhn

Hng dn gii

Do tgA>0,tagB>0,tgC>0 v 12 2 2 2 2 2A B B C C Atg tg tg tg tg tg+ + =

p dng BCS ta c: 2 2 2 2 2 2 12 2 2 2 2 2 3A B B C C Atg tg tg tg tg tg+ + (1)

Mt khc theo BT Cauchy ta c: 2 2 233

2 2 2 2 2 2 2 2 2A B B C C A A B Ctg tg tg tg tg tg tg tg tg+ + (2)

132 2 2 3A B Ctg tg tg

t (1)v(2): 2 2 2 2 2 2 41 4 3

2 2 2 2 2 2 3 2 2 2A B B C C A A B Ctg tg tg tg tg tg tg tg tg+ + +

2 2 2 2 2 21 1 1 1 1 1 8 32 2 2 2 2 2 2 2 2A B C A B C A B Ctg tg tg tg tg tg tg tg tg + + + + 2 2 2

2 2 2 2 2 2

1 1 1 2 2 22 2 2 2 2 21 . . 3 . .

1 1 1 1 1 12 2 2 2 2 2

A B C A B Ctg tg tg tg tg tg

A B C A B Ctg tg tg tg tg tg

+

+ + + + + +

1 cos cos cos 3 sin sin sinA B C A B C + Du = xy ra khi ABC u Bi 3 : Cho a, b, c, l s o 3 cnh .chng minh rng

acbaT += 22 + 12222 +++ cba

cbac

b

Hng dn gii p dng BT Bunhiacpxki cho 6 s:

( ) ( ) ( )cbacbacbacbacba

cbac

bacb

a ++++++ 22;22;22;22;22;22 Ta c: ( ) ( ) ( )[ ] ( )2222222. cbacbacbacbacbaT +++++++ Sau dng bin i tng ng chng minh: (a + b+ c)2 4ab +4bc +4ca a2 b2 - c2 T suy ra pcm.

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Bi 4 : Cho ABC v ng trn ni tip , cc tip tuyn ca ng trn song song vi 3 cnh ca nh v c din tch S1; S2; S3. Gi S l din tch ABC . Chng minh: 3321

SSSS ++ Hng dn gii

Gi s S1= SAMN

Ta c: AMN ng dng ABC vi t s ng dng l: ha

rha 2 vi r l bn knh ng trn ni tip v ha l ng cao k t nh A.

Ta c: 22

1 12

=

=

pa

harha

SS

(V S = pa

harpraha == 2

21 vi p l na chu vi)

Vy: pa

SS =11

Tng t: pb

SS =12 ;

pc

SS =13

Do : 13321 =++=++p

cbaS

SSS

p dng BT Bun ta c:

S = ( ) ( )( )3212222321 111.1.1.1 SSSSSS ++++++ 1 2 3 3

SS S S+ + (pcm). Du = xy ra khi ABC u Bi 5 : Cho ABC v 1 im Q no trong . Qua Q k ng thng song song vi AB ct AC M v ct BC N. Qua im Q k ng thng song song vi AC ct AB F; ct BC E. Qua E k ng thng song song vi BC ct AC P, ct AB R. K hiu S1= dt(QMP); S2 = dt(QEN); S3 = dt(QFR) v S = dt(ABC).Chng minh:

a) ( )21 2 3S S S S= + + b) 1 2 3 13S S S S+ + Hng dn gii

a) Ta c: QMP ng dng BAC (t s MPAC

).

Suy ra 2

11 SS MP MPS AC ACS

= = .

Tng t 32 ;SS PC AM

AC ACS S= =

Do : 1 2 3 1S S S MP PC AM AC

AC ACS+ + + += = =

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Suy ra: ( )21 2 3 1 2 3S S S S S S S S= + + = + + b) p dng BT Bunhiacpxki ta c:

( ) ( )( )2 2 2 21 2 3 1 2 31 2 3

1. 1. 1. 1 1 1

1Suy ra 3

S S S S S S S

S S S S

= + + + + + ++ +

Du = xy ra khi 1 2 3S S S= = Q l trng tm ABC Bi 6 : Cho a , b , c l 3 cnh ca tam gic.Chng minh:

a b c a b cb c a c a b a b c

+ + + ++ + + Hng dn gii

t 000

b c a xc a b ya b c z

+ = > + = > + = >

Khi ta cn chng minh:

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

2 (1)

y z z x x y y z z x x yx y z

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

+ + + + + ++ + + +

+ + + + + + + + + +

D thy ( )(1) 2VT xy yz zx + + (2) Theo BT Bunhiacpxki ta c:

( ) ( )( )

( )

26

6

(2) 2 3 (3)

x y y z z x x y z

x y y z z x x y z

VT xyz x y z

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

+ +

R rng ta c ( )

( ) ( )( )

2 2 2 2 2 2

2 3

3 (4)

x y x y x y xyz x y z

xy yz zx xyz x y z

xy yz zx xyz x y z

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

T (1) (2) (3) (4)pcm. Du = xy ra khi a b c= = Bi 7 : Cho ABC. Chng minh : a2b(a b) +b2c(b a) + c2a(c a) 0

( Trch thi v ch ton quc t 1983 )

Hng dn gii

Gi A; B; C l cc tip im:

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t: ' '' '' '

AB AC x a y zBA BC Y b z xCA CB Z c x y

= = = + = = => = + = = = +

vy: a2b(a b) + b2c(b c) + c2a(c a) 0

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

< => y3z + z3x + x3y xyz(x+y+z) 0

y3z + z3x + x3y xyz (x+y+z) 2 2 2

(*)y z x x y zx y z

+ + + +

p dng BT Bunhiacpxki(bin dng) ta c:

( )22 2 2 x y zx y z x y zz x y x y z

+ ++ + = + ++ +

vy (*) ng ( pcm ) .

Bi 8 : Vi a; b; c l di 3 cnh ca . CMR : 4 9 16 26a bb c a a c b a b c


t: 4 9 16a b cPb c a a c b a b c

= + ++ + +

Ta c: 2 2 22 4. 9 16a b cPb c a a c b a b c

= + ++ + +

4. 1 9 1 16 1a b c a b c a b cb c a a c b a b c+ + + + + + = + + + + + +

( ) 4 9 16 29a b cb c a a c b a b c = + + + + + + +

( ) ( ) ( ) 4 9 16 29b c a a c b a b cb c a a c b a b c = + + + + + + + + + +

p dng BT Bunhiacpxki, ta c:

( )2

2 2 3 481 2 3 4 . .b c a a c b a b cb c a a c b a b c

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

( ) ( ) ( ) 4 9 16b c a a c b a b cb c a a c b a b c + + + + + + + + + +

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=> 2P 81 - 29

=> 2P 52 => P 26

Chn a = 7; b = 6; c = 5 th du ng thc xy ra.

Bi 9 : Cho elip (E): 2 2

116 9x y+ = cc im M; N chuyn ng ln lt trn, cc tia Ox; Oy sao cho MN lun

tip xc vi (E). Xc nh to ca M; N on MN c di nh nht. Tm gi tr nh nht .

Hng dn gii

C1: Gi M(m;O) v N(O,r) vi m; n>0 l 2 im C2 ng trn 2 tia Ox; Oy.

ng thng MN c pt: 1 0x ym n+ =

ng thng ny tx vi (E) khi v ch khi: 2 21 116 9 1

m n + =

Theo BT Bunhiacpxki. Ta c 2

2 2 2 2 22 2

16 9 4 3MN ( ) . . 49m n m n m nm n m n

= + = + + + =

=> MN 7

Du = xy ra 2 2

4 3: :

16 9 1 2 7; 21

0; 0

m nm n

m nm nm n

= + = = = > >

Vy vi (2 7;0; (0; 21)M N th MN t GTNN v GTNN ca Mn l 7

C2: Pt tip tuyn ti im (x0; y0) thuc (E) l 0 0. . 116 9x x y y+ =

Suy ra to ca M v N l 0

16 ;0Mx

v

0

90;Ny

( )2 22 2 2 22 0 02 2 2 20 0 0 0

16 9 16 9 4 3 4916 9x yMN

x y x y = + = + + + =

Khi ( ) ( )2 7;0 ; 0; 21M N= v GTNN ca MN l 7 Bi 10 : Cho ABC. Cho p; q; r >0. CMR: 2 2 2 2 . 3P q ra b c s

q r r p p q+ + + + +

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(Trch tp ch ton hc v tui tr)

Hng dn gii

Trc ht ta chng minh bi ton sau:

Trong ABC ta c: 2 2 2 2 2 24 3 ( ) ( ) ( )a b c s a b b c c a+ + + + + Tht vy:

(2) 2 2 2 2 2 2( ) ( ) ( ) 4 3a b c b c a c a b s + + 4( )( ) 4( )( ) 4( )( ) 4 3p a p b p b p c p c p a s + +

3xy yz zx s + + vi 000

x p ay p bz p c

= > = > = >

3( )xy yz xz x y z xyz + + + + (V theo cng thc Hrng: ( )( )( ) ( )s p p a p b p c xyz x y z= = + +

2 2 2( ) ( ) ( ) 0xy yz yz zx zx xy + + BT ny ng. vy (2) c chng minh:

Mt khc theo BT Bunhiacpxki. Ta c: 2

2( ) a b ca b c q r r p p qq r r p p q

+ + = + + + + + + + +

2 2 2

2 ( )a b c p q rp r r p p q

+ + + + + + +

2 2 2 2 2 22 2( )p q ra b c a b cq r r p p q

+ + + + + + + +

2 2 2 2 2 2 2 22 ( ) 2( )p q ra b c a b c a b cq r r p p q

+ + + + + + + + +

2 2 2 2 2 2( ) ( ) ( )a b c a b b c c a + + + + 4 3s

Vy: 2 2 2 2 3p q ra b c sq r r p p q

+ + + + +

Du = xy ra khi a b cp q r= = = =

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Ch :

+ Qua php chng minh trn, ta c kt qu p trong ABC 2 2 2 2 2 24 3 ( ) ( ) ( ) 4 3a b c s a b b c c a s+ + + + +

+ Ly p = q = r > 0 ta c BT quen thuc 2 2 2 4 3a b c s+ + ( thi Olympic ton quc t ln 3)

+ Ly a = b = c. ta c BT Nesbit:

32

p q rq r r p p q

+ + + + + (3)

Du = xy ra khi p = q = r > 0

+ Nu nhn 2 v ca (3) cho p + q + r > 0 ta c 2 2 2

2p q r p q r

q r r p p q+ ++ + + + +

Bi 11 : Cho t din ABCD c trng tm G, bn knh mt cu ngoi tip R. CMR

( )246

GA GB GC GD R AB AC AD BC CD DB+ + + + + + + + + ( Trch tp ch Ton hc v Tui tr)

Hng dn gii Ta c 2 b : B 1: Nu G l trng tm ca t din ABCD th ( ) ( )2 2 2 2 2 22 3

16AB AC AD CD DB BC

GA+ + + +=

Chng minh: Gi aG l trng tm ca BCD . Ta c:

( )22 29 9 1.16 16 9aGA AG AB AC AD= = + +uuur uuur uuur

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

2 2 22 2 2

2 2 2 2 2 2

3

163

16

AB AC AD AC AD AD AB AB AC

AB AC AD CD DB BC

+ + =

+ + + +=

uuur uuur uuur uuur uuur uuur uuur uuur uuur

B 2: Nu O; G theo th t l tm mt cu ngoi tip v trng tm ca t din ABCD th 2 2 2 2 2 2 2 2 2 2

2 2

4 4GA GB GC GD AB AC AD CD DB BCR OG + + + + + + + + = =

Chng minh: Theo h thc Leibnitz, vi mi im M, ta c

2 2 2 2 2 2 2 2 24MA MB MC MD GA GB GC GD MG+ + + = + + + + T , cho M trng O, ta c

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2 2 2 2 2 2 2 2 24OA OB OC OD GA GB GC GD OG+ + + = + + + + Suy ra:

2 2 2 22 2

4GA GB GC GDR OG + + + = (1)

T b 1 suy ra 2 2 2 2 2 2 2 2 2 2

4 4GA GB GC GD AB AC AD CD DB BC+ + + + + + + += (2)

T (1)(2) suy ra iu phi chng minh Tr li vic gii bi ton trn

Ta c 2 2 2 2 2 2

. . .2 2

OA GA OG GA R OGR GA OAGA OAGA + + = = =uuur uuur T theo cc b 1 v 2, ta c

2 2 2

.8

AB AC ADR GA + + Theo BT Cauchy v Bunhiacpxki, ta c

( ) ( ) ( )22 2 26 2 6 . 3 8 . 3R GA R GA R GA AB AC AD AB AC AD+ = + + + + Suy ra ( )6 R GA AB AC AD+ + +

Tng t

( )( )( )

6

6

6

R GB BC BD BA

R GC CD CA CB

R GD DA DB DC

+ + + + + + + + +

Suy ra ( )246

GA GB GC GD R AB AC AD BC CD DB+ + + + + + + + + ng thc xy ra khi v ch khi t din ABCD l u BBII TTPP :: Bi 1 : Cho na ng trn ( );O R ng knh AB, M l im chuyn ng trn na ng trn.Xc nh v tr ca M 3MA MB+ t gi tr ln nht. Bi 2 : Cho ABC ni tip ng trn bn knh R; ; ;BC a CA b AB c= = = .Gi x;y;z ln lt l khong cch t M thuc min trong ca ABC n cc cnh BC;CA;AB.Chng minh:

2 2 2

2a b cx y z

R+ ++ +

Bi 3 : Cho a , b , c l 3 cnh ca tam gic.Hy tm gi tr nh nht ca biu thc:

2 2 2

2 2 2

a b cb c aP

a b cb c a

+ +=

+ +

Bi 4 : Cho a , b , c l 3 cnh ca tam gic v 2

a b cp + += .Chng minh: 2 2 2 23635

abca b c pp

+ + +

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Bi 5 : im M nm trong ABC .H MA , MB , MC ln lt vung gc vi BC;CA;AB.Xc nh v tr ca M BC CA AB

MA MB MC+ + t gi tr nh nht.

Bi 6 : Cho t gic li ABCD.Cho ; ; ; song song ; song song M AC P BC Q AD MP AB MQ CD . Chng minh : 2 2 2 2

1 1 1MP MQ AB CD

++ . Du = xy ra khi no?

DDNNGG 11:: BBtt nngg tthhcc SScchhwwaarrttzz (( SSvvccxx )) Cho mt s nguyn dng 1n v hai dy s thc 1 2; ;...; na a a v 1 2; ;...; nb b b , trong 0; 0; 1,i ia b i n > = . Khi ta c: ( )222 2 1 21 2

1 2 1 2

......

...nn

n n

a a aaa ab b b b b b

+ + ++ + + + + +

ng thc xy ra khi v ch khi 1 21 2

... nn

aa ab b b= = = .

Chng minh: BT cn chng minh tng ng vi:

( ) ( )22 2 21 2 1 2 1 21 2

... ... ...n n nn

aa a b b b a a ab b b

+ + + + + + + + +

Hay

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

22 22 2 2

1 21 2

1 2

2

1 21 2

1 2

... ...

...

nn

n

nn

n

aa a b b bb b b

aa ab b bb b b

+ + + + + + + + +

p dng BT BCS cho hai dy s thc: 1 21 2

; ;...; nn

aa ab b b

v 1 2; ;...; nb b b ta c BT trn. T ta c

BT cn chng minh.

ng thc xy ra khi v ch khi 1 21 21 2

: : ... :n nn

aa ab b bb b b

= = =

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Hay 1 21 2

... nn

aa ab b b= = =

DDNNGG 22:: Cho 4 s ; ; ;a b c d tu ta c :

( ) ( )2 2 2 2 2 2a c b d a b c d+ + + + + + (1)

Chng minh: Ta c: ( ) ( ) ( )( )2 2 2 2 2 2 2 2 2 2(1) 2a c b d a b a b c d c d + + + + + + + + +

( )( )( )( )

2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2

2 2 2a ac c b bd d a b a b c d c d

ac bd a b c d

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

Bt ng thc cui cng ng theo BT Bunhiacpxki. VN DNG 2 DNG TRN:

Bi 1:Cho , ,a b c l cc s thc dng. Chng minh: 1) 1 1 4a b a b+ + 2)

1 1 1 9a b c a b c+ + + +

Hng dn gii p dng BT BCS ta c cc BT sau:

( )22 2 1 11 11 1 4a b a b a b a b

++ = + =+ + ( )22 2 2 1 1 11 1 11 1 1 9

a b c a b c a b c a b c+ ++ + = + + =+ + + +

Bi 2 : Cho a, b, c dng . Chng minh : 1) 2 2 2a b c a b c

b c a+ + + + 2)

3 3 32 2 2a b c a b c

b c a+ + + +

Hng dn gii

1) p dng BT BCS ta c: ( )22 2 2 a b ca b c a b c

b c a b c a+ ++ + = + ++ +

2) Ta c: ( ) ( ) ( )2 2 22 2 23 3 3 4 4 4 a b ca b c a b c

b c a ab bc ca ab bc ca+ + = + + = + + .

p dng BT BCS ta c: ( ) ( ) ( ) ( )2 2 2 22 2 2 2 2 2a b c a b c

ab bc ca ab bc ca

+ ++ + + + . Mt khc, ta bit: 2 2 2 0a b c ab bc ca+ + + + > T ta suy ra:

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

2 2 2 2 2 2.a b c a b c a b c a b cab bc ca ab bc ca

+ ++ + + + + + + +

n y ta c pcm. Bi 3 : Cho a, b, c l cc s thc dng. Chng minh:

1 1 1 1 1 12 3 2 3 2 3 6 6 6a b c b c a c a b a b c

+ + + ++ + + + + + Hng dn gii

p dng BT BCS ta c:

( )22 2 2 1 2 31 2 3 1 2 3 362 3 2 3 2 3a b c a b c a b c a b c

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

Tng t ta cng chng minh c: 1 2 3 362 3b c a b c a

+ + + +

1 2 3 362 3c a b c a b

+ + + + Cng the v ba BT trn ta nhn c: 6 6 6 36 36 36

2 3 2 3 2 3a b c a b c b c a c a b+ + + ++ + + + + +

T suy ra: 1 1 1 1 1 12 3 2 3 2 3 6 6 6a b c b c a c a b a b c

+ + + ++ + + + + + . Bi 4 : Tm GTNN ca biu thc

4 9 16a b cPb c a c a b a b c

= + ++ + + ( ) ( ) ( )2 2 2

2 2 2 2 2 2a b c a b cP

b c c a a b a b c b c a c a b= + + + ++ + + + + + trong

, ,a b c l di 3 cnh ca mt tam gic. Hng dn gii

Do , ,a b c l di 3 cnh ca mt tam gic nn , ,b c a c a b a b c+ + + l cc s thc dng. Ta c: 1 1 1 294 9 16

2 2 2 2a b cP

b c a c a b a b c = + + + + + + + +

( ) ( )( )( )2 9 16 29

2 2a b c a b c a b c

b c a c a b a b c+ + + + + += + + + + +

4 9 16 292 2

a b cb c a c a b a b c

+ + = + + + + + p dng BT BCS ta c:

2 2 24 9 16 2 3 4

b c a c a b a b c b c a c a b a b c+ + = + ++ + + + + +

( )( ) ( ) ( )22 3 4 81

b c a c a b a b c a b c+ + =+ + + + + + +

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T suy ra: 4 9 16 812 2

a b cb c a c a b a b c

+ + + + + + + Do :

4 9 16 29 81 29 262 2 2 2

a b CPb c a c a b a b c

+ + = + + = + + +

ng thc xy ra khi v ch khi 2 3 4b c a c a b a b c

= =+ + +

Hay 5 7 62 2 2c a b

= =

Vy GTNN ca biu thc P l 26, t c khi 07 6 5a b c= = >

Bi 5 : Tm GTNN ca biu thc 1 1 1

a b cPb a c b a c

= + ++ + + trong a,b,c l cc s thc dng tha mn 1a b c+ + =

Hng dn gii V 1a b c+ + = nn ( )1 2b a a b c b a b c+ = + + + = + Tng t ta cng chng minh c: 1 2c b c a+ + = + v 1 2a c a b+ = + T suy ra:

( ) ( ) ( )2 2 2

2 2 2 2 2 2a b c a b cP

b c c a a b a b c b c a c a b= + + + ++ + + + + +

p dng BT BCS ta c:

( ) ( ) ( )( )

( ) ( ) ( )22 2 2

2 2 2 2 2 2a b ca b c

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

( )( )2

13

a b cab bc ca+ += + +

(v ( ) ( ) ( )2 2 3a b c ab bc ca ab bc ca+ + + + + + + ) Do : 1

2 2 2a b cP

b c c a a b= + + + + +

Vy GTNN ca biu thc P l 1 Bi 6 : Cho , ,a b c l cc s thc dng sao cho 2 2 2 1a b c+ + = Chng minh:

3 3 3 1 .2 3 2 3 2 3 6a b c

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

Hng dn gii

t P l v tri ca BT cn chng minh. Ta cn chng minh: 16

P p dng BT BCS ta c:

( )

( )( )

( )( )

( )2 2 22 2 2

2 3 2 3 2 3a b c

Pa a b c b b c a c c a b

= + ++ + + + + +

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( )

( ) ( ) ( )22 2 2

2 3 2 3 2 3a b c

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

( )

( ) ( )22 2 2

2 2 2 5

a b c

a b c ab bc ca

+ += + + + + +

Mt khc: 2 2 2 0a b c ab bc ca+ + + + > T suy ra:

( ) ( ) ( )( )

( ) ( ) ( )( ) ( )

( ) ( )

22 2 2

2 2 2 2 2 221

2 2

a b ca b cPa b c b c a c a b a b c b c a c a b

a b c ab bc ca a b cab bc ca ab bc ca

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

( )( ) ( )

( )( )

2 22 2 2 2 2 2 2 2 2

2 2 2 2 2 2 65 6

a b c a b c a b cPa b c ab bc ca a b c

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

Thay 2 2 2 1a b c+ + = vo BT trn ta nhn c BT cn chng minh. Bi 7 : Cho , , ,a b c d l cc s thc dng. Chng minh:

1) 32

a b cb c c a a b

+ + + + + ( BT Nesbit ) 2) 2a b c d

b c c d d a a b+ + + + + + + ( BT Nesbit )

Hng dn gii

1) t P l v tri ca BT cho. Ta cn chng minh 32

P p dng BT BCS ta c:

( ) ( ) ( )( )

( ) ( ) ( )22 2 2 a b ca b cP

a b c b c a c a b a b c b c a c a b+ += + + + + + + + + + +

( ) ( )

( ) ( )2 2 2 2 2 22 31

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

ab bc ca ab bc ca+ + + + + + += = + + + + +

( do 2 2 2 0a b c ab bc ca+ + + + > ) 2) t Q l v tri ca BT cho. Ta cn chng minh 2Q . p dng BT BCS ta c:

( ) ( ) ( ) ( )2 2 2 2a b c dQ

a b c b c d c d a d a b= + + ++ + + +

( )( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )

2

2

2 2

2

2 4

0

a b c da b c b c d c d a d a b

a b c d ab bc cd da ac bd

a c b d

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

+

( )( ) ( ) ( ) ( )

2a b c da b c b c d c d a d a b

+ + + + + + + + + + .

Do BT cho ng nu ta chng minh c:

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( )( ) ( ) ( ) ( )2

2a b c d

a b c b c d c d a d a b+ + + + + + + + + +

Hay ( ) ( ) ( )2 2 4a b c d ab bc cd da ac bd+ + + + + + + + ( )2 2 2 2 2a b c d ac bd + + + + ( ) ( )2 2 0a c b d + : BT ng. Bi 8 : Cho a, b, c l cc s thc dng. Chng minh:

( )2 2 22 2 2 32

a b ca b cb c c a a b

+ ++ + + + + Hng dn gii

p dng BT BCS ta c:

( )( )

( )( )

( )( )

2 2 22 2 22 2 2

2 2 2

a b ca b cb c c a a b a b c b c a c a b

+ + = + ++ + + + + +

( )

( ) ( ) ( )22 2 2

2 2 2

a b ca b c b c a c a b

+ + + + + + +

( )

( ) ( ) ( )22 2 2a b c

ab a b bc b c ca c a+ + + + + + + (1)

p dng BT BCS dng thng thng ta c: ( ) ( ) ( ) 2ab a b bc b c ca c a+ + + + + ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2ab bc ca a b b c c a + + + + + + + Mt khc, ta c cc BT sau:

( ) ( ) ( ) ( )22 2 2

2 2 2

3a b c

ab bc ca+ + + +

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2 2 2 22 2 4a b b c c a a b c ab bc ca a b c + + + + + = + + + + + + + T suy ra : ( ) ( ) ( ) ( ) ( )

22 2 22 2 2 2.4

3a b c

ab a b bc b c ca c a a b c+ ++ + + + + + + ( )32 2 243 a b c= + +

Hay ( ) ( ) ( ) ( )2 2 2 2 2 223

ab a b bc b c ca c a a b c a b c+ + + + + + + + + Kt hp vi (1) ta suy ra:

( )

( ) ( ) ( )22 2 22 2 2 a b ca b c

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

( )

( )( )2 2 2 22 2 2

2 2 2 2 2 2

32 23

a b ca b c

a b c a b c

+ ++ + =+ + + +

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Bi 9 : Cho , ,a b c l cc s thc dng. Chng minh : 25 16 8a b cb c c a a b

+ + >+ + + Hng dn gii

BT cn chng minh tng ng vi:

25 1 16 1 1 8 25 16 1 50a b cb c c a a b + + + + + > + + + = + + +

Hay 25 16 1 50b c c a a b a b c

+ + + + + + + (1) K hiu P l v tri ca (1). p dng BT BCS ta c:

( )( ) ( ) ( )22 2 2 5 4 15 4 1 50P

b c c a a b b c c a a b a b c+ += + + =+ + + + + + + + + +

ng thc xy ra khi v ch khi 5 4 1

b c c a a b+ + += =

Suy ra ( ) ( ) 25 4 1 5

c a a bb c b c a+ + ++ + += =+ , hay 0a = : tri vi gi thit 0a >

T suy ra: 50Pa b c

> + + Do BT (1) ng v ta c BT cn chng minh. Bi 10 : Cho , ,a b c l cc s thc dng. Chng minh:

2 2 2

3 .2

a b cab b bc c ca a


K hiu P l c tri ca BT cn chng minh. p dng BT BCS ta c:

2

1 1 1 1 1 1

a b ca b cb c ab c aP

a b c a b cb c a b c a

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

Hay ( )2

1 1 1

x y zP

x y z

+ + + + + + + (1) vi , ,a b cx y zb c a

= = = ( ch 1xyz = ). S dng BT Cauchy cho ba s khng m ta c:

3 33. . . 3. 3.xy yz zx xy yz zx xyz+ + = = Suy ra:

( ) ( ) ( )2 2 6x y z x y z xy yz zx x y z+ + = + + + + + + + + Mt khc, p dng BT BCS (dng thng thng ta c): ( )1 1 1 3 3x y z x y z+ + + + + + + + Kt hp hai BT va c vi BT (1) ta nhn c:

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( )_ 6

3 3x y zP

x y z+ + + + +

Hay 33

SPS+ vi 33 3. 3 6S x y z xyz= + + + + =

T suy ra BT cn chng minh ng nu ta c: 3 33 2

SS+

Hay 3 3 32

SS

+ (2). Ch : 6S nn ta c cc bin i nh sau: 3 6 3 3 2 3 3(2) 3 2 .

2 2 2 2 2 2 2S S SVT

S S = + + + = + =

T suy ra BT (2) ng v ta c BT cn chng minh Bi 11 : Cho , ,a b c l cc s thc dng. Chng minh:

2

2 2 21

8 8 8a b c

a bc b ca c ab+ + + + + ( IMO 2001 )

Hng dn gii

K hiu P l v tri ca BT BCS ta c:2

2 2 28 8 8a b cP

a a bc b b ca c c ab= + ++ + +

( )22 2 28 8 8

a b c

a a bc b b ca c c ab

+ + + + + + + T suy ra BT cho ng nu ta chng minh c:

( )22 2 2

18 8 8

a b c

a a bc b b ca c c ab

+ + + + + + +

Hay ( )22 2 28 8 8a a bc b b ca c c ab a b c+ + + + + + + (1) K hiu Q l v tri ca BT (1). p dng BT BCS ta c:

( ) ( ) ( ) 22 2 2 28 8 8Q a a a bc b b b ca c c c ab = + + + + ( ) ( ) ( ) ( )2 2 28 8 8a b c a a bc b b ca c c ab + + + + + + + ( )( )3 3 3 24a b c a b c abc= + + + + + Do BT (1) ng nu ta c: ( )33 3 3 24a b c abc a b c+ + + + + Ta bit: ( ) ( ) ( ) ( )3 3 3 3 2 2 2 2 2 23 3 3 6 .a b c a b c a b c b c a c a b abc+ + = + + + + + + + + + T suy ra BT trn tng ng vi: ( ) ( ) ( )2 2 2 2 2 2 6 .a b c b c a c a b abc+ + + + +

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Hay ( ) ( ) ( )2 2 2 2 2 22 2 2 0a b c bc b c a ca c a b ab+ + + + + BT cui cng ng v n tng ng vi BT ng: ( ) ( ) ( )2 2 2 0a b c b c a c a b + + pcm Bi 12 : Chng minh bt ng thc sau ng vi mi s thc dng , ,a b c :

( ) ( ) ( )3 3 3

2 2 2 2 2 2 3a b c ab bc ca

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

Hng dn gii K hiu P l v tri ca BT BCS ta c:

p dng BT BCS ta c: ( )

( )( )

( )( )

( )2 22 2 2

2 2 2 2 2 2

a b cP

a b bc c b c ca a c a ab b= + + + + +

( )

( ) ( ) ( )22 2 2

2 2 2 2 2 2

a b c

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

+ + + + + + +

( )

( ) ( ) ( )22 2 2

3a b c

ab a b bc b c ca c a abc+ += + + + + +

Mt khc, p dng BT: ( ) ( )23 xy yz zx x y z+ + + + ta c: 3 ab bc ca a b c

a b c+ + + ++ +

Do c BT cho ta ch cn chng minh:

( )

( ) ( ) ( )22 2 2

3a b c

a b cab a b bc b c ca c a abc

+ + + ++ + + + + Hay

( ) ( ) ( ) ( ) ( )22 2 2 3a b c ab a b bc b c ca c a abc a b c+ + + + + + + + + ( ) ( )4 4 4 2 2 2 2 2 22a b c a b b c c a + + + + +

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 3ab a b bc b c ca c a abc a b b c c a abc a b c + + + + + + + + + + + + + ( ) ( ) ( ) ( )4 4 4 3 3 3a b c abc a b c a b c b c a c a b + + + + + + + + +

( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2 0a a bc a b c b b ca b c a c c ab a a b + + + + + + + + ( )( ) ( )( ) ( )( )2 2 2 0a a b a c b b c b a c c a c b + + (1)

Do vai tr ca , ,a b c trong BT (1) l nh nhau nn khng nhn mt tnh tng qut ta c th gi s 0a b c > . Khi ta c:

VT (1) ( )( ) ( )( )2 2a a b a c b b c b a + ( ) ( ) ( )2 2a b a a c b b c = ( ) ( ) ( )3 3 2 2a b a b a c b c =

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( ) ( )2 2 2a b a b ab ca cb= + + ( ) ( ) ( )2 0a b a a c b b c ab= + +

T suy ra BT (1) ng. Do ta c BT cn chng minh.

Bi 13 : Cho , ,a b c l cc s thc dng. Chng minh:

1a b c a b b cb c a b c a b

+ ++ + + ++ + Hng dn gii

Ta ch cn chng minh BT sau ng: ( )2 1a b c a b b cab bc ca b c a b

+ + + + + ++ + + +

Hay ( )2

3 2a b c a b b c

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

( ) ( ) ( ) ( ) ( )( )( )( )2 2 23 2a b c ab bc ca a b b c a b b c

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

( ) ( ) ( )( )( )

( )( )2 2 2 2

2a b b c c a c a

ab bc ca a b b c + + + + + + (1)

Ta c: ( ) ( ) ( ) ( ) ( ) ( )22 2 2a b b c a b b c a b b c + = + + ( ) ( ) ( )2 2c a a b b c= + T suy ra BT (1) tng ng vi:

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

( )( )2 2 22

2c a a b b c c a c a

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

Hay ( ) ( )( ) ( )( )( )2 2c a a b b c c a

ab bc ca a b b c + + + +

( ) ( )( ) ( )( ) ( ) ( )2 22 2 2 2c a a b b c a b b c c a ab bc ca + + + + ( ) ( )( )2 2 2 2 2 2 0c a b a b b c ( )24 2 2 2 22 0 0b a c b ac b ac + : BT ng. T ta c BT cn chng minh. Bi 16 : Cho :f R R+ + l mt hm s tha mn iu kin:

( ) ( ) ( )f x f z f y+ vi mi 0x y z > Chng minh BT sau ng vi mi s thc dng , ,a b c : ( )( )( ) ( )( )( ) ( )( )( ) 0f a a b a c f b b c b a f c c a c b + + (1) Hng dn gii

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Khng mt tnh tng qut ta c th gi s 0a b c > . Theo gi thit ta c: ( ) ( ) ( )f a f c f b+ (2)

D dng chng minh rng nu a b= hoc b c= th (1) ng. Do ta ch cn xt trng hp 0a b c> > > . Khi ta vit BT (1) di dng: ( )( )( ) ( )( )( ) ( )( )( )f a a b a c f c a c b c f b b c a b + Hay ( ) ( ) ( )f a f c f b

b c a b a c+ (3) v 0, 0, 0a c a b b c > > >

p dng BT BCS ta c:

( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( )2 2 2 2f a f c f a c f a cf a f cb c a b b c a b b c a b a c

+ ++ = + = +

Kt hp BT trn vi BT (2) ta nhn c:

( ) ( ) ( ) ( )( ) ( )2f a cf a f c f bb c a b a c a c

++ = BT (3) ng v ta c PCM.

Nhn xt: Nu hm s :f R R+ + xc nh bi ( ) rf x x= vi r l mt s thc th f tha mn tnh cht ca bi ton. Tht vy, vi 0x y z > ta c: i) Nu 0r th r rx y nn r r r rx z x y+ > ii) Nu 0r < th r rz y nn r r r rx z z y+ > Do trong c hai trng hp ta u c: ( ) ( ) ( )f x z f y+ Khi ta c BT:

( )( ) ( )( ) ( )( ) 0r r ra a b a c b b c b a c c a c b + + vi mi s thc dng , ,a b c

BBII TTPP ::

Bi 1: Cho 1 2, ,..., na a a l cc s thc dng. Chng minh: 2

1 2 1 2

1 1 1......n n

na a a a a a+ + + + + +

Bi 2: Cho , ,a b c l cc s thc dng. Chng minh;

1) 2 2 2

2a b c a b c

b c c a a b+ ++ + + + + 2)

3 3 3 2 2 2

2a b c a b c

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

Bi 3: Cho , ,a b c l cc s thc dng. Chng minh : 1 1 1 12 2 3 3a b b a a b

+ ++ +

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Bi 4: Cho , ,a b c l cc s thc dng tha mn iu kin 1a b c+ + = . Tm GTNN ca biu thc 2 2 2

1 1 1 1Pa b c ab bc ca

= + + ++ + Bi 5: Cho , , , , ,a b c d e f l cc s thc dng. Chng minh:

3a b c d e fb c c d d e e f f a a b

+ + + + + + + + + + +

Bi 6: Cho ,a b l cc s thc dng. Chng minh : ( )2 2 2 22a b a bb a+ + Bi 7: Cho , , , , ,a b c x y z l cc s thc dng. Chng minh:

2

xa yb zc x y zxy yz zxb c c a a b

+ ++ + + + + + + Bi 8: Cho , ,a b c l cc s thc dng. Chng minh

1) ( ) ( ) ( )2 2 2 2 2 2 2 2 2 13 2 3 2 3 2a b c

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

+ + + + + + + + +

2) ( )3

1

a b ca b ca xb b xc c xa x

+ ++ + + + + + vi 2x

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Bất đẳng thức Bunhiacopxki và ứng dụng

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