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TRANSCRIPT
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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
1.3 M rng 2 ca bt ng thc Holder ..............................................................30
1.4 M rng 3 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
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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
ng thc xy ra khi v ch khi 1 2 ... na a a .
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
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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
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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
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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
ng thc xy ra khi v ch khi 1 2 ... 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
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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
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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
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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
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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.
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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
Theo bt ng thc AM-GM ta c:
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 :
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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.
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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
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Bt ng thc c chng minh.
ng thc xy ra khi v ch khi a b c .
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.
ng thc xy ra khi v ch khi 1a b c .
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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
Theo bt ng thc AM-GM ta c:
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
Bi 6. Cho cc s thc dng , ,a b c . Chng minh:
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
Bi 8. Cho cc s thc dng , ,a b c . Chng minh:
3
3 . .3 2 3
a ab abc a b a b ca
Bi 9. Cho cc s thc dng , ,a b c . Chng minh:
3 31 1 1 3
1 1 1 1a b b c c a abc abc
-
- 24 -
BT NG THC MINKOWSKI V NG DNG
on Quc t Ng Hong Thanh Quang
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
ng thc xy ra khi v ch khi 1 2
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
Theo bt ng thc AM-GM ta c:
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)
ng thc xy ra khi v ch khi 1 2
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,
ng thc xy ra khi v ch khi a b c .
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
Theo bt ng thc AM-GM ta c:
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
ng thc xy ra khi v ch khi 1 21
... 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 -
ng thc xy ra khi v ch khi 3a b c .
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.
ng thc xy ra khi v ch khi a b c .
3. Bi tp t gii
Bi 1. Cho cc s thc dng , ,a b c . Chng minh:
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
on Quc t Ng Hong Thanh Quang
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
Phn tch v nh hng li gii
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)
Phn tch v nh hng li gii
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
Cho , , 0a b c .Chng minh rng:
2 2 2 2 2 2
3
23 3 3
a b c
b c c a a b
Phn tch v nh hng li gii
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
Phn tch v nh hng li gii
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 -
Theo bt ng thc AM-GM ta c:
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.
Cho , , 0a b c .Chng minh rng:
35 2 5 2 5 23 3 3a a b b c c a b c
(USAMO 2004)
Phn tch v nh hng li gii
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
Phn tch v nh hng li gii
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 -
Phn tch v nh hng li gii
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)
Cho , , 0a b c .Chng minh rng:
2 2 2
2 2 2 2 2 21
7 7 7
a b c
a ab b b bc c c ca a
Cho , , 0a b c .Chng minh rng:
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
b b vi quy c l nu 0kb th a 0k
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
b b vi quy c l nu 0kb th a 0k
2. V d
By gi chng ta s khng cp n nhng bi ton kh c bn nh:
Cho , , 0a b c .Chng minh rng:
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)
Phn tch v nh hng li gii
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)
Phn tch v nh hng li gii
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)
Phn tch v nh hng li gii
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
Phn tch v nh hng li gii
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.
S dng bt ng thc Cauchy-Schwarz:
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
Phn tch v nh hng li gii
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
Phn tch v nh hng li gii
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)
Phn tch v nh hng li gii
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)
Phn tch v nh hng li gii
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 .
S dng bt ng thc Cauchy-Schwarz ta 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
Kt thc chng minh. Du ng thc xy ra khi 1a b c
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
Phn tch v nh hng li gii
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
S dng bt ng thc Cauchy-Schwarz ta c:
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)
S dng bt ng thc Cauchy-Schwarz ta c
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
V d 11 Cho a,b,c l cc s thc dng. Chng minh rng:
2 2 23
4
a b c
a b b c c a
Phn tch v nh hng li gii:
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
S dng bt ng thc Cauchy-Schwarz:
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 -
V d 12 Cho a,b,c l cc s thc dng. Chng minh rng:
1 1 1 3
1 1 1 1a b b c c a abc
Phn tch v nh hng li gii
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)
S dng bt ng thc Cauchy-Schwarz:
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
Phn tch v nh hng li gii:
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
S dng bt ng thc Cauchy-Schwarz ta 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)
Phn tch v nh hng li gii
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
S dng bt ng thc Cauchy-Schwarz ta 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)
Phn tch v nh hng li gii:
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)
S dng bt ng thc Cauchy-Schwarz:
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)
Phn tch v nh hng li gii
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
S dng bt ng thc Cauchy-Schwarz ta c
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)
Phn tch v nh hng li gii
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
Phn tch v nh hng li gii
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
a bc ab ac bc ab bc ca a 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
S dng bt ng thc Cauchy-Schwarz ta c:
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
Kt thc chng minh. Du ng thc xy ra khi 1a 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
a bc ab ac bc ab bc ca a 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)
Phn tch v nh hng li gii
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
S dng bt ng thc Cauchy-Schwarz ta 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)
Phn tch v nh hng li gii
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
S dng bt ng thc Cauchy-Schwarz ta c:
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)
Phn tch v nh hng li gii
S dng bt ng thc Cauchy-Schwarz ta c:
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)
Phn tch v nh hng li gii
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