55001014 Bd Hsg Thcs Chuyen de Cuc Tri

5/24/2018 55001014 Bd Hsg Thcs Chuyen de Cuc Tri

1/23

A. Mt s$v&n ()v)b&t (+ng th.c (1i s$:Bt $%ng th(c l m+t trong nh.ng vn $0l th nht trong gi2i tan ph4thng. Trong m5c

ny chng ta s7 n l8i m+t s9 bt $%ng th(c c4 $i:n v ti;p cng ki;n th(c l t)=ng $9i l?n nn m+t s9khini@m,tnh cht c=b2n $0u $)>c bAqua. Cc b8n c th:tm thy nh.ng tnh cht ny nySch Gio Khoa cBa B+Gio D5c v Co T8o.D)i $y l cc n+i dung trong chuyn $0ny.a)Bt $%ng th(c Cauchy

i)Bt $%ng th(c Cauchy c l7 l $ quen thu+c v?i nhi0u b8n . Ngay tDnFm l?p8,cc b8n $ bGt gHp cc bt $%ng th(c nh):

3

4

2

3

4

x yxy

x y z

xyz

x y z txyzt

+

+ +

+ + +

Trong $ , , ,y z tl cc s9thIc khng mNh.ng bt $%ng th(c c d8ng ny $)>c gJi l bt $%ng th(c Cauchy. Bt $%ng

th(c Cauchy t4ng qut c d8ng nh)sau:Cho 1 2, ,..., nx x x l cc s9thIc khng m. Khi $ ta c bt $%ng th(c sau:

Du bKng x2y ra khi v chLkhi 1 2 ... nx x x= = =

C8i l)>ng 1 2... nx x

n

+ + +$)>c gJi l trung bnh c+ng cBa cc s9 1 2, ,..., .nx x x

C8i l)>ng 1 2...n nx x x $)>c gJi l trung bnh nhn cBa cc s9 1 2, ,..., .nx x

Do $ bt $%ng th(c Cauchy cn c tn gJi khc l bt $%ng th(c TBC-TBN (bt$%ng th(c gi.a $8i l)>ng trung bnh c+ng v $8i l)>ng trung bnh nhn).

Bt $%ng th(c Cauchy c nh nhi0u cch ch(ng minh. Tuy nhin do khun kh4quy:n sch nn N$y,tc gi2chLnu ra cch ch(ng minh $i:n hnh nht. Ph)=ng php

ch(ng minh ny cOng $a gGn li0n v?i m+t tn gJi: Quy n8p Cauchy. Cc b8n c th:tham kh2o thm v0ph)=ng php ny trong phPn ph)=ng php Quy N8p.

Ta s7ch(ng minh bt $%ng th(c cPn ch(ng minh $ng khi 2kn= Tr)?c h;t ta ch(ng minh cho tr)Qng h>p c=sN, 1.k= Ta cPn ch(ng minh 22 ( ) 0.x y xy x y+ = Bt $%ng th(c t)=ng $)=ng l $ng do $ bt $%ng th(c ban $Pu cOng $ng.

Gi2i sRbt $%ng th(c $ $ng cho k m= , t(c l

1 21 2

......n n n

x x xx x x

n

+ + +


2/23

1 2 2 21 2 2

......

2

mm

mm

x x xx x

+ + +

Ta s7ch(ng minh bt $%ng th(c cOng $ng cho 1.k m= +

Ta c:1 11 1

1

1 2 3 41 2 2 1 22 21 21 2

.........

2 2

m mmm

mm m

x x x x x xx x xx x

+ ++ ++

+

+ + ++ + +

(S trn ta $ sR d5ng bt $%ng th(c Cauchy cho tDng cHp s9

2 1 2 2 2 1 2 22 , 1,2 1m

k k k k x x x x k+ + + ++ = sau $ sR d5ng bt $%ng th(c Cauchy

cho 2m s9 1 11 2 3 4 2 1 2, ,..., m mx x x x x x+ + .

Nh)v


3/23

ad bcad bc

a cx y

adx bcyb d x ya c ad bc

b d

++ +

= ++

Strn, ta $ p d5ng bt $%ng th(c Cauchy cho ad bc+ s9h8ng, bao gWm ads9x , bc s9y .Nh)vc ch(ng minh.

BKng m+t t)Nng t)=ng tI,ta c th: pht bi:u bt $%ng th(c Cauchy trong s9trong tr)Qng h>p t4ng qut nh)sau

Cho 2n s9nguyn d)=ng 1 2 1 2, ,..., , , , ...,n na a a b b b v n s9thIc d)=ng 1 2, ,..., nx x .

Khi $ ta c bt $%ng th(c sau:

Du bKng x2y ra khi v chLkhi 1 2 ... .nx x x= = =

t)Nng ch(ng minh han tan t)=ng tI trong tr)Qng h>p hai s9, do $ xinnh)Qng l8i cho b8n $oc J.

By giQta hy xt m+t s9v d5(ng d5ng bt $%ng th(c Cauchy.Bi tan : Tm gi trUl?n nht v nhAnht cBa :

2 (4 )Z x y x y=

Trong $ x,y 0.x+y 6.

(C0thi vo l?p 10 Chuyn Tan CHTH H N+i nFm 1993)

Tr)?c h;t ta nh


4/23

4

24

2 2(4 ) 4 (4 ) 4 42 2 4

x xy x y

x xZ x y x y y x y

+ + + = = =

(Bt $%ng th(c Cauchy)Du bKng x2y ra ch%ng h8n nh) 2, 1.x y= = Xt 4 y + ,ta c: 0 4.Z < Nh)vp ta $0u c 4.Z Vp gi trU nhA nht, cc b8n c th: nhc sRd5ng. V $y l lc $:ta sRd5ng $i0u ki@n ny.

N;u cc b8n thay 6x y+ = vo Z ,cc b8n c th: thy 0Z< . Do $ gi trUnhAnht cBa Z cOng ph2i nhc k;t qu2 mong mu9n:3 3

32

2 2( ). .2 43 3

x 32.2 2 2 2

x x y x y

x x yy

+ + + = = =

TD$y ta $i t?i lQi gi2i:

Xt 4 6.x y + Ta c: 4 6 4 2.x y+ 3

32

2

2( ). .2 43

322 2 2

32

x y

x x yx y

x y

+ =

Nhn cc bt $%ng th(c trn v;theo v;ta thu $)>c:2 2(4 ) .2 32.2 64x y x y x y = (Nhn hai v;cho s9khng d)=ng, bt $%ng th(c

$4i chi0u)Du bKng x2y ra ch%ng h8n nh) 4, 2.x y= = Xt 0 4x y + ,ta c: 0 64.Z > Vp ta $0u c min64 64.Z Z = C%ng th(c x2y ra nh) 4, 2.x y= = Sau $y l m+t s9bi t


5/23

Tm gi trUl?n nht v nhAnht cBa 2 2.t x y= + (C0thi HSG l?p 9 TP.HCM nFm 1995)Bi 2:Cho , 1a b> . Tm gi trUnhAnht cBa :

2 2

1 1

a bP b a= + (C0ki:m tra l?p 9 Chuyn Tan TP.HCM nFm 1994)Bi 3:Tm gi trUnhAnht cBa:

2 2

1 11 1S

y

=

, bi;t, 0

1.

x y

x y

> + =

(C0th vo l?p 10 PTTH chuyn L HWng Phong TP.HCM nFm 1994)Bi 4:Cho , , 0.a b c Ch(ng minh rKng:

4 4 4 ( )a b c abc a b c+ + + + (C0thi hJc sinh giAi l?p 9,b2ng B,tan qu9c nFm 1994)Bi 5:Cho a,b,c l cc s9thIc khng m. Ch(ng minh rKng:

2 2 2 3

3 3 33 2

2

a b c

b c a c a b

+ + + + +

(T8p chi Ton hJc v Tu4i Tr[).b)Bt $%ng th(c Bouniakovskii)Bt $%ng th(c Bouniakovski cOng l m+t trong nh.ng bt c4$i:n n4i ti;ng nht.

Bt $%ng th(c cn gGn v?i nhi0u tn gJi khc,nh)Cauchy,Schwarz. COng xin ch v?ib8n $Jc rKng, nh.ng bt $%ng th(c c4$i:n th)Qng $)>c hnh thnh trong cc vn $0cu+cs9ng,trong cc vn $0 v0 thin vFn,v


6/23

Bt $%ng th(c cPn ch(ng minh t)=ng $)=ng v?i:

1 1 2 2

2 2 2 2 2 21 2 1 2

...1

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

n n

a b a b a b

a a a b b b

+ + +

+ + + + + +

Ta c th:gi2sRcc s9 , , 1,i ia b i n= $0u l cc s9thIc d)=ng. BNi l7khi $ chng

ta chLcPn sRd5ng bt $%ng th(c trUtuy@t $9i:1 1 1 1... | || | ... | || |n n n na b a b a b a b+ + + +

V l8i p d5ng bt $%ng th(c Bouniakovski cho cc s9 thIc d)=ng

| |,| |, 1,i ia b i n= v ta s7c $i0u ph2i ch(ng minh.Quay l8i vn $0chnh. p d5ng bt $%ng th(c Cauchy cho hai s9thIc d)=ng ta

$)>c:2 2

2 2 2 2 2 22 2 2 2 2 21 2 1 21 2 1 2

,2( ... ) 2( ... )( ... )( ... )

i i i i

n nn n

a b a bi

a a a b b ba a a b b b + =

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

C+ng cc bt $%ng th(c trn v;theo v;ta thu $)>c:2 2 2 2 2 2

1 1 2 2 1 2 1 22 2 2 2 2 22 2 2 2 2 2

1 2 1 21 2 1 2

... ... ...1.

2( ... ) 2( ... )( ... )( ... )n n n n

n nn n

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

a a a b b ba a a b b b

+ + + + + + + + + + =

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

V nh)vc ch(ng minh xong.

Du bKng x2y ra khi v chLkhi

2 2 21 2

2 2 21 2

1 2

1 2

..., 1,

...

... 0.

ni

i n

n

n

a a aai n

b b b b

aa a

b b b

+ + += =

+ + +

= = =

T)=ng tInh)bt $%ng th(c Cauchy, bt $%ng th(c Bouniakovski cOng c nhi0ucch nhn nhng c t4ng bnhph)=ng l m+t hKng s9, hoHc cOng c khi l t4ng cc cFn bng nKm Nv;b h=n trong bt $%ng th(c cPn ch(ng minh. Cy l nh.ng du hi@u $: sRd5ng bt$%ng th(c Bouniakovski.

Ngai ra bt $%ng th(c Bouniakovski cOng th)Qng hay $)>c sRd5ng trong cc bt$%ng th(c c d8ng phn th(c.Do $ cc b8n nn ch khi gHp nh.ng d8ng ny.

ii) Bt $%ng th(c Bouniakovski mNr+ng.Cho m dy s9 thIc khng m 1 2( );( );...( ).ma a a M\i dy gWm n s9 h8ng

1 2,, ...., .

ni i ia a a Khi $ ta c bt $%ng th(c sau:

Du bKng x2y ra khi v chLkhi 1 2

1 2

... n

n

aa a

b b b= = = .

Ch(ng minh bt $%ng th(c Bouniakovski mNr+ng c th:lm bKng t)Qng t)=ngtItrong tr)Qng h>p 2m= ,do $ phPn ny xin dnh cho b8n $Jc.

1 2 1 2 1 1 11 1 1 1 2 1 2( ... )...( ... ) ... ... ...

n n n n n

m m m m m mmm m m m ma a a a a a a a a a a a+ + + + + + + +


7/23

Ch thm v?i cc b8n rKng trong tr)Qng h>p m l s9tInhin ch]n th ta c chocc dy s9thIc l bt k, khng cPn khng m,tuy nhin khi y du bKng x2y ra th cc tLs9vYn ph2i bKng nhau v bKng m+t $8i l)>ng khng m.

D)?i $y ta s7xt qua m+t v d5(ng d5ng bt $%ng th(c Bouniakovski.Bi tan:

a) Cho ,x y thAa: 2 21 1 1x y y x + = (1)Ch(ng minh: 2 2 1x y+ = b) TD (2)c th:suy ra $)>c (1) hay khng.(C0thi vo l?p 10 PT NFng Khi;u TP.HCM nFm 1999)

Bi tan $)>c pht bi:u d)?i d8ng $%ng th(c, tuy nhin bi0u th(c trong (1) l8ikhi;n cho ta c c2m gic quen thu+c. R rng trong bi:u th(c y,ta c:

2 2 2

2 2 2

( 1 ) 1

( 1 ) 1

y y

x x

+ =

+ =

Cy l nh.ng du hi@u r rng nht cho sI hi@n di@n cBa bt $%ng th(cBouniakovski. TD$y ta $)a ra lQi gi2i:

a)p d5ng bt $%ng th(c Bouniakovski cho hai dy: 2( , 1 )x v 2( 1 , )y ta thu $)>c:

( )2 2 2 2 2 2

2 2

1 1 x 1 (1 )

1 1 1

x y x y x y y

x y y x

+ + +

+

Du bKng x2y ra khi v chLkhi2

2

2 2

2 2

2 2 2 2

2 2 2 2

2 2

2 2

10

11

1

1 11

1 1

1

1

yx

yxx y

x y

x x x x

y y y y

x y

x y

=

=

+ = = =

+

=

+ =

b) TDvi@c xt du bKng, chng ta thy ngay n;u , 0x y th (1) (2) .Tuy nhinN$y do $0bi khng cho ,y l cc s9 thIc d)=ng nn ta d^dng chL ra tr)Qng h=p

(2) (1) ,ch%ng h8n nh) 0, 1.x y= = Bi tan:Cho , ,a b c l cc s9thIc d)=ng.Ch(ng minh rKng:

1.2 2 2

a b c

a b b c c a+ +

+ + +


8/23

Bi tan $)>c nu ra d)?i d8ng phn th(c, $y l du hi@u khi;n ta c2nh gic v?ibt $%ng th(c Bouniakovski.Thng th)Qng, ta sRd5ng bt $%ng th(c ny $:tri@t tiu mYuth(c.S$y ta s7p d5ng bt $%ng th(c ny bKng cch nh)sau:

Ta c:

[ ] 2

2 2

( 2 ) ( 2 ) ( 2 ) ( )2 2 2

( ) ( )2 2 2

1.2 2 2

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

a b b c c a

a b ca b c a b c

a b b c c a

a b c

a b b c c a

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

+ + + + + + + + +

+ + + + +

Nh)vc gi2i quy;t xong.

Sau $y l cc bi t v d+da=1ab bc c+ + .

Ch(ng minh rKng:3 3 3 3 1

3

a b c d

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

+ + + + + + + +.

(C0thi chJn HSG kh9i PTCT-CHSP H N+i nFm 1995)Bi 2:Cho , , 0a b c> v 2 2 2 1a b c+ + = .Tm gi trUnhAnht cBa:

3 3 3

.2 3 2 3 2 3

a b cA

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

+ + + + + +

(C0thi $0nghUOlympic 30-4 lPn 6,nFm 2000)

Bi 3:Cho , , , , 0a b c p q> .Ch(ng minh rKng:1 1 1

.p q p q p q

a b c pa qb pb qc pc qa

+ + ++ + + +

+ + +

c)Bt $%ng th(c Chebysevi)Cho hai dy s9cng tnh $=n $i@u

1 2

1 2

...

...

n

n

a a a

b b b

hay 1 2

1 2

...

...

n

n

a a a

b b b

Khi $ ta c bt $%ng th(c sau:

1 1 2 2 1 2 1 2... ... ...n n n na b a b a b a a a b b b

n n n

+ + + + + + + + +


9/23

Trong tr)Qng h>p m+t dy tFng m+t dy gi2m

1 2

1 2

...

...n

n

a a a

b b b

Ta c bt $%ng th(c ng)>c l8i nh)sau:

Bt $%ng trn cOng c nhi0u cch ch(ng minh, nh)ng cch ch(ng minh sau lngGn gJn nht m tc gi2bi;t $)>c.

Ta c:

1 1 2 2 1 2 1 2

1 1 2 2 1 2 1 22

1

2

1

2

... ... ...

( ... ) ( ... )( ... )

( )

( )( )

.

n n n n

n n n n

i i i j j i j j

i j n

i i j j

i j n

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

n n n

n a b a b a b a a a b b b

n

a b a b a b a b

n

a b a b

n

<

<

+ + + + + + + + +

+ + + + + + + + +=

+=

=

Trong tr)Qng h>p hai dy cng tnh $=n $i@u ta c cc $8i l)>ng( )( )i j i ja a b b l khng m, do $ ta thu $)>c bt $%ng th(c nh)$ ni.

Trong tr)Qng h>p hai dy khc tnh $=n $i@u ta c cc $8i l)>ng( )( )i j i ja a b b l khng m, do $ ta thu $)>c bt $%ng th(c ng)>c chi0u.

Du bKng cBa bt $%ng th(c l t)=ng $9i ph(c t8p,ta chLc th:ni du bKng c2y

ra khi v chLkhi 1 1

1 1

... ; ... ;...; ...

... ; .. ;...; ...k k k t a b

l l l c n b

i i i i i i

j j j r j j

a a a a a a

b b b b b b

+ +

+ +

= = = = = =

= = = = = =trong $ , ( , )m ni j i j .

Ta c th:hi:u m+t cch nm na l dy ( )a chia thnh tDng $o8n bKng nhau.Cnnh.ng $o8n cn l8i t)=ng (ng l cc $o8n bKng nhau cBa dy ( )b .

ii) Bt $%ng th(c v?i dy han vU.Trong phPn i) cBa m5c ny, ta $ $0cc $9i v?i hai dy:1 2

1 2

...

...n

n

a a a

b b b

Th

1 2 1 21 1 2 2 1 2 1 1

( ... )( ... )... ... .n nn n n n n

a a a b b ba b a b a b a b a b a b

n

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

1 1 2 2 1 2 1 2... ... ...n n n na b a b a b a a a b b b

n n n

+ + + + + + + + +


10/23

Th;cn $9i v?i t4ng1 21 2

...ni i n i

a b a b a b+ + + trong $1 2

( , ,..., )ni i i

b b b l m+t han vU

cBa cc s9 1 2( , ,..., )nb b b (nghXa l cc s9h8ng 1 2, ,..., nb b b thay $4i vUtr) th sao nhL? Cy

l m+t cu hAi kh tInhin, v cu9n ht chng ta vo vi@c gi2i quy;t chng. Cy chnh lci vng xoy v tc bt $%ng th(c sau,$)>c gJi l bt $%ng th(c hon vU:

t)Nng trong ch(ng minh bt $%ng th(c ny l quy n8p.V?i 2n= ,ta chLcPn ch(ng minh bt $%ng th(c:

1 1 2 2 1 2 2 1

1 2 1 2( )( ) 0.

a b a b a b a b

a a b b

+ +

Bt $%ng th(c t)=ng $)=ng cu9i cng l $ng, do $ ta c $i0u ph2i ch(ng minh.Gi2sRbt $%ng th(c $ $ng cho n k= ,t(c l

1 21 1 2 2 1 2 1 2 1 1... ... ... .kk k i i k i k k k a b a b a b a b a b a b a b a b a b+ + + + + + + + + Ta cPn ch(ng minh bt $%ng th(c cOng $ng cho 1.n k= + C9i v?i bPt $%ng th(c $Pu tin, sRd5ng bt $%ng th(c quy n8p v ta chLcPn ch(ng

minh:

1 1 1 1

1 1( )( ) 0.

k k i j i k k j

k i k j

a b a b a b a b

a a b b

+ + + +

+ +

+ +

C9i v?i bt $%ng th(c th(hai, sRd5ng bt $%ng th(c quy n8p v ta chLcPn ch(ngminh:

1 1 1 1

1 1

( )( ) 0.

k i j k j i

k i j

a b a b a b a b

a a b b

+ +

+

+ +

Hai bt $%ng th(c t)=ng $)=ng cu9i cng l $ng, do $ cc bt $%ng th(c ban$Pu cOng vp $ nu, tuy nhin n s7cn gipch cc b8n rt nhi0u v0mHt t)Nng trong qu trnh hJc cp ba cBa mnh.

N;u 1 ax 0+ th bt $%ng th(c hi:n nhin $ng do (1 ) 0.ax+ Xt 1 ax 0+ .Bt $%ng th(c $ cho t)=ng $)=ng v?i:

1 1a ax+ +

1 21 1 2 2 1 2 1 2 1 1... ... ...

nn n i i n i n n na b a b a b a b a b a b a b a b a b+ + + + + + + + +

(1 ) 1a ax+ +


11/23

p d5ng bt $%ng th(c Cauchy cho a s9h8ng,gWm m+t s91 ax+ v a-1 s91,ta thu$)>c:

1 ax+(a-1)1 axa

a

+ +

1 1 axax + + .Nh)vc d_u bKng.

e)M+t s9 t)Nng tDbt $%ng th(c x2 `0Trong cc phPn Ntrn, chng ta $ bi;t cch (ng d5ng cc bt $%ng th(c c4$i:n

vo vi@c ch(ng minh cc bt $%ng th(c. Tuy nhin trong thIc t;, khng ph2i bao giQchng ta cOng c th: p d5ng bt $%ng th(c m+t cch d^ dng nh) vc v0d8ng ny th ta chLcPn ch(ng minh ( , ) 0g a b n.a

l xong. Bt $%ng th(c ( , ) 0g a b , thng th)Qng l khng c du bKng,t(c l khngchHt han tan, hoHc l khng my ph(c t8p.Do $ vi@c gi2i quy;t chng l t)=ng $9id^dng.

Ta hy bGt $Pu bKng m+t s9hKng $%ng th(c sau:2 2 2

2 2

2 2 2 2

3 3 2

4 4 2 2 2 2 2

2 ( )

( ) 4 ( )

2( ) ( ) ( )

( ) ( )( )

( ) ( )( )

a b ab a b

a b ab a b

a b a b a b

a b ab a b a b a b

a b ab a b a ab b a b

+ =

+ =

+ + =

+ + = +

+ + = + +

Trong tr)Qng h>p gHp phn th(c hay cFn th(c,cc b8n c th:quy $Wng mYu s9hay nhn v?i l)>ng lien h>p $:$)a v0d8ng quen thu+c $ bi;t. Ta xt m+t s9v d5:

2 2 2 22 2

2 2 2 2

2( ) ( ) ( )2( ) ( ) (1)

2( ) 2( )

a b a b a ba b a b

a b a b a b a b

+ + + + = =

+ + + + + +

2 21 1 4 ( ) 4 ( )

( ) ( )

a b ab a b

a b a b ab a b ab a b

+ + = =

+ + +


12/23

t)Nng cBa ph)=ng php thc bt $%ngth(c:

2

2 2

2 2 22 2

22 2

2 2

( )2( ) ( ) 2( )

( ) 2( )2( )

2( )

( )2( )

2 2(a )

a ba b a b a b

a b a ba b

a b a b

a ba b a b

b

+ + +

+ + +

+ +

+ + +

+

Nh) vc cc bt $%ng th(c m8nh h=n bt $%ng th(cBouniakovski rt l th sau :

2 22 2

2 2

( ) ( )2( )

2( )2 2( )

a b a ba b a b a b

a ba b

+ + + + +

++

Cc b8n c l7$ thy $)>c phPn no sIl th v s(c m8nh cBa ph)=ng php nyrWi ch(. J

Chng ta hy xt thm m+t v d5n.a m chng ta bUp vo th;tm lQi gi2i ch(khng ph2i l sng t8o ton hJc xem no.

Bi tan:Ch(ng minh rKng v?i mJi s9thIc d)=ng a,b ta c bt $%ng th(c sau:

2

3

( ) ( 3 )( 3 )

2 16( )

a b a b a b b aab

a b

+ + + +

+.

(T8p ch Ton HJc v Tu4i Tr[)

S$y du bKng x2y ra khi a b= , tuy nhin bi:u th(c2

3( ) ( 3 )( 3 )8( )a b a b b a

a b + ++

$

c ch(a 2( )a b rWi, do $ ta chLcPn ph2i $)a2

a bab

+ v0d8ng 2( ) ( , ).a b f a b

Ta c:2 2

2

( ) ( )

2 2 2( )

a b a b a bab

a b

+ = =

+. Nh)vc:2

22

(1. 1. ) 2( )

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

4

a b a b

a b b aa b b a a b

+ +

+ + ++ + = +

Nhn v;theo v;ta thu $)>c $i0u ph2i ch(ng minh.D)?i $y l m+t s9bi t


13/23

Bi 1:Cho cc s9thIc d)=ng ,a b .Ch(ng minh rKng:

2 22 27( ) 8 2( )

a ba b a b

b a+ + + +

Bi 2:Cho 1.a b c+ + = Ch(ng minh rKng:

2 23 3 3 23 .

4

a b aba b c abc

+ + +

Xc $Unh tr)Qng h>p $%ng th(c x2y ra.G>i : SR d5ng hKng $%ng th(c

3 3 3 2 2 2( )3 ( ) ( ) ( )2

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

+ + + + = + +

f)S(c m8nh cBa php bi;n $4i t)=ng $)=ng.

Thng th)Qng khi gHp cc bi tan v0 bt $%ng th(c d8ng phn th(c, ng)Qi talun nghX$;n cc bt $%ng th(c c4$i:n nh)Cauchy, Bouniakovski.Tuy nhin vi@c pd5ng chng $i khi rt rGc r4i v khng ph2i lc no cOng thIc hi@n $)>c.

Ton hJc ngy nay $ c nhi0u bt $%ng th(c t9t h=n, thuc $0c


14/23

[ ]2

( )( ) ( )( ) ( )( ) 0

( ) ( ) ( ) ( )( ) 0.

( ) ( ) ( )( ) 0.

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

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

a b a b c c a c a b

+ +

+

+ +

Bt $%ng th(c t)=ng $)=ng cu9i cng l $ng, do $ bt $%ng th(c ban $Pu cOng

$ng. D)?i $y ta s7xem xt v d5:

Bi tan:Ch(ng minh rKng v?i mJi b+s9th)c d)=ng ( , , )a b c ,ta c bt $%ng th(c sau:9( )( )( ) 8( )( )a b a c b c a b c ab ac bc+ + + + + + + Bt $%ng th(c trn th


15/23

Trong phPn ny chng ti s7trnh by cho cc b8n m+t ph)=ng php ti;p cp nh)v


16/23

( , , ) ( , ,1)f a b c f a b . ThIc v >

Do $: ( , ,1) (2, ,1)f a b f b .By giQta xt (2, ,1).f b

( )3 1 11 3 3

(2, ,1) 32 2 2

11 1 11 ( 1)( 2) 3 11 93 3 10.

2 2 2 2 2 2 2

bf b b

b b

b b b

b b

= + + = + +

= + + = + + + =

V


17/23

Bi 1:Cho [ ], , 1, 2a b c . Tm gi trUl?n nht cBa:

3 3 3

3

a b cA

abc

+ +=

T8p ch Ton HJc v Tu4i Tr[.Bi 2:Cho [ ], , , , 0,1a b c d e .Tm gi trUl?n nht cBa

(1 ) (1 ) (1 ) (1 ).B a b b c c d d a= + + + Bi 3:Cho [ ], , 1, 2x y z . Tm gi trUl?n nht cBa bi:u th(c sau:

y xz yzC

z yz xy yz xy xz= + +

+ + +

Bi 4:Cho [ ], , 0,1a b c .Ch(ng minh rKng:

3 3 3 2 2 22( ) ( ) 3.a b c a b b c c a+ + + + T8p ch Ton HJc v Tu4i Tr[.

h)SRd5ng tam th(c b


18/23

Bt $%ng th(c cu9i cng l $ng,nh)vc gi2i quy;t.ii)Ph)=ng php t8o tam th(c.

C:ch(ng minh f g . Ta c th:chuy:n2 ,2

4 0f g b ac b ac = = Khi $ tas7t8o ra tam th(c bp 2 2 2 21 2 3 ... na a a a= + + + ,khi $ bt $%ng th(c l hi:nnhin v v;ph2i lun khng m.

Tr)Qng h>p 2 2 2 21 2 3 ... na a a a> + + + ,ta xt tam th(c b


19/23

s?m nht s7$)>c gRi tHng m+t mn qu cBa chng ti,cc b8n nh?ghi $Ua chL r rngtrong th)gRi $;n $:thup, n ch]n v n l[.V?i n ch]n,$Ht 2n k= . Ta c:

1 2 1 2 2

1 2 1 2 2

1 2 1 2 2

... ...

... ...

... ... .

k k k k

k k k k

k k k k

A x a x a x a a x a x a x

x a x a x a a x a x a x

a a a a a a

+ +

+ +

+ +

= + + + + + + +

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

Du bKng c th:x2y ra, ch]n h8n nh) .ka= V?i n l[, $Ht 2 1n k= + . Ta c:

1 2 1 2 3 2 1

1 2 2 3 2 1

1 2 2 3 2 1

... ...

... 0 ...

... ...

k k k k k

k k k k

k k k k

A x a x a x a x a a x a x a x

x a x a x a a x a x a x

a a a a a a

+ + + +

+ + +

+ + +

= + + + + + + + +

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

Du bKng c th:x2y ra, ch]n h8n nh) 1ka += .LQi gi2i cBa bi tan trn c l7 l kh kb thup n nhArWi m?i c th:gi2i m+t ccht4ng qut. Cy cOng l m+t kinh nghi@m trong hJc v lm Tan, chng ta nn bGt $Pu tDnh.ng ci nhAv khi qut ln cho ci l?n.

Ngai vi@c p d5ng bt $%ng th(c x x , ta cn c th:sRd5ng bt $%ng th(c:1 2 1 2... ...n nx x x x x x+ + + + + + $:gi2i quy;t bi tan trn.

M+t ph)=ng php cOng hay sRd5ng $9i v?i du gi trUtuy@t $9i ni dhung l xttDng khAang $:bAdu gi trUtuy@t $9i.

Ci khi,chng ta cOng th)Qng xuyn dRd5ng cc bt $%ng th(c c4$i:n trong vi@cch(ng minh cc bt $%ng th(c ch(a du gi trU tuy@t $9i, nht l bt $%ng th(c


20/23

Bouniakovski. BNi l[ p d5ng bt $%ng th(c ny, cc gi trU tuy@t $9i s7 $)>c bnhph)=ng lm mt du gi trUtuy@t $9i.

Ta thRxt m+t v d5Bi ton:Cho ,a b l cc s9thIc thAa mn 2 2 1.a b+ = Ch(ng minh rKng:

( ) ( )1 1 1 1 2 6a b a b+ + + + + LQi gi2i dIa trn bt $%ng th(c Bouniakovski. Ta lm nh)sau:

2 2 21. 1 1. 1 2 ( 1) ( 1) 2 1a a a a a + + + = +

2 2 21. 1 1. 1 2 ( 1) ( 1) 2 1b b b b b + + + + = +

2 2 2 22. 1 2. 1 8( 2) 2 6.a b a b+ + + + + = C+ng cc bt $%ng th(c trn v;theo v;,ta c $i0u ph2i ch(ng minh.

Sau $y l cc bi t


21/23

i) D8ng 2( ) axf x bx c= + + Ph)=ng php gi2i v cc k;t qu2v0d8ng bt $%ng th(c ny $ $)>c nu trong phPn m+ts9ki;n th(c cPn nh?.D)?i $y l m+t s9bi tc $i0u cPn

tm v?i ( )

m

g x .D)?i $y l m+t s9bi t


22/23

iv) D8ng2

2( ) , 0.

ax

mx nx d f x am

bx c

+ +=

+ +

Ph)=ng php gi2i l $)a v0d8ng iii), c5th:nh)sau:

( )222 2

2

ax

ax ax

ax

m mb mcbx c n x d

mx nx d a a a

bx c bx c

mb mcn x d

m a a

a bx c

+ + + + + + =+ + + +

+ = +

+ +

Ta xt qua m+t s9bi tc hnh thnh tDcc $%ngth(c $8i s9:Tr)?c h;t ta n;u l8i m+t s9$%ng th(c hay cBa $8i s9:

1( )( ) ( )( ) ( )( )

( )( ) ( )( ) ( )( )1

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

ab bc ca

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

x y y z y z z x z x x y

x y y z y x z x z x x y

+ + = + + + + + +

+ + =

(1)

(2)

Trong $ ( , , )a b c v ( , , )x y z l cc b+s9khc nhau phPn [email protected])u v?i cc b8n rKng hai $%ng th(c ny l t)=ng $)=ng nhau, ch]n h8n trong (2) , $Ht

, ,x y a y z b z x c+ = + = + = ta s7thu $)>c $%ng th(c (1)TDcc $%ng th(c trn ta rt ra m+t s9h@qu2trIc ti;p sau $y:

2 2 2

2 2 2

2

2

a b c

b c c a a b

x y y z z x

x y y z z x

+ +

+ + + + +

C0thi vo l?p 10 PTTH Chuyn L HWng Phong TP.HCM nFm 1999.2 2 2 2 2 2

2 2 2

2 2 2

5.

( ) ( ) ( ) 21

( ) ( ) ( ) 4

x y y z z x

x y y z z x

xy yz zx

x y y z z x

+ + +

+ +

+ +

Ta hy xt m+t s9bi tan lien quan $;n cc bt $%ng th(c th vUny.Bi tan d)?i $y l bi tan v0bt $%ng th(c Nesbit Nd8ng hi@u:Bi ton:Cho , ,a b c l cc s9thIc phn bi@t. Ch(ng minh rKng:


23/23

2.a b c

b c c a a b+ +

R rang cc b8n cOng thy $)>c m9i lien h@v?i cc bt $%ng th(c ta $ang xt rWich(, $: m9i quan h@ thm r rang ta bnh ph)=ng hai v; cBa bt $%ng th(c, v ta thu$)>c:

2 2 2

2 4( )( ) ( )( ) ( )( )

a b c ab bc ca

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

+ + + + +

MHc khc ta $ c:2 2 2

2a b c

b c c a a b

+ +

2 2( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )(

ab bc ca ab bc ca

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

+ + + +

C+ng cc bt $%ng th(c trn v;theo v;v ta thu $)>c $i0u cPn ch(ng minh.Sau $y l m+t s9bi t

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55001014 Bd Hsg Thcs Chuyen de Cuc Tri