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Tr-ng THPT CHUYN QUNG BNH

ti nghin cu khoa hc

PHNG PHP CHNG MINH

BT NG THC

Gio vin hng dn : Nguyn Chin Thng

Nhm tc gi: Tp th chuyn Ton kha 2012-2015

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LI NI U

Trong mn Ton trng THPT, bt ng thc ngy cng c quan

tm ng mc v t ra c sc hp dn mnh m nh v p v tnh c o

ca phng php v k thut gii chng cng nh yu cu cao v t duy cho

ngi gii. Bt ng thc l mt trong nhng dng ton hay v kh i vi

hc sinh trong qu trnh hc tp cng nh trong cc k thi, trc ht l k thi

i hc m hu ht hc sinh THPT u phi vt qua. Ngoi ra bt ng thc

cng l mt dng thng gp trong cc k thi hc sinh gii ton cc cp

tnh, Quc gia, Olympic khu vc v Olympic quc t.

Cc bi ton bt ng thc khng nhng rn luyn t duy sng to, tr

thng minh m cn em li say m v yu thch mn Ton ca ngi hc.

Trong ti nghin cu khoa hc ny, tp th lp 10 Ton trng THPT

Chuyn Qung Bnh xin trnh by mt s vn v bt ng thc, mt s

phng php chng minh bt ng thc. ti gm cc bi vit ca cc

nhm tc gi c trnh by di dng cc chuyn .

Nhm tc gi

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MC LC

LI NI U ........................................................................................................ 2

MC LC .................................................................................................................. 3

BT NG THC AM-GM V NG DNG...................................... 7

1. Bt ng thc AM-GM ........................................................................................... 7

1.1. nh l ................................................................................................................... 7

1.2. Chng minh ........................................................................................................ 7

1.3. Cc dng thng gp ......................................................................................... 8

2. V d.............................................................................................................................. 8

3. Bi tp t gii ............................................................................................................23

BT NG THC MINKOWSKI V NG DNG...................... 24

1. Bt ng thc Minkowski ......................................................................................24

1.1 Bt ng thc Minkowski dng 1 ....................................................................24

1.1.1 nh l ..........................................................................................................24

1.1.2 Chng minh................................................................................................24

1.2 Bt ng thc Minkowski dng 2......................................................................25

1.2.1 nh l .........................................................................................................25

1.2.2 Chng minh................................................................................................25

2. V d.............................................................................................................................25

3. Bi tp t gii ............................................................................................................28

BT NG THC HOLDER V NG DNG ............................... 29

1. Bt ng thc Holder .............................................................................................29

1.1 Dng tng qut ....................................................................................................29

1.1.1 nh l ..........................................................................................................29

1.1.2 Chng minh................................................................................................29

1.2 M rng 1 ca bt ng thc Holder ..............................................................30



2. V d.............................................................................................................................30

3. Bi tp t gii ............................................................................................................41

BT NG THC CAUCHY-SCHWARZ .......................................... 43

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1.Bt ng thc Cauchy-schwarz .............................................................................43

1.1. nh l ..................................................................................................................43

1.2. Chng minh .......................................................................................................43

1.3. H qu .................................................................................................................45

2. V d.............................................................................................................................45

3. Bi tp t gii ............................................................................................................78

BT NG THC CHEBYSHEV ..............................................................82

1.Bt ng thc Cheybyshev .....................................................................................82

1.1. nh l ..................................................................................................................82

1.2. Chng minh .......................................................................................................82

2. V d.............................................................................................................................83

3. Bi tp t gii ............................................................................................................96

BT NG THC MUIRHEAD ...................................................... 97

1. Gii thiu bt ng thc Muirhead......................................................................97

2. Mt s khi nim lin quan n Bt ng thc Muirhead .............................97

2.1. B tri ..................................................................................................................97

2.2. Trung bnh loi .............................................................................................98

2.3. Tng hon v .......................................................................................................98

2.4. Tng i xng ....................................................................................................98

2.5. Lc Young ...................................................................................................99

3. nh l Muirhead .....................................................................................................99

4. K thut s dng nh l Muirhead ................................................................... 101

Phng php chung ............................................................................................... 101

5. S dng nh l Muirhead vi AM GM, Holder, ASYM, Schur ............ 102

5.1. Bt ng thc AM GM ................................................................................ 102

5.2. Bt ng thc Holder ...................................................................................... 102

5.3. Bt ng thc ASYM ...................................................................................... 102

5.4. S dng nh l Muirhead vi bt ng thc Schur ................................. 102

6. V d........................................................................................................................... 103

7. Bi tp t gii .......................................................................................................... 112

[ ]a

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PHNG PHP PQR ...................................................................................... 114

1. Kin thc lin quan ................................................................................................ 114

1.1. nh ngha v cc php bin i ................................................................... 114

1.2. Phng php pqr kt hp bt ng thc Schur .................................. 114

1.3. M rng phng php pqr kt hp hm s ................................................ 117

2. Bi tp t gii .......................................................................................................... 119

PHNG PHP PHN TCH TNG BNH PHNG S.O.S ............................................................................................................................................ 124

1. L thuyt v v d .................................................................................................. 124

1.1 nh l v cc k thut phn tch ................................................................... 124

1.2. Cc tiu chun v k thut sp xp bin ...................................................... 130

1.3. ng dng tm hng s k tt nht .................................................................. 135

2. Bi tp t gii .......................................................................................................... 137

3. M rng..................................................................................................................... 141

S DNG PHNG PHP S.O.S TRONG CHNG MINH

BT NG THC .............................................................................................. 142

1. Li ni u .............................................................................................................. 142

2. Xy dng nh l, tiu chun ............................................................................... 142

3. Phn tch c s ........................................................................................................ 143

4. Cc ng dng ca phng php S.O.S ............................................................. 144

5. Bi tp vn dng .................................................................................................... 149

6. Bi tp dnh cho bn c ..................................................................................... 151

PHNG PHP DN BIN ....................................................................... 153

1. Kin thc lin quan ............................................................................................... 153

2. V d minh ha ....................................................................................................... 157

3. Bi tp vn dng .................................................................................................... 184

S DNG TIP TUYN TRONG VIC CHNG MINH BT

NG THC .......................................................................................................... 187

1. Phng trnh tip tuyn tng qut .................................................................... 187

2. S dng tip tuyn chng minh bt ng thc ......................................... 187

3. V d .......................................................................................................................... 188

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PHNG PHP NHN T LAGRANGE ......................................... 203

1. C s l thuyt ......................................................................................................... 203

2. Mt s v d ............................................................................................................. 204

3. Bi tp vn dng .................................................................................................... 215

KT LUN ............................................................................................................... 218

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BT NG THC AM-GM V NG DNG

on Quc t Ng Hong Thanh Quang

1. Bt ng thc AM-GM

1.1. nh l

nh l (Bt ng thc AM-GM). Vi mi s thc dng 1 2, ,..., na a a ta c bt ng

thc

1 21 2

......n n n

a a aa a a

n

ng thc xy ra khi v ch khi 1 2 ... na a a .

1.2. Chng minh

Phng php Quy np Cauchy

Vi

2

1 21 2 1 2

1 2 1 22 : 02 2 2

a aa a a an a a a a

(ng)

Gi s bt ng thc ng vi n k ta s chng minh bt ng thc ng vi

2n k . S dng gi thit quy np ta c:

1 2 2 1 2 1 2 2... ... ...1

2 2 2

k k k k ka a a a a a a a a

k k k

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

k k k k kk k k k k k k k ka a a a a a a a a a a a a a

Gi s bt ng thc ng vi n p ta s chng minh bt ng thc ng vi

1n p .

Tht vy, xt 1p s: 1 2 1, ,..., 0.pa a a S dng gi thit quy np vi n p ta c:

11 2 1 1 2 1

1 11 1 1 1 1 2 1

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

pp p p p p

p p p

a a a a a aa a a a a a a

p

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

p pp p pa a a a a a p a a a

- 8 -

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

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

pp pp p p

a a aa a a p a a a a a

p

Theo nguyn l quy np ta c bt ng thc ng vi mi 2, .n n


1.3. Cc dng thng gp

n 2n 3n 4n

iu kin , 0a b , , 0a b c , , , 0a b c d

Dng 1 2

a bab

3

3

a b cabc

4

4

a b c dabcd

Dng 2

2

2

a bab

3

3

a b cabc

4

4

a b c dabcd

Du bng a b a b c a b c d

2. V d

V d 1: (Bt ng thc Nesbit) Chng minh rng vi mi s thc khng m , ,a b c

ta c

3

2

a b c

b c a c a b

Gii: Xt cc biu thc sau

a b cS

b c a c a b

b c aM

b c a c a b

c a bN

b c a c a b

Ta c 3M N . Mt khc theo bt ng thc AM-GM th

- 9 -

3

3

a b b c c aM S

b c a c a b

a c a b b cN S

b c a c a b

Vy 2 6 2 3M N S S hay

3

2

a b c

b c a c a b

ng thc xy ra khi v ch khi a b c (pcm)

Nhn xt: Bi ny cn nhiu cch gii khc nhng c l y l cch hay nht v

vic ngh ra cc biu thc ,M N khng phi l d dng.

V d trn phn no cho ta thy c sc mnh v s tinh t ca bt ng thc AM-

GM, nhng ch mi l mt v d n gin. Chng ta s xt n k thut thm bt

trong bt ng thc AM-GM qua v d sau.

V d 2: Chng minh rng vi mi s thc khng m , ,a b c ta c

2 2 2

2

a b c a b c

b c a c a b

Gii: S dng bt ng thc AM-GM, ta c:

2 2

2 .4 4

a b c a b ca

b c b c

2 2

2 2

2 .4 4

2 .4 4

b a c b a cb

a c a c

c a b c a bc

a b a b

Cng theo v 3 bt ng thc trn ta c:

2 2 2

2

a b c a b ca b c

b c a c a b

Hay 2 2 2

2 a b c a b c

b c a c a b

- 10 -

ng thc xy ra khi v ch khi a b c (pcm)

Nhn xt: y l dng bi tp nh gi im ri t AM sang GM. Nu nhng ai

mi ch tip xc qua bt ng thc AM-GM th c th nhn xt rng vic tm ra

nh gi 2 2

2 .4 4

a b c a b ca

b c b c

c v mang nhiu tnh may mn. Nhng

khng phi vy, chng ta cng , im ri ca bt ng thc trn ti a b c .

Khi 2

2

a a

b c

, chng ta phi to ra mt biu thc va c gi tr bng

2

a, va

c th loi c mu ca biu thc 2a

b c. Hn na, 2 v ca bt ng thc l ng

bc 1, t d dng nhn ra biu thc thm vo phi l 4

b c.

S dng kt qu bi ny ta c th lm bi ton sau:

V d 3: [IMO 1995] Cho , , 0a b c tha mn 1abc . Chng minh rng:

3 3 3

1 1 1 3

2a b c b a c c a b

(1)

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

3 3 31 1 1 1

2

abc abc abc

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

2 2 2

1 1 1

1 1 1 1

1 1 1 1 1 1 2

a b c

a b c

b c a c a b

t 1 1 1

, ,x y za b c

, ta quay tr li v d 2.

Nhn xt: Bi ny c th gii bng bt ng thc Cauchy Schwarz m chng ta s

xt trong phn sau.

V d 4: Cho , , 0a b c . Chng minh rng:

2 2 2 4

ab bc ca a b c

a b c b c a c a b

- 11 -

Gii: Ta c:

1 1 1.

2 4

1 1 1.

2 4

1 1 1.

2 4

ab abab

a b c a c b c a c b c

bc bcbc

b c a a b b c a b b c

ca caca

c a b a b b c a b b c

Cng theo v 3 bt ng thc trn ta c iu phi chng minh.

ng thc xy ra khi v ch khi a b c

Nhn xt: Trong v d trn chng ta s dng bt ng thc AM-GM dng cng

mu s: Cho 1 2, ,..., na a a l cc s thc dng. Ta c:

21 21 2

1 1 1... ...n

n

a a a na a a


V d 5: Cho 3 s , ,a b c khng m, chng minh rng:

3 3 3

3 3 33 3 31

a b c

a b c b a c c a b

Gii: Xt bt ng thc ph sau:

2

31 1 02

xx x

Tht vy, theo bt ng thc AM-GM, ta c:

2 2

3 2 1 11 1 1 12 2

x x x xx x x x

(1)

p dng vo bi ton ta c:

3 2

3 3 2 2 2 23

1 1

11 1

2

a a

a b ca b c b c b c

a a

Tng t ta c

- 12 -

3 2

3 2 2 23

b b

a b cb a c

3 2

3 2 2 23

c c

a b cc a b

Cng ba bt ng thc theo v ta c iu phi chng minh.

ng thc xy ra khi v ch khi a b c .

Nhn xt: Bi ton trn thuc dng bi tp nh gi im ri ca bt ng thc t

biu thc GM sang AM. im kh ca v d trn l nm ch i bin v tm ra bt

ng thc ph (1). Bi tp trn cn c th gii bng bt ng thc Cauchy-Schwarz.

V d 6 [diendantoanhoc.net] Cho 3 s thc dng , ,a b c tha mn 1ab bc ca

.Chng minh rng:

2 2 2

1 1 1 1 1 13 1 1 1

ab bc ca a b c

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

2

23

cyc

ab bc ca ab bc ca ab bc ca a ab bc ca

ab bc ca a

3 3

.cyc cyc cyc

a b a ca b

b a a a

M theo bt ng thc AM-GM th

16

. 2cyc cyc cyc

a b a c a b

a a b a

Cn chng minh 6cyc cyc

a b

b a (hin nhin ng theo AM-GM)

Vy bt ng thc cho c chng minh.

ng thc xy ra khi v ch khi 1

3a b c

- 13 -

Nhn xt: Vi bi ton trn, nu kho lo s dng gi thit 1ab bc ca th bi

ton s tr nn n gin.

V d 7: Cho cc s thc dng , ,a b c . Chng minh:

a b c a b b c c a

b c a c a a b b c

Gii: t , ,a b c

x y zb c a . Khi , ta c:

1 1

1 1

a b yz yy

c a z z

Bi ton quy v vic chng minh:

1 1 10

1 1 1

x y z

y z x

2 2 21 1 1 1 1 1 0x z y x z y

2 2 2 2 2 2 3x z z y y x x y z x y z

D thy theo bt ng thc AM-GM ta c:

2 2 2 3 3 333 3x z z y y x x y z

`

2

2 2 2

3

x y zx y z x y z

(v 3x y z )

Kt thc chng minh.ng thc xy ra khi v ch khi a b c .

Nhn xt: rng biu thc v phi ca bt ng thc cha php cng gia 2

bin c t v mu nn vic s dng bt ng thc AM-GM mt cch trc tip l

v cng kh khn. Do phng n kh d nht l i bin to ra bt ng thc

mi.

By gi, chng ta s xt ti mt k thut mi trong vic chng minh bt ng thc

bng AM-GM, l k thut nh gi ph nh. K thut ny c dng chng

- 14 -

minh mt s bt ng thc khi p dng trc tip AM-GM th b ngc du rt hiu

qu.

V d 8 [ Bulgarian TST 2003] Cho cc s thc dng , ,a b c tha mn 3a b c .

Chng minh:

2 2 2

3

1 1 1 2

a b cS

b c a

Gii: Bin i v s dng bt ng thc AM-GM ta c:

2 2

2 2

2 2

2 2

2 2

2 2

1 1 2 2

1 1 2 2

1 1 2 2

a ab ab aba a a

b b b

b bc bc bcb b b

c c c

c ca ca cac c a

a a a

Cng theo v 3 bt ng thc trn ta c:

1 1

32 2

S a b c ab bc ca ab bc ca

Mt khc: 2

9 3 3a b c ab bc ca ab bc ca

T suy ra 3

2S

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

Nhn xt: 1. bt ng thc ban u, nu ta p dng trc tip bt ng thc AM-

GM th s b ngc du. V d:

3

3 2 2 2

33. 3.

2 .2 .2 21 1 1

abc abcS

b c ab c a

(sai)

2. Ta c bi ton tng qut ca bi ton trn:

Cho cc s thc dng 1 2, ,..., na a a tha mn 1 2 ... na a a n . Chng minh rng:

1 2

2 2 2

2 3 1

...1 1 1 2

naa a n

a a a

- 15 -

V d 9: Cho , ,a b c l cc s thc dng. Chng minh:

3

2 2 228

a b c ab bc ca

abc a b c

Gii: Theo bt ng thc AM-GM ta c:

362 2 2

2 2 2 22

3 27

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

Suy ra:

3 3

2 62 2 2 2 2 2

27ab bc ca ab bc caab bc ca

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

Cn chng minh:

3 62

12

2728

a b c ab bc ca

abc a b c

Theo bt ng thc AM-GM ta c:

63 2 2 23 6 62

55

12 4 42 2

34 275 5 5

27 27 27

a b ca b c ab bc ca ab bc ca

abc a b c abc abc

(1)

Mt khc, ta c:

3

23. 2327

a b c

abc

(2)

T (1) v (2) ta c iu phi chng minh.

ng thc xy ra khi v ch khi 0a b c

Nhn xt: Trong bi ton trn nu khng quan st k lng m p dng ngay bt

ng thc AM-GM th s dn n ngc du v

3

27a b c

abc

nhng

2 2 21

ab bc ca

a b c

. Qua cho chng ta thy c v p v sc mnh ca phi hp

hai bt ng thc ng bc ngc chiu.

- 16 -

V d 10 [IMO 2005]: Cho cc s dng , ,x y z tha mn 2 2 2 3x y z . Chng

minh rng:

5 2 5 2 5 2

5 2 2 5 2 2 5 2 20

x x y y z z

x y z y z x z x y

Gii: Bt ng thc cho c vit li nh sau:

5 2 2 2 2 2

1 3

cyc x y z x y z

T y ta suy ra ch cn xt trng hp 2 2 2 3x y z .

Bt ng thc cn chng minh tng ng vi

5 2

11

3cyc x x


6 65

2

2

1

x xx

x x

t 2 2 2, ,a x b y c z . Suy ra: 3a b c .

Bt ng thc cn chng minh tr thnh

3

11

23

1

cyc a aa

3 2

11

2 2 3cyc

a

a a a

2 2

3 2

1 2 3 30

2 2 3cyc

a a a

a a a

(1)

Khng mt tnh tng qut, gi s a b c , suy ra 1a c . Xt 2 trng hp:

+TH1: 1b c , suy ra 2a , khi :

- 17 -

3

3

3

2 3 3 0

2 3 3 0

2 3 3 0

a a

b b

c c

Suy ra, (1) ng.

+TH2: 1b c , suy ra 2a , khi :

3 2 3 22 2 3 5 1 2 3 2a a a a a a a

33 3

2 3 2 3

1 3 2 1 3 22 2 0

2 2 2 2

aa a

a a a

Suy ra 3 2

1 1

2 2 3 5

a

a a a

. Cn chng minh:

3 2 3 2

1 1 4

2 2 3 2 2 3 5

b c

b b b c c c

Ta c b : Vi mi 0 1x , ta c:

3 2

1 2

2 2 3 5

x

x x x

(2)

Ta c (2) tng ng vi: 34 1 2 1x x x

+ Nu 1

2x , ta c iu phi chng minh.

+ Nu 1

2x , ta c:

3 3 34 1 2 1 4 2 2 1 2 2 2 1x x x x x x x

222 2 1 2 1 0x x x (pcm)

Bt ng thc (1) c chng minh.

ng thc xy ra khi v ch khi 1a b c .

Nhn xt: 1. im kh ca bi ton ny l vic a bt ng thc v dng (1) nh

bt ng thc AM-GM.

- 18 -

2. Bi ton ny c th gii bng mt s cc khc nh Cauchy-Schwarz, S.O.S,

U.C.T.

Tip theo, chng ta s xt mt s v d v s kt hp gia bt ng thc AM-GM

vi mt s bt ng thc cng nh phng php khc.

u tin chng ta s xt ti s kt hp gia 2 bt ng thc AM-GM v Cauchy-

Schwarz:

V d 11 [diendantoanhoc.net] Cho 3 s thc dng , ,a b c . Chng minh rng:

1 1 1 3

3 2 3 2 3 2 5a a b b b c c c a abc

Gii: t 1 1 1

, ,a b cx y z

. Bt ng thc cn chng minh tr thnh:

3

3 2 3 2 3 2 5

x x x

zx yz xy zx yz xy

3

55 . 3 2 5 . 3 2 5 . 3 2

x y z

z x y x y z y z x

Theo bt ng thc AM-GM v Cauchy-Schwarz, ta c:

23 2 55 . 3 2cyc cyc

x x

x y zz x y

2

2

2 2 2

2

2 2 2

2

3 2 5 5 3 2 2 5 3

2

3 7

2

1 203

3 3

x y z

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

x y z

x y z xy yz zx

x y z

x y z xy yz zx xy yz zx

2

2 2 2 2 2 2

2

2 2 2

2

1 203

3 3

3 3

55 2

x y z

x y z x y z xy yz zx

x y z

x y z xy yz zx

- 19 -

Bt ng thc c chng minh.


Tip theo s l s kt hp y ngon mc gia 2 bt ng thc AM-GM v Schur

qua v d sau y:

V d 12 [Vasile Cirtoaje]: Cho cc s khng m , ,a b c sao cho 3 3 3 3a b c .

Chng minh rng:

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

Gii: Theo bt ng thc AM-GM ta c:

3 3 31 4

3 3

b c abc

(1)

T suy ra: 3 3 3 3 3

4 4 4

3

b c a b cb c

Tng t ta c: 3 3 3 3 3

4 4 4

3

a b a b ca b

3 3 3 3 3

4 4 4

3

c a a b cc a

Cng 3 bt ng thc trn theo v ta c:

3 3 3 3 3 34 4 4 4 4 4 3 3 3

4

3

a b b c c aa b b c c a a b c

Cn chng minh: 3 3 3 3 3 3

3 3 34

33

a b b c c aa b c

3 3 3 3 3 3 3 3 34 3 9a b b c c a a b c

Mt khc, theo bt ng thc Schur, ta c:

3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 34 9a b b c c a a b c a b c a b c

3 3 3 3 3 3 3 3 34 3 9a b b c c a a b c

Vy bt ng thc trn c chng minh.


- 20 -

Nhn xt: Trong v d trn, nu khng pht hin ra bt ng thc ph (1) th vic

gii l rt kh khn. V d trn cn c th gii quyt bng phng php dn bin.

Cui cng, ta s xt n s kt hp gia bt ng thc AM-GM v phng php

kho st hm s.

V d 13 [Vit Nam TST 2005]: Cho cc s , , 0a b c . Chng minh:

3 3 3

3 3 3

3

8

a b c

a b b c c a

Gii: t , , , 1.b c a

x y z xyza b c

Bt ng thc cn chng minh tr thnh:

3 3 3

1 1 1 3

81 1 1x y z


33 3 6 2

33 3 6 2

33 3 6 2

1 1 1 1 33

81 1 8 1 2 1

1 1 1 1 33

81 1 8 1 2 1

1 1 1 1 33

81 1 8 1 2 1

x x x x

y y y y

z z z z

Ta cn chng minh:

2 2 2

1 1 1 3

41 1 1x y z

(1)

Ta c :

2 21 1 1

, 011 1

x yxyx y

2 2

1 0xy x y xy (lun ng)

Suy ra: VT(1)

2

2 2 2

1 1 1 1

1 1 2 11 1

z z z

xy z z zz z

Gi s max , , 1z x y z z .

Xt hm s: 2

2

1( )

2 1

z zf z

z z

- 21 -

Ta c:

2

4

1'( ) 0, 1

1

zf z z

z

T suy ra: 3

( ) (1)4

f z f

Bt ng thc c chng minh.

ng thc xy ra khi v ch khi a b c

Nhn xt: V d trn l mt bi ton hay v kh. gii c bt ng thc trn

cn phi hp rt nhiu k thut m li gii trn nm trong nhng li gii nhanh v

hay nht cho bi ny.

Sau y, chng ta s xt thm 2 v d v du bng khng i xng trong bt ng

thc AM-GM, qua , ta s thy ht c v p v s tinh t ca bt ng thc.

V d 14: Cho cc s , ,a b c tha mn 3a b c . Chng minh rng:

3 3 31 1 1 5a b b c c a

Gii: Ta c: 3 3 31 1 1a b b c c a

2 2 2

2 2 2

2 2 2

1 1 1 1 1 1

2 2 2. . .

2 2 2

32

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

b c aa b c

ab bc ca

Cn chng minh: 2 2 2 4ab bc ca (1)

Gi s b l s nm gia 2 s ,a c .Ta c:

2 2 2

2 2 2 2 2 2 2

0a b a b c

ab a c a b abc

ab bc ca a b abc bc b a ac c

2

2 21 1 2 3 3.2 . 3 . 4

2 2 3

b b bb a c b b

Suy ra iu phi chng minh.

ng thc xy ra khi v ch khi 0, 1, 2a b c v cc hon v.

Nhn xt: Ci kh trong v d ny l nh gi c bt ng thc (1). Ngoi cch

nh gi nh trn, chng minh (1) c th dng phng php dn bin v bin.

- 22 -

V d 15 [Tp ch TH&TT]: Cho , ,a b c l cc s thc i mt khc nhau thuc

[0;2]. Chng minh:

2 2 2

1 1 1 9

4P

a b b c c a

Gii: Khng mt tnh tng qut gi s 2 0a b c . Theo bt ng thc AM-GM

ta c:

32 21 1

3 . . 3a b a b a b a ba b a b

(*)

32 2

1 13 . . 3b c b c b c b c

b c b c

Cng 2 bt ng thc trn theo v ta c:

2 2

2

1 12 6

12 6

a ca b b c

P a ca c

Cn chng minh:

21 9

2 64

P a ca c

. (1)

V 2 0a b c nn 2

1 90 2 2.2 6

2 4a c P

Vy 9

4P . ng thc xy ra khi v ch khi 2, 1, 0a b c v cc hon v.

Nhn xt: Trong bi ton trn, nu ta p dng 3 ln bt ng thc (*) cho 3 bin

, ,a b b c c a th bt ng thc s ri vo ng ct, khng th i tip. n lc

dn n bt ng thc (1) l bt ng thc mt bin th bi ton tr nn n gin,

ta ngh ngay n phng php kho st hm s trn on.

Vy l chng ta cng nhau i ht chng ng khm ph bt ng thc AM-GM.

Pht biu v chng minh bt ng thc c a ra trong mc 1. Cc k thut

chuyn i qua li gia trung bnh cng v trung bnh nhn c trnh by trong

cc v d 2, 3, 4, 5. K thut phi hp gia bt ng thc AM-GM v bin i i s

thng thng c cp trong cc v d 6 ,7. Cc k thut nh gi ph nh v

phi hp cc bt ng thc ng bc ngc chiu c gii thiu qua cc v d

8, 9. S kt hp gia bt ng thc AM-GM v cc bt ng thc khc c gii

thiu trong cc v d 11, 12, 13. Cui cng, phng php cn bng h s hay du

- 23 -

bng khng i xng trong bt ng thc AM-GM c cp trong hai v d

14, 15. Qua cc v d trn phn no cho chng ta thy v p, sc mnh, s linh

hot ca bt ng thc AM-GM trong vic chng minh bt ng thc. Sau y l

mt s bi tp gip cc bn cng c kin thc:

3. Bi tp t gii

Bi 1. Cho cc s thc dng , ,a b c tha mn 1abc . Chng minh:

a b ca b c

b c a

Bi 2. Cho cc s thc dng , ,a b c tha mn 1abc . Chng minh:

3b c c a a b

a b ca b c

Bi 3. [Russia MO] Cho , ,a b c 0 tha mn 3a b c . Chng minh:

a b c ab bc ca

Bi 4. Cho cc s thc dng , ,a b c . Chng minh:

35 2 5 2 5 23 3 3a a b b b b a b c

Bi 5. Chng minh rng vi mi s thc , , 1x y z :

2 2 2

2 2 2

1 1 12

1 1 1

x y z

y z z x x y


32 2 2 2 2 23 x y y z z x xy yz zx xyz x y z

Bi 7. [MOSP 2001] Cho cc s thc dng , ,a b c tha mn 1abc . Chng minh:

4 1a b b c c a a b c


3

3 . .3 2 3

a ab abc a b a b ca


3 31 1 1 3

1 1 1 1a b b c c a abc abc

- 24 -

BT NG THC MINKOWSKI V NG DNG


1. Bt ng thc Minkowski

1.1 Bt ng thc Minkowski dng 1

1.1.1 nh l

Cho 1 2

1 2

, ,...,

, ,...,

n

n

a a a

b b b

v 1 p , khi

1 1 1

1 1 1

n n np q ppp p

k k k k

k k k

a b a b

ng thc xy ra khi v ch khi 1 2

1 2

... n

n

aa a

b b b .

c bit: 2 22 2 2 2a b c d a c b d

2 2 22 2 2 2 2 2a b c m n p a m b n c p

1.1.2 Chng minh

Ly q sao cho 1 1

1p q . S dng bt ng thc Holder cho 2 b dy s:

1 2

1 1 1

1 1 2 2

, ,...,

, ,...,

n

p p p

n n

a a a

a b a b a b

v

1 2

1 1 1

1 1 2 2

, ,...,

, ,...,

n

p p p

n n

b b b

a b a b a b

Ta c: 11

1 1 1

1 2 1 1

1

... ...n

p q p q pqp p p pn n n k k k

k

a a a a b a b a a b

11

1 1 1

1 2 1 1

1

... ...n

p q p q pqp p p pn n n k k k

k

b b b a b a b b a b

Li c: 1 1

1 1p p qp q , nn cng 2 bt ng thc trn ta c:

1 1 1

1 1 1

n n np q ppp p

k k k k

k k k

a b a b

- 25 -

1.2 Bt ng thc Minkowski dng 2:

1.2.1 nh l

Cho

1 2

1 2

1 2

, ,...,

, ,...,

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

, ,...,

n

n

n

a a a

b b b

l l l

khi ta c bt ng thc

1 2 1 2 1 21

... ... ... ... ...n

n n n nn n n i i i

i

a a a bb b l l l a b l


1 2

... n

n

aa a

b b b .

1.2.2 Chng minh:

1 2 1 2 1 21

... ... ... ... ...n

n n n nn n n i i i

i

a a a bb b l l l a b l

1 2 1 2

1 1 1 1 1 1

... ...... 1

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

n nn n

n n n n n n

a a a l l l

a b l a b l a b l a b l


1 2 1

1 1 1 1 1 1

1 2 1

1 1 1 1 1 1

... 1...

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

...

... 1...

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

n nn

n n n n n n

n nn

n n n n n n

a a a aa

a b l a b l n a b l a b l

l l l ll

a b l a b l n a b l a b l

T suy ra:

1 2 1 2

1 1 1 1 1 1

... ...... 1

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

n nn n

n n n n n n

a a a l l l

a b l a b l a b l a b l

(pcm)


1 2

... n

n

aa a

b b b .

2. V d:

V d 1: Cho cc s thc dng ,a b .Chng minh:

3 3 32 2a b

a bb a a b

(1)

- 26 -

HD: a bt ng thc (1) v dng:

3 3 31 1

1 1a b

a bb a a b

S dng bt ng thc Minkowski loi 2 ta c iu phi chng minh.

V d 2: Cho cc s thc dng , ,a b c . Chng minh rng:

2 2 22 2 2 3

2

a b ca b c b c a c a b

HD: V bt ng thc trn l thun nht nn ta c th chun ha: 1a b c .

Bt ng thc cn chng minh tr thnh:

2 2 22 2 2 31 1 1

2a a b b c c

Ta c:

2

2 2 3 33

2 2

a b c a b cVT a b c a b c VP

Vy ta c iu phi chng minh,


V d 3: Cho cc s thc dng , ,a b c sao cho 1a b c . Tm min ca:

2 2 2

2 2 2

1 1 1P a b c

b c a

Gii: Ta c: 2 2 22 2 2

1 1 1 1 1 12P a b c ab bc ca

a b c ab bc ca

2

2 2

2

1 80 1 1 1 1 12 2

81 81 81 81 81

2 80 1 1 1 482

3 81 3

a ab bc caa a ab ab bc ca

a b c

Vy min 82P khi v ch khi 1

3a b c .

Nhn xt: Vi bi ton trn nu vi vng p dng ngay bt ng thc AM-GM th

s khng tha mn iu kin 1a b c dn n sai. Ta c bi ton tng qut ca

bi trn: Cho cc s thc dng 1 2, ,..., na a a tha mn 1 2, ,...,2

n

na a a . Tm min:

- 27 -

2 2 2

1 22 2 2

2 3 1

1 1 1... na a a

a a a

V d 4: Cho

1 2

1 2

, ,..., 0

... 1

, 2

n

n

a a a

a a a

n n

. Chng minh:

1 2

1 1 11 1 ... 1 1

n

n

na a a

Gii: p dng bt ng thc Minkowski loi 2 ta c:

1 2 1 2

1 1 1 11 1 ... 1 1

...

n

nn n

a a a a a a


1 21 2

... 1... nn n

a a aa a a

n n

Do : 1 2

1 1 11 1 ... 1 1

n

n

na a a


... na a an

.

V d 5: Cho cc s thc dng , ,a b c tha mn ab bc ca abc . Chng minh:

2 2 2 2 2 22 2 23

a b b c c a

ab bc ca

Gii: Theo bi ra ta c: 1 1 1

1ab bc ca abca b c

Bt ng thc cn chng minh tng ng vi:

2 2 2 2 2 2

1 2 1 2 1 23

a b b c c a (1)

p dng bt ng thc Minkowski, ta c:

2 2

2 2 2 2 2 2

1 2 1 2 1 2 1 1 1 1 1 12 3

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

Bt ng thc trn chng minh.

- 28 -


Nhn xt: Bi ny khng kh, ch cn tinh a bt ng thc v dng (1) l bi

ton tr nn rt d.

V d 6: Cho cc s thc dng , ,a b c . Chng minh:

2 2 22 1 1 1 1 1 1 1a b c a b c abc

Gii: B : 3 32 32 1 1 1 , 0u u u u (1) 4 21 1 0u u u

Quay tr li bi ton, ta c:

3 3 3 3

2 2 2 2 2 22 1 1 1 2 1 .2 1 .2 1a b c a b c

3 3 33 3 3

33 3 3

3

1 1 1 1 1 1

1 1 1 1 1 1

1 1 1 1

a a b b c c

a b c a b c

abc a b c

Vy bt ng thc c chng minh.


3. Bi tp t gii


2 2 2

2 2 2 2 2 21 1 1 1 1 13 3 3

a b b c c aa b c

Bi 2. Chng minh rng:

1 2 1 21 1 ... 1 ... 1 , 0n

nn n ia a a a a a a

Bi 3. Chng minh rng:

,2. ,

, , 0

nn nk k

k

n km p m p m

m p m p

Bi 4. Cho cc s thc dng , ,a b c tha mn 3

2a b c . Tm min:

3 3 33 3 3

3 3 3

1 1 1S a b c

b c a

- 29 -

BT NG THC HOLDER V NG DNG


1. Bt ng thc Holder

1.1 Dng tng qut

1.1.1 nh l

Cho 2 b s 1 2

1 2

, ,...,

, ,...,

n

n

a a a

b b b

v ,p q sao cho

1 11

p q . Khi , ta c:

1 1

1 2 1 2 1 1 2 2... . ... ...p p p q q qp q

n n n na a a b b b a b a b a b

1.1.2 Chng minh

B : Cho ,a b v ,p q sao cho 1 1

1p q . Khi :

p qa aab

p q .

Chng minh: V ,p q , *1 1

, , ,m n kp q

Sao cho 1 1

,m n

p k q k vi m n k . S dng bt ng thc AM-GM ta c:

. .. .

k kk kp q m n

k k km na a m n ma nb

a b a b abp q k k k

ng thc xy ra khi v ch khi p qa b .

p dng b 1 1

1 1

,j j

n np qp q

i i

i i

a ba b

a b

vi 1,j n . Ta c:

1 1 1 11 1

1 1 1 11 1 1 1

1

1

1

. .1 1 1 1. . . .

. .

.1 1

1

.

p q p qn nj j j j j j j j

n n n np q p qj jn n n np q p q

p q p qi i i ii i i ii i i i

i i i i

n

j j

j

n pp q

i i

i i

a b a b a b a b

p q p qa b a b

a b a b

a b

p q

a b

1 1

11 1 1

1

.n n np q

p q

i i i i

i i in q

a b a b

- 30 -

1.2. M rng 1 ca bt ng thc Holder [ Bt ng thc Francis-Lithewood]

Cho 2 b s 1 2

1 2

, ,...,

, ,...,

n

n

a a a

b b b

v ,p q sao cho 1 1

1p q . Khi , ta c:

1 1 2 2 1 2 1 2... ... ... , 0q ppq p p p q q q

n n n na b a b a b a a a b b b pq

1.3 M rng 2 ca bt ng thc Holder

Cho m b s

1 2

1 2

1 2

, ,...,

, ,...,

...

, ,...,

n

n

n

a a a

b b b

l l l

v 1 2

1 2

, ,...,

... 1

n

n

p p p

p p p

. Khi ta c:

1 1 1

1 1 1 1

... ...i i i

i i i

n n n np p pp p p

i i i i i i

i i i i

a b l a b l

1.4 M rng 3 ca bt ng thc Holder [Bt ng thc Jensen]

Cho m b s

1 2

1 2

1 2

, ,...,

, ,...,

...

, ,...,

n

n

n

a a a

b b b

l l l

v , ,...,

... 1

. Khi ta c:

1 1 1 1

... ...n n n n

i i i i i i

i i i i

a b l a b l

2. V d

Trong th gii bt ng thc, cc bt ng thc c cha cn thc hoc cc ly tha

bc cao lun l chng ngi vt cn bc chng ta. Vic chng minh cc bt ng

thc nh vy lun gp kh khn v thng lm chng ta tn rt nhiu thi gian.

Nhng ngh nh ly tha kh cn thc trong nhiu trng hp s a ta n

vi nhng bi ton phc tp v kh hn bi ton gc. Tuy nhin, khng hn l

khng c cch gii quyt vn ny; mt trong nhng cch x l tt chnh l s

dng bt ng thc Holder:

- 31 -

hiu r hn v bt ng thc ny, chng ta s n vi v d sau:

Cho , , 0a b c .Chng minh:

43

a b c a b c a b ca b c

c b a a b b c c a

Phn tch v nh hng li gii

Mt cu hi c t ra l: Ti sao li ngh n vic s dng bt ng thc Holder?

- Nh ni trn th vic cn thc xut hin cn thc c 2 v gi cho chng ta

tng bnh phng c 2 v. Khi , ta cn chng minh:

2 316

3

a b ca b c a b c

c b a a b b c c a

Xem ra, bc u chng ta kh thnh cng trong vic kh i du cn thc v

phi. Tuy nhin, n y, nu nh bin i tng ng th s tn kh nhiu thi

gian, v v tri cn c s xut hin ca ; ;a b c di mu ca cc phn thc.

iu cn thit by gi l phi trit tiu c cc i lng ny.

Nu nh s dng bt ng thc Holder kiu :

2

33 3 3a b c a b c c b a a b c a b c

c b a

Th nhng cch ny vn cha ph hp; nu lm nh trn, ta cn phi chng minh

33 3 3 3

16

3

a b c a b c a b c

a b c a b b c c a

Tuy nhin, vic chng minh bt ng thc ny li b tc do s xut hin ca cc

i lng 3 3 3; ;a b c a b c v tri. Do , chng ta s khng i theo con

ng ny. Vy, by gi phi lm th no? Cc tng kh cn thc hu nh

c s dng nhng cng vic chng minh vn khng thnh cng. R rng l

chng ta cn phi tinh t hn cht na. rng trong bc kh ; ;a b c bng

V d 1

- 32 -

vic s dng bt ng thc Holder nu nh thay i lng nhn thm c b a

bng 2 2 2

c a b b c a a b c

th ta s m a b ;c a ;b a s l s nguyn.

Khi ta c :

2

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

c b a

Ta cn chng minh

3 3

2 2 2

8 16

3

a b c a b c

a b b c c ac a b b c a a b c

2 2 2

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

9a b c ab bc ca abc

y l mt kt qu quen thuc theo bt ng thc AM-GM:

33a b c abc

3 2 2 23ab bc ca a b c

T ta c iu phi chng minh. Du ng thc xy ra khi a b c

Cho , , 0a b c .Chng minh rng:

2a b c

b c c a a b


p dng bt ng thc Holder, ta c:

2

32a a b c a b cb c

Ta cn chng minh:

V d 2

- 33 -

3 2 2 24a b c a b c b c a c a b

3 3 3 6a b c abc ab a b bc b c ca c a

Bt ng thc ny hin nhin ng theo bt ng thc Schur:

3 3 3 3a b c abc ab a b bc b c ca c a v 3 0abc

Kt thc chng minh. Du ng thc xy ra khi ; 0a b c v cc hon v

Cho , , , 0a b c d tha mn 1abcd .Chng minh:

4

4 4 4 4 4 1 1 1 14 1 1 1 1a b c d a b c da b c d

( Gabriel Dospinescu)


Do 1abcd nn bt ng thc cn chng minh c th vit li thnh

44 4 4 4 44 1 1 1 1a b c d a b c d abc bcd cda dab

n lc ny th tng kh r. p dng bt ng thc Holder, ta c:

4 4 4 44 1 1 1 1a b c d a bcd

4 4 4 44 1 1 1 1a b c d b cda

4 4 4 44 1 1 1 1a b c d c dab

4 4 4 44 1 1 1 1a b c d d abc

Cng v theo v cc bt ng thc trn, ta c:

4 4 4 444 1 1 1 1a b c d a b c d abc bcd cda dab

44 4 4 4 44 1 1 1 1a b c d a b c d abc bcd cda dab

V d 3

- 34 -

y chnh l iu cn phi chng minh. Du ng thc xy ra khi 1a b c d


2 2 2 2 2 2

3

23 3 3

a b c

b c c a a b


Cng tng t v d , ta s tm cch kh cn thc di mu v tri bng vic s

dng bt ng thc Holder :

2

32 2

2 23

3

aa b c a b c

b c

Khi ta cn chng minh:

3

2 2

9

43

a b c

a b c

3 2 24 9 3a b c a b c

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

y l mt kt qu quen thuc theo bt ng thc AM-GM.

3 3 3 23a a b a b

3 3 3 23b b a b a

3a ab a b

Kt thc chng minh.Du ng thc xy ra khi a b c

V d 4

- 35 -

Qua cc v d trn ta thy c sc mnh ca bt ng thc Holder trc nhng

bi ton c dng phn thc: a bi ton t dng phc tp v dng n gin hn.

Bn cnh , Holder cn rt hiu qu i vi cc dng bt ng thc thng thng:

Cho , , 0a b c tha mn 3a b c . Chng minh:

31 1 1

a b c

b bc c ca a ab


Li gii 1: Khng mt tnh tng qut, gi s ; ;a max a b c

rng du ng thc xy ra ti 2 b l 1a b c v 3; 0a b c Do ta s

s dng bt ng thc Holder vi cc tham s , ,m n p nh sau:

2

332 211

aa ma nb pc b bc m a n p ab

b bc

(*)

Du ng thc (*) xy ra khi

3 3 32 2 2

1 1 1

1 1 1

a b c

b bc c ca a ab

a ma nb pc b bc b mb nc pa c ca c mc na pb a ab

By gi ta s chn b s , ,m n p tha mn ng thi 2 du bng xy ra. vic

chng minh nng nhc, ta s chn , ,m n p sao cho 2m a n p ab c dng

2

k a b c ; c th :chn 2, 1, 3m n p . Khi (*) tr thnh:

2

3 62 2 3 1 81

aa a b c b bc a b c

b bc

Ta cn chng minh

6 328 3 2 3 1a b c a a b c b bc (**)

7 2 328 3 9 2 3a b c a a b c b a b c bc a b c

V d 5

- 36 -

4 4 2 4 4 2 3 3 2 2 24 3 2 2 3

4 26 39 54 261

24 93 97

ab a b a b a b a b a b c

a bc a b c a b c

Bt ng thc cui ng theo bt ng thc Schur v AM-GM

Kt thc chng minh.Du ng thc xy ra khi 1a b c hoc 3; 0a b c v

cc hon v

Tuy nhin, vic bin i t (**) v bt ng thc cui l mt bc tn kh nhiu

thi gian v ch cn mt cht s sut trong vic tnh ton th ton b cng trnh ca

ta s "tan vo my khi". Chng ta hy cng xem xt li gii sau:

Li gii 2:

S dng bt ng thc Holder ta c:

2 3

2

311

aa b bc a

b bc

Khi ta cn chng minh:

3

2 2 2

3 3 3 9 3 9a b c ab bc ca abc

2 2 2 2 2

2 3 3 3 3 33 6 9 3 9a a b a b abc ab abc

S dng bt ng thc AM-GM ta c

2 2 2 2

3 3 3 3 2a b a b ab

2 2 2 2

2 23 3 3 33 6 9 4a a b a b a ab ab

Do ta ch cn chng minh:

2

36 9ab bc ca abc abc

- 37 -


2 2 2 2

3 3 3 36 3 6 9ab bc ca abc abc abc abc

V t gi thit 3a b c ta c

2 2 2 2

3 3 3 39 3 9 9abc a b c abc abc abc abc

2

36 9ab bc ca abc abc

Vy ta c iu phi chng minh.Du ng thc xy ra khi 1a b c hoc

3; 0a b c v cc hon v.


35 2 5 2 5 23 3 3a a b b c c a b c

(USAMO 2004)


Cng nh v d u tin ,mt cu hi c t ra l: Ti sao li s dng bt ng

thc Holder?-Du hiu no nhn bit n?

D thy l bt ng thc cn chng minh khng thun nht, hn na, cc bin hon

ton c lp vi nhau; tng l ta s "p" cc i lng ring bit 5 2 3a a ;

5 2 3b b ; 5 2 3c c ra h bt s bin. Tuy nhin, vic ny khng kh thi lm

bi v rt kh to ra c cc i lng ni trn, do ta s i tm mt con

ng khc. Vn da trn tng ban u, do vai tr ca , ,a b c nh nhau nn nu

x l c i din 5 2 3a a th bi ton s c gii quyt. Nhn thy du ng

thc ca bt ng thc ti 1a b c ; bc ca v phi l 3 v , ,a b c c lp vi

nhau. Nn ta s s dng bt ng thc Holder nh sau:

3 3 3 31 1 1 1 1 1a b c a b c

Ta s chng minh bt ng thc sau:

V d 6

- 38 -

5 2 33 2a a a

2 21 1 1 0a a a a (ng)

Tng t ta c:

5 3 33 2b b b

5 2 33 2c c c

Nhn v theo v 3 bt ng thc trn, ta c

5 2 5 2 5 2 3 3 33 3 3 1 1 1 1 1 1a a b b c c a b c

35 2 5 2 5 23 3 3a a b b c c a b c

y chnh l iu cn phi chng minh.Du ng thc xy ra khi 1a b c

Cho , , 0a b c tha mn 3a b c .Chng minh rng:

3 3 3a b c ab bc ca


Mt bt ng thc c nu ln trong cun Sng to bt ng thc ca Phm

Kim Hng. Sau y l li gii:

S dng bt ng thc Holder ta c

8

33 53 4a a b c a

Ta s chng minh:

8

3 3 3354 4 4 3a c c ab bc ca

(*)

t

3 3 3

4 4 4; ;a x b y c z

V d 7

- 39 -

8 3

3 3 3 5 4 4 4 4 4 4(*) 3x y z x y y z z x

V y l mt bt ng thc thun nht nn ta c th b qua gi thit u bi

chun ha 3 3 3 3x y z

Khi ta cn chng minh

4 4 4 4 4 43 x y y z z x

Theo bt ng thc AM-GM th

3 3 31 4

3 3

x y zxy

3 3 3 3 34 4 4

3

x y x y zx y

3 3 3 3 3

4 44 3

3

x y x y zx y

Do ta cn chng minh

3 3 3 3 34 3 9x y x y z (**)

Do 3 3 3 3x y z nn (**) tng ng

3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 34 9x y z x y y z z x x y z x y z

9 9 9 3 3 3 3 3 3 3 3 3 3 3 3 3 3 33x y z x y z x y x y y z y z z x z x

y chnh l bt ng thc Schur

Kt thc chng minh.Du ng thc xy ra khi 1a b c

Cho , , 0a b c tha mn 1 1 1

a b ca b c

. Chng minh rng:

2

27ab bc ca ab bc ca

V d 8

- 40 -


Kh khn ca bi ton chnh l gi thit ca n, do ta s x l iu kin u

tin. Ta c:

1 1 1a b c

a b c

abc a b c ab bc ca

t ; ;ab x bc y ca z . Khi bi ton tr thnh:

Cho , , 0x y z

x y z xy yz zx

. Chng minh rng:

2

27x y z x y z

p dng bt ng thc Holder ta c

2 32 2 2x y z x y z x y z

4

2

2 2 2

x y zx y z x y z

x y z

Ta s chng minh

4 2 2 227x y z x y z

Mt khc, t gi thit x y z xy yz zx ta c

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

Do ta cn chng minh

- 41 -

4

27 2x y z x y z x y z

3

27 54x y z x y z

Hin nhin ng theo bt ng thc AM-GM

3

27 27 27x y z x y z

T suy ra iu phi chng minh. Du ng thc xy ra khi 1a b c

3. Bi tp t gii

Cho , , 0a b c tha mn 3a b c .Chng minh:

a b c ab bc ca

( Russian Mathematical Olympiad 2002)

Cho , , 0a b c tha mn 4 4 4 3a b c . Chng minh:

2 2 2

3a b c

b c a

( Alexey Gladkich)


2 2 2

2 2 2 2 2 21

7 7 7

a b c

a ab b b bc c c ca a


Bi 1

Bi 2

Bi 3

Bi 4

- 42 -

2 2 2

1 1 1 4

4 4 4 a b ca bc b ca c ab

Cho , , , 0a b c d . Chng minh:

2 2 2 24

9

a b c d

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

Cho , , , 0a b c d . Chng minh:

3 4

4 . . .4 2 3 4

a ab abc abcd a b a b c a b c da

Cho , , 0a b c v 0ab bc ca .Chng minh rng:

48 48 481 1 1 15

a b c

b c c a a b

( Vasile Cirtoaje)

Bi 5

Bi 6

Bi 7

- 43 -

BT NG THC CAUCHY-SCHWARZ

on Quc t - Ng Hong Thanh Quang

1.Bt ng thc Cauchy-schwarz

1.1. nh l

Vi 2 dy s thc ty 1,..., na a v 1,..., nb b ta lun c bt ng thc:

22 2 2 21 1 1 1... ... ...n n n na a b b a b a b

ng thc xy ra khi 1

1

... n

n

aa

b b vi quy c l nu 0kb th a 0k

1.2. Chng minh:

Cch 1: S dng bt ng thc AM-GM:

Chia c hai v cho 2 2

1 1

n n

i i

i i

a b

ta cn chng minh:

1 2 2

1 1

1n

i i

n ni

i i

i i

a b

a b

S dng bt ng thc AM-GM ta c:

2 2

2 2

1 1

2 2 2 21 1 12 2 2 2

1 1 1 11 1 1 1

1 11

2 2

n n

i in n ni ii i i i i i

n n n nn n n n

i i ii i i i

i i i ii i i i

i i i i

a ba ba b a b

a b a ba b a b

Vy ta c iu phi chng minh

Cch 2: S dng tam thc bc hai:

- 44 -

Nu 2

1

0n

i

i

a

th ta c 1a ... 0na . Khi bt ng thc tr thnh ng thc.Do

ta ch cn xt vi 2

1

0n

i

i

a

.

Xt tam thc bc hai

2 2 2

1 1 1

( ) 2n n n

i i i i

i i i

f x a x a b x b

Ta thy

2

1

( ) 0;n

i i

i

f x a x b x

nn 0f

Hay

22 2 2 21 1 1 1... ... ...n n n na a b b a b a b

Cch 3: S dng bt ng thc Jensen

Cng nh cch th 2, ta ch cn xt trng hp 2

1

0n

i

i

a

Nhn xt: Nu c 1 s bng 0 th ta c th quy v chng minh bt ng thc cho

trng hp 1n bin s, v nu vn cn mt s no bng 0 th ta s tin hnh

tng t cho n khi khng cn s no bng 0.

Do ta ch cn chng minh trong trng hp 0ia l .

Xt 2( ) ;f x x x . Ta c '( ) 2xf x

''( ) 2 0f x .Do 2( )f x x l hm li trn .

Do theo bt ng thc Jensen, vi mi 0ic tha mn 1

0n

i

i

c

v ix , ta c

1 1

1 1

n n

i i i i

i i

n n

i i

i i

c x c f x

f

c c

- 45 -

Suy ra

2 2 21 1 1 1 1... ... ...n n n n nc x c x c c c x c x

Chn 2; ii i ii

ac b x

b ta c iu phi chng minh

1.3.H qu

H qu 1.3.1. Vi 2 dy s 1,..., na a v 1,..., nb b 0, 1,ib i n ta c:

222

11

1 `

.....

...

nn

n n

a aaa

b b b b

ng thc xy ra khi 1

1

... n

n

aa


Bt ng thc ny c gi l bt ng thc Cauchy-Schwarz dng Engel, gi tt l

Schwarz

H qu 1.3.2.Vi 2 dy s 1,..., na a v 1,..., nb b ta c:

2 22 2 2 2

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

Bt ng thc ny c gi l bt ng thc Minkowski

ng thc xy ra khi 1

1

... n

n

aa


2. V d

By gi chng ta s khng cp n nhng bi ton kh c bn nh:


2 2 2

2 2 21

2 2 2

a b c

a bc b ca c ab

- 46 -

M s i vo mt s v d kh hn thy c v p ca bt ng thc quan

trng ny:

V d 1.Cho , , 0a b c . Chng minh rng:

22 2 2 4 a ba b ca b c

b c a a b c

(BMO 2005)


Bi ton ny gy kh khn cho khng t ngi bi s xut hin kh l "v duyn"

ca i lng

24 a b

a b c

v phi. V th y l mt bt ng thc mnh hn bt

ng thc c bn: 2 2 2a b c

a b cb c a do

2

4 a ba b c a b c

a b c

.

Thng thng, khi gp nhng bi ton cht nh th ny, php bin i tng ng

lun cho ta hiu qu cao nht v khng cn phi s dng bt k s nh gi no

tuy nhin iu yu cu vic tnh ton qu kh khn v mt nhiu thi gian

cha k n vic d gy ra sai st . V vy, ta s i tm mt con ng khc, mt

hng suy ngh khc. u tin ta thy v tri ca bt ng thc trn c dng phn

thc, hn na, v phi s xut hin ca 2

a b

24 a b

a b c

phn no gi

cho chng ta v s dng bt ng thc Cauchy-Schwarz dng Engel. V vic cn

thit by gi l phi to c cc phn thc m t s c dng bnh phng

rng:

22

2a ba

a bb b

22

2b cb

b cc c

- 47 -

22

2c ac

c aa a

Do chng ta s thm cc i lng 2a b ; 2b c ; 2c a v tri to ra

cc i lng nh trn.Bt ng thc cn chng minh tng ng

22 2 2 4

2 2 2a ba b c

a b b c c ab c a a b c

2 2 2 24a b b c c a a b

b c a a b c

S dng bt ng thc Cauchy-Schwarz:

22 2 2

a b b c c aa b b c c a

b c a a b c

p dng bt ng thc c bn:

x y x y

Ta c 2 2 2 2

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

2 2

4a b b c c a a b

a b c a b c

Hay

2 2 2 2

4a b b c c a a b

b c a a b c

Kt thc chng minh, du ng thc xy ra khi

0

a b b c c a

b c a

b c c a

6 2 5 5 1; ; 0

4 2

a b c

a b c b b

- 48 -

V d 2 Cho , , 0a b c .Chng minh rng:

2 2 22 2 2 6 a b ca b ca b c

b c a a b c

(Phm Hu c)


tng hon ton tng t v d 1,ta s thm cc i lng 2a b

; 2b c ; 2c a v tri , bt ng thc c vit li thnh:

2 2 22 2 2 62 2 2 2

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

b c a a b c

22 2 2 2 2 26 a b c a b ca b b c c a

b c a a b c

Khng mt tnh tng qut gi s b l s nm gia a v c. p dng bt ng thc

Cauchy-Schwarz ta c:

22 2 2 2

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

b c a a b c a b c

Do ta cn chng minh

2 22 2 22 3a c a b c a b c

2 0b c b a

iu ny ng do b l s nm gia a v c.

Bi ton c chng minh .Du ng thc xy ra khi a b c

- 49 -

V d 3: Cho , , 0a b c tha mn a b c

a b cb c a

.Chng minh rng

3 3 3 3

2

a c b a c b

b c a c a b a b c

(Romania 2005)


tng s dng Cauchy-Schwarz dng Engel kh r rng. Nhng vic s dng

trc tip Cauchy-Schwarz:

23 3 3

3 3 3

2

a c b a c ba c b a c b

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

c

em li hiu qu?

Chng ta cha bit liu bt ng thc

23 3 3

3

2 2

a c b a c b

ab bc ca

ng hay cha

v con ng chng minh xem ra khng n gin cht no. R rng l ta vn

cha s dng ht d kin ca bi ton :a b c

a b cb c a

.

ng nhin, s dng c d kin ny ta phi to ra s xut hin ca cc i

lng ; ;a b c

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

2 2 2 3

21 1 1

a b c

b a c b a c

a c b a c b

S dng bt ng thc Cauchy-Schwarz ta c

22 2 2

1 1 1

a b ca b c

b c a a b cb a c b a c

a b c b c aa c b a c b

- 50 -

Ta cn chng minh

2

2 3b c a a b c

a b ca b c b c a

Mt khc ta c:

2

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

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

V theo gi thit a b c

a b cb c a

Nn

23 ( ) 3( )b c a a b c

a b c a b ca b c b c a

Do bi ton s c chng minh nu

2 2

2 3a b c a b c a b c

3 0a b c a b c

Bt ng thc ny hin nhin ng v theo AM-GM v iu kin bi ton ta c:

33 . . 3a b c a b c

a b cb c a b c a

Kt thc chng minh.Du ng thc xy ra khi 1a b c

V d 4 Cho a,b,c l cc s thc dng. Chng minh:

22 2 216 1 1 1 5 1a b c a b c


y l mt bt ng thc khng thun nht v cc bin hon ton c lp vi nhau,

do ta s tm cch nh gi s bin gim i. Tc l nu s dng Cauchy-

- 51 -

Schwarz sao cho 1 trong 3 i lng 2 1b ; 2 1c ; 2( 1)a xut hin th khi

cng vic chng minh bt ng thc ban u s c quy v chng minh mt bt

ng thc ch cn 2 bin, hin nhin vic chng minh s nng nhc v d dng

hn. Vic by gi cn lm l nhm cc s hng 1 cch ph hp.


Do ta cn chng minh:

2 2 25 1 1 16 1 1b c b c

2 2 2 216 10 11 10 6 0b c bc b c b c

S dng Cauchy-Schwarz:

2 2 22 2 2 216 10 11 10 6 4 1 5 1 0b c bc b c b c bc b c b c

Vy ta c iu phi chng minh. Du ng thc xy ra khi

0

4 1 0

1 0

1 1

1

b c

bc

b c

b c

a

1

2a b c

i khi, cc bi ton s dng cch chng minh trn s khng l r nh th m phi

qua mt s bc bin i, ci chng ta cn mi hin ra:

V d 5 Cho , , 0a b c . Chng minh rng:

2 2 25 1 5 1 1 1a b c b c a

22 210 5b c b c

- 52 -

2 2 22 1 2 1 1 1 1 1 1abc a b c a b c


Vic xut hin ca 2 2 22 1 1 1a b c lm cho bi ton kh l rc ri, chnh

v vy, ta s chuyn n sang mt v, sau bnh phng c 2 v ln kh i cn

thc. Da trn tng , bt ng thc c th vit li thnh:

2 2 22 1 1 1 1a b c a b c ab bc ca abc

Nu 1 0a b c ab bc ca abc th bt ng thc hin nhin ng do

2 2 22 1 1 1 2a b c

Nu 1 0a b c ab bc ca abc , bt ng thc cn chng minh tng

ng:

22 2 22 1 1 1 1a b c a b c ab bc ca abc

Cng nh bi trc, ta s s dng Cauchy-Schwarz gim bt s bin i, c th:

2 2 221 1 1 1 1a b c bc b c bc a b c bc b c bc

Do ta cn chng minh

2 22 22 1 1 1 1b c b c bc b c bc

Nhng thc cht y ch l mt hng ng thc.

Kt thc chng minh. Du ng thc xy ra khi 1a b c

Nhn xt: Cch nh gi trn kh l hiu qu trong vic chng minh mt lp cc bt

ng thc m cc bin c lp vi nhau. Th nhng, chng qua ch l mt trng

hp c bit trong th gii bt ng thc. i vi cc dng khc th vic s dng n

hon ton khng kh thi, chng ta hy cng xem xt v d di y:

- 53 -

V d 6 Cho , , 0a b c tha mn 2 2 2a 1b c . Chng minh rng:

2

2 2 2

3

1 1 1 4

a b ca a b b c c

b c a


R rng, trong bi ton ny, chng ta khng th bin i a v mt bt ng

thc m cc bin c lp vi nhau; do ta s i tm con ng chng minh

khc.Nhn thy rng nu ta s s dng Cauchy-Schwarz sao cho biu thc sau khi

nh gi c s xut hin ca 2

a a b b c c th bi ton s khng cn kh

khn na, cng vic chng minh hon ton s nh nhng hn.

S dng bt ng thc Cauchy-Schwarz ta c:

2

3 3 3

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

a a b b c ca b c a b c

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

Do bi ton s c chng minh nu ta ch ra:

2 2 2 2 2 2 2 2 2

1 3

4a b b c c a a b c

2 2 2 2 2 21 3 a b b c c a

2

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

Bt ng thc cui l mt kt qu ca bt ng thc quen thuc:

2

3 xx y z xy yz z

Do ta c iu phi chng minh. Du ng thc xy ra khi 1

3a b c

- 54 -

Nhn xt: Vic s dng bt ng thc Cauchy-Schwarz cho mt v ( thng l v

phc tp hn) sao cho biu thc sau khi nh gi c s xut hin ca mt i lng

lin quan n v cn li th s gip vic chng minh nh nhng v hiu qu hn.

V d 7 Cho , ,a b c l cc s dng. Chng minh rng:

2

1 1 1a b ca b c

b c a a b c

(British Mathematical Olympiad 2005)


Nh ni trn, chng ta s nh gi sao cho v tri xut hin cc i lng

lin quan n v phi, c th l ta s to ra s c mt ca a b c v 1 1 1

a b c . S

dng bt ng thc Cauchy-Schwarz ta c:

2a b c

ab bc ca a b cb c a

(1)

V

21 1 1 1 1 1a b c

b c a ab bc ca a b c

(2)

Nhn v theo v (1) v (2) ta c

22

1 1 1 1 1 1a b cab bc ca a b c

b c a ab bc ca a b c

Ta cn chng minh

1 1 1 1 1 1

a b c ab bc caa b c ab bc ca

Nhng thc cht y ch l mt hng ng thc.

- 55 -

Kt thc chng minh.Du ng thc xy ra khi a b c

V d 8 Cho , ,a b c l cc s thc dng tha mn 1abc . Chng minh rng:

5 2 5 2 5 2

5 2 2 5 2 2 5 2 20

a a b b c c

a b c b c a c a b

(International Mathematic Olympiad 2005)


Do5 2 2 2 2

5 2 2 5 2 21

a a a b c

a b c a b c

nn bt ng thc cn chng minh c th vit li

thnh:

2 2 2 2 2 2 2 2 2

5 2 2 5 2 2 5 2 23

a b c a b c a b c

a b c b c a c a b

5 2 2 5 2 2 5 2 2 2 2 2

1 1 1 3

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

By gi chng ta cn s dng Cauchy-Schwarz sao cho cc phn thc 5 2 2

1

a b c ;

5 2 2

1

b c a ;

5 2 2

1

c a b v chung mt i lng lin quan n v phi , c th l

phn thc c mu l 2 2 2a b c .


2 2 2 2

25 2 22 2 2

5 2 2 2 2

1 1

1

1

b c b ca a

a b c a b ca b c b ca

Cng bt ng thc ny vi cc bt ng thc tng t, ta c

2 2 2

25 2 22 2 2

1 1 12

1b c a

a b c

a b c a b c

- 56 -

Ta cn chng minh

2 2 21 1 1

2 3 b c aa b c

2 2 2

2 2 2

1 1 1a b c

a b c

ab bc caa b c

abc

iu ny ng do

2 2 21ab bc ca

abc ab bc ca a b cabc


V d 9 Cho a,b,c l cc s thc dng tha mn 1abc . Chng minh rng:

2 3 2 3 2 3

1 1 1 11 1 1 a b ca b b c c a

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


Mc d trng c v phc tp v rc ri nhng thc ra bi ny cng chung mt

tng chng minh vi 2 bi va ri


22 31

1a b c a a b cc

22 3

11 1

1a b a

a b c

a b c a b c

Cng cc bt ng thc tng t li, ta cn chng minh

- 57 -

1

1 1 1 1 1 1a b a a b c a b cc

Th nhng y thc ra ch l mt hng ng thc do 1abc

Vy ta c iu phi chng minh. Du ng thc xy ra khi 1a b c

By gi chng ta s i n mt k thut chng minh khc: K thut i bin.

Mt cu hi c t ra l Ti sao li phi i bin? Nu khng i bin th c

c khng. tm kim cu tr li, ta s xem qua v d di y:

V d 10 Cho a,b,c l cc s thc dng. Chng minh rng:

2 2 2

2 2 2 2 2 21

a b c

a ab b b bc c c ca a

Phn tch v nh hng li gii:

Sai lm thng gp khi gp bi ny l vic s dng bt ng thc Cauchy-

Schwarz mt cch thng thng:

22 2 2

2 2 2 2 2 2 2 2 22

a b ca b c

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

s dn n mt kt qu sai do

2

2 2 2

2 2 21 ??

2

a b cab bc ca a b c

a b c ab bc ca

.

R rng l vic s dng bt ng thc Cauchy-Schwarz trc tip nh vy s khng

th p dng trong trng hp ny. Ta s cn mt cht bin i sng to cng

vic chng minh hiu qu hn. rng nu chia

- 58 -

2 2a ab b ; 2 2b bc c ; 2 2c ca a tng ng cho 2a ; 2b ; 2c th s xut hin cc i

lng ; ;b c a

a b c v nu t ; ;

b c ax y z

a b c th bt ng thc cn chng minh by

gi l

2 2 2

1 1 11

1 1 1x x y y z z

v khi ta c 1xyz

Vy l t cc bin c lp hon ton , ta bin i c v cc bin ph c mi

quan h mt thit vi nhau. Vic a c v bin ph nh vy v cng quan trng,

n l mt bc tin ln trong vic nh hng ti tng i bin. V 1xyz nn

ta i bin 2 2 2, , , ,np mp mn

x y zm n p

;trong , , 0m n p

Bt ng thc c vit li thnh :

4 4 4

4 2 2 2 4 2 2 2 4 2 2 21

m n p

m m np n p n mn p m p p mnp m n

(1)


22 2 2

4 2 2(1)

m n pVT

m m n mnp m n p

.

Do , bi ton s c chng minh nu ta ch ra:

2

2 4 2 2m m m n mnp m n p

2 2m n mnp m n p

Bt ng thc cui l mt kt qu quen thuc:

2 2 2 xx y z xy yz z

Vy ta c iu phi chng minh. Du ng thc xy ra khi a b c

- 59 -

R rng l trong nhiu trng hp, vic s dng Cauchy-Schwarz mt cch

thng thng s khng th gii quyt c bi ton, hoc nu gii quyt c th

cng s kh vt v. Lc , cc php i bin lun l mt gii php tt.

Thng thng, vi cc bt ng thc 3 bin a,b,c th cc php i bin thng c

s dng l:

I. 1 1 1

, , , ,a b ca b c

II. , , , ,ka kb kc

a b cb c a

III. , , , ,kb kc ka

a b ca b c

IV. 2 2 2, , , ,kbc kca kab

a b ca b c

V. 2 2 2

, , , ,ka kb kc

a b cbc ca ab

Trong v d 10 ta s dng php i bin (V) vi 1k


2 2 23

4

a b c

a b b c c a


Thng thng, ta s ngh n vic h bc ca biu thc v bn tri bng cch s

dng Cauchy-Schwarz:

2 2 2 2

1 1 1a b c a

a b b c c a a b

v khi cn chng minh

- 60 -

3

2

a b c

a b b c c a

.

Bt ng thc ny c dng "kh ging" vi bt ng thc Nesbitt ni ting nhng

rt tic l tnh ng n ca n li khng c nh th. Trong trng hpa c b

hoc c b a th bt ng thc hon ton b ngc du. n y, ta s ngh n

vic s dng k thut i bin nh v d 10 va ri. t ; ;b c a

x y za b c

khi

1xyz v bt ng thc ban u tng ng:

2 2 2

1 1 1 3

41 1 1x y z

(1)

V 1xyz nn ta i bin 2 2 2, , , ,np mp mn

x y zm n p

;trong , , 0m n p

4

4 2 2 2

3(1)

2 4

m

m m np n p


2

24

4 2 2 2 4 2 22 2

mm

m m np n p m mnp m m n

Ta cn chng minh

2

2 4 2 24 3 2m m mnp m m n

4 2 25 6m m n mnp m n p

Bt ng thc cui ng theo 2 bt ng thc c bn sau:

4 4 4 2 2 2 2 2 2m n p m n n p p m

v

2 2 2 2 2 2m n n p p m mnp m n p

T ta c iu phi chng minh. Du ng thc xy ra khi a b c .

- 61 -


1 1 1 3

1 1 1 1a b b c c a abc


Do bt ng thc trn khng thun nht nn r ngi ln nht ca chng ta chnh l

i lng 1 abc di mu v phi. Do ta s tm cch "kh" i lng ny;

cch hiu qu nht trong trng hp ny l tng i bin x

, , , ,k ky kz

a b cy z y

. trong 0k . Bt ng thc cn chng minh tr thnh:

3

z 3

z z 1

zy x xy k

xy kx yz kxy zx k y k

(1)


22 2

2 2(1)

(1 )

xyz yVT

xy z kxyz k xyz x y z

.

Ta cn chng minh

2

3

3

11

xy k

kk xyz x y z

22 1 3k k xy kxyz x y z

Bt ng thc ny ng do

2

x 3xxy yz z yz x y z

v

- 62 -

22 1 1 0k k k k

Vy ta c iu phi chng minh. Du ng thc xy ra khi x y z v 1k hay

a b c

By gi chng ta s n vi mt k thut khng th thiu khi s dng bt ng thc

Cauchy-Schwarz: K thut thm-bt . tng kh n gin: chng minh mt bt

ng thc c dng , , , ,f a b c g a b c m vic chng minh thng thng c l s

khng em li hiu qu th khi ta s tm 1 biu thc , , , ,h a b c f a b c sao cho

, , , , , , , ,f a b c h a b c g a b c h a b c . V nh gi ny cng cht cng tt .

, ,h a b c trong mt s trng hp c th l cc s, cng c th l cc biu thc lin

quan ti bin. r hn v k thut ny, ta s xem xt v d sau y:

V d 13 Cho , , 0a b c .Chng minh rng:

2 2 22 2 2 2 2 2

22 2 2 2 2 2

6 a b ca b b c c a

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


Khng t bn khi nhn vo bi ton ny s ngh ngay n phng php S.O.S; nhng

bi ny hon ton c th gii c ch vi k thut thm -bt ni trn. Nhn thy v

tri ca bt ng thc c dng phn thc- mt du hiu chng ta ngh n vic s

dng Cauchy-Schwarz dng

22211

1 `

.....

...

nn

n n

a aaa

b b b b

; th nhng chiu ca n li

l ch khng phi nh mong mun. Do ta ngh n vic i chiu bt ng

thc li bng vic thm bt 2 v vi mt s thc 0k . Bt ng thc cho c th

vit li thnh:

2 2 22 2

22 2

63

a b ca bk k

a ab b a b c

- 63 -

22 2

22 2

2 21 13

k a k abk a kab k b

a ab b a b c

m bo vic s dng bt ng thc Cauchy-Schwarz hiu qu v an ton nht,

chng ta phi tm s k sao cho 2 21 1k a kab k b ; 2 21 1k b kbc k c ;

2 21 1k c kca k a u l cc s khng m. chc chn

2 21 1 0k a kab k b vi , 0a b th

22 4 1 0

1 0

k k

k

2k

Trong trng hp ny hng s k tt nht chnh l hng s nh nht.Vy ta s chn

2k . Khi bt ng thc cn chng minh tr thnh

2

22 2

12a b ab bc ca

a ab b a b c


2 2

2 2 2

4

2

a b a b c

a ab b a ab

Ta cn chng minh

2

22

4 12

2

a b c ab bc ca

a ab a b c

Hay

4 23 2a b c ab a ab

Bt ng thc ny ng theo AM-GM:

22

4223 2

3 24 4

a b cab a abab a ab a b c

Vy ta c iu phi chng minh. Du ng thc xy ra khi a b c

- 64 -

V d 14 Cho , , 0a b c tha mn 3a b c .Chng minh rng:

2 2 2 2 2 2 2 2 2

1 1 1 1

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

(Vasile Cirtoaje)


Li gii 1: rng 2 2 2 2

2 2 2 2 2 2

31

4 4

a b c a

a b c a b c

; chnh chi tit ny phn no

khi gy tng trong ta: nhn c 2 v vi 2 2 2a b c . khi bt ng thc cn

chng minh l:

2 2 2 2 2 2

2 2 24 2

a b c a b c

a b c

hay

2 2 2 2

2 2 23 3

4 2

a a b c

a b c


22

2 2 2 2 2 2 2 2 2

3

4 4 2

a b ca

a b c a b c a b c

V theo bt ng thc AM-GM:

2 2 2

2 2 2

93

22

a b c

a b c

T hai bt ng thc trn ta c iu phi chng minh. Du ng thc xy ra khi

1a b c

Li gii 2: Do bc ca 2 v lch nhau nn ta s ngh n vic ng bc bt ng

thc nh sau: Bt ng thc cn chng minh c th vit li thnh:

2 2 2

2 2 2 2 2 2 2 2 2

9

4 4 4 2

a b c a b c a b c

a b c b c a c a b

- 65 -

Theo bt ng thc Cauchy-Schwarz ta c:

2 2 2 2 2

2 2 2 2 2 2 2 22 2 2 2 24 22

a b c a b c a b c

a b c a a b c aa a b a c

Cng cc bt ng thc tng t li vi nhau, ta c

2 2 2 2 2 2

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

1 9

4 2 2 2

a b c a b c b a

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

Vy ta c iu phi chng minh.Du ng thc xy ra khi 1a b c

Bn cnh vic thm bt th li gii 2 ca v d va ri s dng mt k thut khc

song song v khng th thiu trong vic s dng bt ng thc Cauchy-Schwarz : k

thut tch ghp.

V d 15 Cho a,b,c dng . Chng minh rng:

1 1 1

1 22 2 2 2 2 2

a b b c a ca b c

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

(diendantoanhoc.net)


im kh khn ca bi ton ny chnh l s hin din h s t do v tri, trong khi

v phi li l mt biu thc cha bin kh phc tp. tng ca ta by gi l

phi loi b c s 1 ; mt ngh kh t nhin l to ra h s t do v phi.

rng 12 2

a b a

b a b a

. Do bt ng thc cn chng minh c vit li

thnh:

22 2

a b c b

b a b a

22 2

a b c b

b a b a

hay

- 66 -

22 2 2

b c c a a b

b a c b a c

. (1)


2

(1)2

a cVT

b a a c

Ta cn chng minh

2

4 2 2a b c b a a c hay 2 2 2a b c ab bc ca

Bt ng thc cui l mt kt qu quen thuc. Vy ta c iu phi chng minh. Du

ng thc xy ra khi a b c

V d 16 Cho , , 0a b c tha mn 2 2 2 1a b c .Chng minh rng:

1 1 1 9

1 1 1 2ab bc ca

( Vasile Cirtoaje)


Li gii 1:Cng nh nhng bi trc, ta s ngh n vic s dng mt s 0k v

bin i bt ng thc v dng

1 93

1 2k k

ab

Hay

1 3

31 2

k kabk

ab

Ta cn tm s k sao cho 1 0k kab , v nh gi ny cng cht cng tt. Bng bt

ng thc AM-GM, ta c 2 2 2 21= 2 2a b c ab c ab , do ta s chn 2k .

Khi , ta cn chng minh bt ng thc

1 2 1 2 1 2 3

1 1 1 2

ab bc ca

ab bc ca

- 67 -

22 3

1 1 2

a bc

ab ab

Khng mt tnh tng qut gi s a b c .

Khi , s dng bt ng thc Cauchy-Schwarz ta c

22

1 3

a b cc

ab ab bc ca

2 2 24

1 3 3

a b a b b c a c a c

ab ab bc ca ab bc ca

Do ta cn chng minh

2 2

2 4 3 3a c a b c ab bc ca (1)

ng bc bt ng thc:

2 2 2 2 2(1) 2 4 3 3a c a b c a b c ab bc ca

2

7 0b a b a c a c

Bt ng thc cui ng do a b c

Vy ta c iu phi chng minh. Du ng thc xy ra khi 1

3a b c

Li gii 2: Cch gii trn l mt cch gii mang m tnh k thut thm bt, tuy

nhin cch gii trn cha phi l ngn nht .By gi chng ta s tm mt li gii

ngn hn thng qua k thut tch ghp: V tri ca bt ng thc c dng phn thc

v chiu ca bt ng thc l . iu ny gi cho ta ngh n vic s dng bt

ng thc Cauchy-Schwarz dng

2 221 1

` 1

.....

...

n n

n n

a a aa

b b b b

. Nu s dng AM-GM

:

2 2 2 21 1 2

1 1 11

2

a bab a b

- 68 -

ri s dng Cauchy-Schwarz:

2 22 22 1 1 1

2 1 11 1 a ba b

Khi cng vic chng minh s hon tt nu nh ch ra c

2

1 9

1 2a

Tuy nhin chiu ca bt ng thc ny li ngc li do:

2 2 2 21 9 9

1 23a a b c

Vy, tng trn khng cho kt qu nh mong mun; chng ta s li tip tc i

tm mt con ng mi: Mun s dng c bt ng thc

2 22

1 1

` 1

.....

...

n n

n n

a a aa

b b b b

th ta phi c i lng bnh phng t ca phn thc.

V con ng trn khng thnh cng khi i lng trn t l h s t do nn by gi

ta s th i lng bnh phng lin quan ti cc bin. Do 1

11 1

ab

ab ab

nn bt

ng thc cn chng minh c th vit li thnh

3

1 1 1 2

ab bc ca

ab bc ca

22

43

1

ab

c a b

Mt khc ta c 2

4ab a b nn

2

2 2 2 2 22

4

1

a bab

a c b cc a b


2 2 2

2 2 2 22 2 2 2

a b a b

a c b ca c b c

- 69 -

2 2 2

2 2 2 22 2 2 2

b c b c

a b a ca b a c

2 2 2

2 2 2 22 2 2 2

c a c a

c b a bb c b a

2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 22 2 2 23

a b a b b c c a

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

T suy ra iu phi chng minh. Du ng thc xy ra khi 1

3a b c

V d 17 Cho , , 0a b c . Chng minh rng:

2 2 2

1 1 1 a b b c c a

a b c c ab a bc b ac

(Hojoo Lee)


tng nh bi trn, ta s to ra i lng bnh phng t ca phn thc v bn

phi, sau dng k thut tch ghp ph hp. Theo bt ng thc Cauchy-Schwarz

ta c:

2 2 2

2 2 2 2 2 2 2 2 2

a ba b a b

c ab b c a a b c b c a a b c

Tng t ta c

2 2

2 2 2 2 2

b c b c

a bc c a b c c a

2 2

2 2 2 2 2

c a c a

b ac a b c c a b

- 70 -

2 2

2 2 2 2 2

1 1 1a b a c

c ab a b cb c a c c a

Kt thc chng minh. Du ng thc xy ra khi a b c

Trong mt s trng hp, ngoi pht hin ra du hiu tch ghp, chng ta cn

phi tinh t v kho lo s dng k thut ny sao cho vic chng minh tr nn n

gin nht. Chng hn nh v d di y:

V d 18 Cho , , 0a b c tha mn 3a b c Chng minh rng:

3

2

a b c

a bc b ca c ab


Do 1a bc

a bc a bc

nn bt ng thc cn chng minh c th vit li thnh:

3

2

bc ca ab

a bc b ca c ab

1

3 3 3 2

bc ca ab

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

21

2 2

bc

a bc ab ac bc

Theo bt ng thc Cauchy-Schwarz ta c:

2 21

2 4 2

bc bc bc

a bc ab ac bc ab bc ca a bc

2 21

2 4 2

bc bc bc


Do 1bc

ab bc ca

nn ta cn chng minh

2 2 21

2 2 2

bc ca ab

a bc b ca c ab

- 71 -

2 2 2

2 2 22

2 2 2

bc ca ab

a bc b ca c ab

2 2 2

2 2 21

2 2 2

a b c

a bc b ca c ab


2 22 2 2

22 2 2 2 2 21

2 2 2 2 2 2

a b c a b ca b c

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


Nhn xt :im mu cht ca li gii trn chnh l vic tch kh k thut:

2 21

2 4 2

bc bc bc


.

Bi v nu tch thng thng: 2 2

13

2 36

bc bc b c

a bc ab ac bc a a a

s lm

cho vic chng minh i vo ng ct.

V d 19 Cho , , 0a b c tha mn 2a b c . Chng minh rng:

4

2 2 24 4 4

2 3

33 4 3 4 3 4

bc ca ab

a b c

(Internatinonal Mathematical Archimede Olympiad 2010)


B : Vi a, , 0b c tha mn 2a b c , ta c:

1

2 2 2 2

bc ca ab

a b c

Chng minh:

Bt ng thc cn chng minh tng ng

- 72 -

1

2

bc ca ab

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


1

4

bc bc bc

a b a c a b c a

1

4

ca ca ca

b a c b a b b c

1

4

ab ab ab

c b a c b c c a

1 1 1

4 4 4 2

bc bc bc bc ca a b c

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

B c chng minh. Du ng thc xy ra khi 2

3a b c

Tr li bi ton:

cho n gin, ta vit bt ng thc cn chng minh li thnh:

2 2 24 4 4

2

39 12 9 12 9 12

bc ca ab

a b c

Tr ngi ln nht khi ng trc bi ton ny chnh l s hin din ca 4 mu.

Do , ngh n gin l phi lm sao loi b n i, hoc l tng bc ca n ln.

D on c du ng thc ca bi ton xy ra khi 2

3a b c . Mt khc ta thy

2

2

29

9 13

12 12 3

a

nn ta d on rng, nu bin i hp l th bt ng thc c th

a c v dng 1

2 2 2 2

bc ca ab

a b c

. Da trn tng ta s gii quyt bi

ton nh sau:

t

- 73 -

2 2 24 4 49 12 9 12 9 12

bc ca abA

a b c

4 2 2 24 4 44 4 9 12 4 9 12 4 9 12

A bc ca ab

a b c

.

Khi , theo bt ng thc Cauchy-Schwarz:

229 12 1 3 3 6a a

4 4 3 6 3 2

A bc bc

a a

Tip tc s dng bt ng thc Cauchy-Schwarz ta c

2

3 2 2 23 2

bc bc ca ab bc ca ab

a b ca

p dng b v bt ng thc c bn 2

3 4ab bc ca a b c , ta c

2

2

93 2

bc

a

2

4

2

94

A

2

3A

Hay

4

2 2 24 4 4

2 3

33 4 3 4 3 4

bc ca ab

a b c

y chnh l iu cn phi chng minh. Du ng thc xy ra khi 2

3a b c

- 74 -

Trong th gii bt ng thc, nu , ta s thy rng cc bt ng thc kinh in

nh AM-GM, Cauchy-Schwarz, Chebyshev, Holder, Minkowxki,.. hay l cc bt

ng thc hin i nh Muirhead, S.O.S,... khi s dng chng minh cc bt ng

thc i xng em li hiu qu rt cao th nhng khi trm chn vi cc bt ng

thc dng hon v vng quanh th cha hn nh vy. Do , trong nhiu bi ton,

chng ta thng ngh n vic a bt ng thc cn chng minh v dng i xng.

Trong

nhng phng php s dng bt ng thc Cauchy-Schwarz, c mt phng php

rt hiu qu, l :k thut i xng ha.

m u k thut ny, ta s n vi mt bi ton kh ni ting sau:

V d 20 Cho , , 0a b c . Chng minh rng :

3

2

a b c

a b b c c a

( Vasile Cirtoaje)


y l mt bt ng thc dng hon v vng quanh, do vic chng minh s kh

kh khn. Nh ni trn, ta s s dng bt ng thc Cauchy-Schwarz a

bt ng thc ny v dng i xng. C th:

Bt ng thc cn chng minh tng ng:

2 2 23

a b c

a b b c c a


2

42 a b c ab bc caa aa c

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

Do ta cn chng minh:

- 75 -

8 9a b c ab bc ca a b b c c a

9a b c ab bc ac abc

y l mt kt qu quen thuc. Bi ton c chng minh. Du ng thc xy ra

khi a b c

V du 21 Cho , , 0a b c .Chng minh rng:

3

22 2 2

a b ca b c

a b b c c a

(Phm Kim Hng)



2

22 22

a aa a c

a b a ca b

Do ta cn chng minh:

2

3

2 2 2

a a b c a b c

a b a c

Ta c

2 2 2

2 2 2 2 2

a a b c a b c a b ca a

a b a c a b a c ab bc ca

nn

2 2

2 2 2

a a b c a b ca b c

a b a c ab bc ca

M

2 9 3

2 2 2 2

a ab abc a aba b ca b c a a a

ab bc ca ab ab

- 76 -

T suy ra iu phi chng minh. Du ng thc khng xy ra

Sau khi tri qua nhng bi ton, v d c th trn ta c th thy c th gii bt

ng thc tht phong ph, k o, cc phng php cng nh k thut cng rt nhiu,

nhng v d va qua ch l nhng v d ca cc phng php ni bt v quan trng

m ta cn bit. kt thc bi vit ny, chng ta hy n vi mt bi ton ni ting

v rt kh sau y:

V d 22 Cho cc s thc khng m a,b,c tha mn 0ab bc ac .Chng minh

rng:

5

4

a b ca b c

a b b c c a

( Jack Garfunket)


y l mt bt ng thc rt kh v c nhiu ng dng ca Jack Garfunket, hin c

3 li gii cho bi ton ny. Ngoi li gii ca tc gi, xin trch dn li gii ca V

Quc B Cn-SV lp YY0647A1, Kha 32, H Y Dc Cn Th:

Bt ng thc ny c ng thc xy ra ti 3 ; 0a b c v cc hon v tng ng.

Nhn thy ci kh ca bt ng thc ny chnh l ch n c cha cn thc, v vy

ra s tm cch dng Cauchy-Schwarz loi b cn thc.

T bi ton s tr nn n gin hn. Hn na ta cng phi thm vo cc tham s

thch hp. Th nhng, nh va ch ra trn, ta khng th thm vo cc tham s c

nh khi nh gi bt ng thc ny. V vy ta s dng cc tham s chy. C th l

ta s thm vo cc i lng c dng ; ;ma nb pc mb nc pa mc na pb vi

, , 0m n p .

Ta p dng bt ng thc Cauchy-Schwarz nh sau:

2 aVT a ma nb pca b ma nb pc

(*)

- 77 -

nh gi ny xy ra du bng khi

a b c

a b ma nb pc b c mb nc pa c a mc na pb

a ma nb pc b mb nc pa c mc na pb

Mt khc, do ta d on c 0c nn ta cn tm m,n,p tha mn

a b

a b ma nb pc b c mb nc pa

a ma nb pc b mb nc pa

v 3a b (1)

Mt khc, ta

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