chinh ph c b t ng th c trong thi i h c

CHINH PHC BT NG THC TRONG THI QUC GIA THPT - Pht hnh: 26/10

Nh sch gio dc LOVEBOOK 101 Nguyn Ngc Ni, Thanh Xun, H Ni 046.6860.849

Gi cc em v cc qu thy c trch on n phm CHINH

PHC BT NG THC TRONG THI QUC GIA THPT



Ta xt qua mt s v d nng cao hn thy c s ng dng ca bt ng thc CAUCHY-SWARCH.

V d 5: Chng minh rng nu l cc s thc tha mn iu kin .Th bt ng

thc sau y l ng:

Li gii

Ta s gii bi ton ny bng bt ng thc CAUCHY-SWARCH, nhng l do v sao li ngh n ?

iu ta cn l nh gi tng thng qua tng

Nh vy, dng ca CAUCHY-SWARCH xut hin, vn l ta chn nh gi nh th no m thi.

Ta th chn nh gi nh sau:

iu ny khng mang li g cho chng ta c, vy hy tm cch kt hp cc b s khc.

th du ng thc y xy ra khi .Nh vy c mt quan h kh r rng l

.Nh th th xem l mt s hng xem sao. Ta s nh gi:

Nh vy, ta cn ch ra rng:

iu ny ng v ta c:

V d 6:Chng minh rng nu thuc on v tha mn iu kin . Th

ta c:

Li gii

Ta p dng CAUCHY-SWARCH mt cch th s nht:

By gi xt xem c nh hn 3 khng ? Cha hn ng khng ? iu ch xy ra khi

.Vy by gi ta xt .

Ta c th gi s . Lc ny ta c .Lc ny ta c bin nh tch khi ci chung. T

t duy l ta s nh gi 2 s hng u tin.

By ta cn chng minh:

By gi ta rt

Ta quy v vic chng minh bt ng thc sau:

p dng CAUCHY-SWARCH, ta c:

x, y, z2 2 2x y z 2

x y z 2 xyz

x(1 yz) y z 2 2 2x y z

2 2 2 2 2(x(1 yz) y z) (x y z )((1 yz) 1 1)

x y 1;z 0

x y z y z

2 2 2 2(x(1 yz) y z) (x (y z) )(1 (1 yz) )

2 22(1 yz)(2 2yz y z ) 4 3 3 2 2y z y z

2 2y z 2 yz 1

x, y, z 1;1 x y z xyz 0

x 1 y 1 z 1 3

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

3(x y z 3)

x y z 0 x y z 0

z 0 x, y (0;1] z

x 1 y 1 z 1 2x 2y 4 z 1

2x 2y 4 z 1 3

2(x y) z

2 2x 2y 4 1 1 z

2xy 2(1 xy) 1 z 2x 2y 4

(1 x)(1 y)z 1

1 xy

x yxy (1 x)(1 y)(1 xy) 1

2

2 2

xy (1 x)(1 y)(1 xy)

x yx xy 1 x 1 xy y xy 1 xy y 1 1

2



V d 7: Chng minh rng vi 3 s dng , ta c bt ng thc sau:

Li gii

Mt iu d dng nhn thy y l bc ca t s bng bc ca mu s. V th ta s chia xung a

v mt biu thc mi nh sau:

cho gn ta t n ph

Ta cn chng minh:

iu ny tng ng vi:

Chng ta chng minh c:

Nh vy bt ng thc cn chng minh tng ng vi:

y l bt ng thc ng v

V d 8:Vi mi s dng a, b, c dng ta c:

Li gii

Ta c:

Theo BT Svacx ta c:

Ta c

p dng BT Bunhiacpxki ta c:

Do :

Ta cn chng minh:

Bt ng thc cui cng ng theo BT AM-GM. T suy ra pcm.

a;b;c

2 2 2

2 2 2 2 2 2

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

a (b c) b (c a) c (a b) 5

2 22

2 22 2

b c b c( 1) ( 3)

(b c a) a ab c b ca (b c)

1 ( ) 3 ( )a a

b c c a a bx; y; z

a b c

2 2 2 23(x y z 3) (x y z 3)52(x y z) 15(x y z) 3(xy yz zx) 18 0

xy yz zx 2(x y z)

2(x y z) 9(x y z) 18 0

x y z 6

2 2 2

a b c1

a 8bc b 8ac c 8ab

2 2 2

2 2 2 2 2 2

a b c a b c1 1

a 8bc b 8ac c 8ab a a 8bc b b 8ac c c 8ab

22 2 2

2 2 2 2 2 2

a b ca b c

a a 8bc b b 8ac c c 8ab a a 8bc b b 8ac c c 8ab

2 2 2 3 3 3a a 8bc b b 8ac c c 8ab a. a 8abc b. b 8abc c. c 8abc

3 3 3 3 3 3a. a 8abc b. b 8abc c. c 8abc a b c a b c 24abc

22 2 2

2 2 2 2 2 2

a b ca b c

a a 8bc b b 8ac c c 8ab a a 8bc b b 8ac c c 8ab

33 3 3 2 2 2 2 2 2a b c 24abc a b c a b b c c a ab bc ca 6abc



V d 9: Cho l cc s thc dng c tng l 3. Chng minh rng:

Li gii

u tin, ta tch

Lc ny s dng bt ng thc CAUCHY-SWARCH, ta c:

Nh vy, ta thao tc tng t vi 2 s hng cn li ri cng li, ch rng:

Nh vy, ta c c iu phi chng minh

Qua nhng v d trn, ta thy bt ng thc CAUCHY-SWARCH c nhng ng dng rng ri kh ln.

Cng xin lu bn c rng bt ng thc ny thc s gii quyt c rt nhiu dng ton kh v cht.

Trong gii hn thi i hc th bt ng thc CAUCHY-SWARCH kt hp vi o hm l mt cng c

thc s mnh. By gi chng ta s lm r ng dng ca bt ng thc CAUCHY-SWARCH dng cng

mu s.

Trc ht, ta xin nhc li bt ng thc CAUCHY-SWARCH dng cng mu s,hay cn c tn gi l

Dng Engel:

Vi hai dy s vi ta c:

By gi, ta s i vo nhng v d c th c th ha cho phng php ny. Hy c tht k, tht k

nhng iu sp vit, hy vng rng bn c s c mt ci nhn mi v cc bi ton bt ng thc. Quan

im ca chng ti l: Nhng cng c n gin nht l cng c mnh nht

V d 1: (A-2013): Cho cc s thc dng tha mn .Tm gi tr nh nht ca

biu thc

Li gii

Ta thy vai tr ca trong biu thc l ging nhau nn c th d on rng GTNN t ti

,thay vo iu kin ta c . Nh vy ta s on c

Quan st tip ta thy iu kin cho dng ng bc nn ta s t duy ngay c cch t quen thuc sau:

.

Ta i vo gii bi ton.

t vi , th ta c .

p dng bt ng thc vo iu kin ta c:

V

a,b,c

2 2 2 2 2 2 2 2 2

1 1 1 1

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

2 2 2 2 2 2 2 24a b c 2a (a b ) (a c )

2 2 2 2

2 2 2 2 2 2 2 2

(a b c) a b c

4a b c 2a a b a c

2 2

2 2 2 2

b c3

a b c a

1 2 n 1 2 na ,a ,...,a , b ,b ,..., b ib 0, i 1,n

22 2 21 2 n1 2 n

1 2 n 1 2 n

a a ... aa a a...

b b b b b ... b

a,b,c 2(a c)(b c) 4c

3 3 2 2

3 3

32a 32b a bP

(b 3c) (a 3c) c

a;b a b

a b c P min 1 2

a xc;b yc

a xc;b yc x, y 0 (x 1)(y 1) 4

3 32 2

3 3

32x 32yP x y

(y 3) (x 3)

2(u v) 4uv 216 (x y 2) x y 2

2 216 (x 1) (y 1) 4x.4y 16xy xy 1



p dng bt ng thc CAUCHY-SWARCH dng cng mu s ta c:

Mt khc, ta c:

Do

:

Php chng minh c hon tt.

V d 2: Cho l cc s thc dng Chng minh rng:

Li gii

y l mt bt ng thc dng phn thc, t duy ngh ngay n l p dng bt ng thc CAUCHY-

SWARCH dng cng mu s. lm c iu , trc ht ta lm chn bc ca t s.

Theo bt ng thc CAUCHY-SWARCH, ta c:

Ta cn x l mu s. rng ta c ng thc:

Nh vy ta chng minh c:

ng nhin, ta s c:

Vy ta c iu phi chng minh.

Qua 2 v d trn, phn no khng nh c sc mnh ca bt ng thc ny. Qua 2 v d ny, ta s

nhn mt cch tng qut hn v cch p dng phng php ny.

+ C dng mu s i xng

+Lm chn bc t s, nu nh bc l ta s lm tng ln mt bc ri nh gi. ng nhin s c trng

hp bt ng thc ngc du, ng lo lng, hy mnh dn lm tng bc cao hn na. Lc nh gi

ca chng ta s cht hn

+ Gii quyt bt ng thc i xng cn li .

V d 3: Cho cc s thc dng .Chng minh rng:

Li gii

3 3 4 4 2 2 2

3 3 3 3 3 3

x y x y (x y )

(y 3) (x 3) x(y 3) y(x 3) x(y 3) y(x 3)

3 3 2 2

2 2 2 2 2

x(y 3) y(x 3) xy(x y ) 9xy(x y) 54xy 27(x y)

(x y ) 36(x y) 27(x y ) 64(x y)

2 2 22 2x y tP x y t; t 2

2 2

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

2 2

a;b;c

3 3 3

2 2 2 2 2 2

a b c a b c

a ab b b bc c c ca a 3

3 3 3 4 4 4

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

a b c a b c

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

4 4 4 2 2 2 2

2 2 2 2 2 2 3 3 3

a b c (a b c )

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

3 3 3a b c ab(a b) bc(b c) ca(c a) (a b c)(ab bc ca)

3 3 3 2 2 2

2 2 2 2 2 2

a b c a b c

a ab b b bc c c ca a a b c2 2 2 2(a b c) 3(a b c )

a, b,c,d

a b c d 2

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



Trc ht, ta lm chn bc ca t s:

p dng tng t v d 2, ta c:

By gi, ta cn chng minh:

Khai trin ra, th ta cn chng minh:

May mn l iu ny hin nhin ng. Ta kt thc chng minh ti y

V d tip theo l bi ton s 4 k thi Olympiad ton quc t nm 1995. Nm m on Vit Nam t kt

qu rt cao ti Hong Kong. Bi ton ny c xut hin trong rt nhiu ti liu, v ng nhin, n

cng s c gii quyt bng bt ng thc CAUCHY-SWARCH.

V d 4: Cho l cc s thc dng tha mn . Chng minh rng:

Li gii

u tin, ta s vit li bt ng thc di dng d nhn hn:

Nh vy ta c iu phi chng minh.

Bnh lun: Ngoi ta da vo iu kin tch ta c th ngh ngay n vic i bin

. Cc bn t gii quyt theo hng ny xem nh l mt bi tp

V d tip theo c xut bi Peter Scholze ( ngi 3 ln t HCV ton quc t, trong c nm

2007 IMO c t chc Vit Nam )

V d 5: Cho l cc s thc dng.Chng minh rng:

Li gii

Cng tng t nh cc v d trn, p dng bt ng thc CAUCHY-SWARCH ta c:

Rt tic, chng ta li b ngc chiu. Nh ni, by gi chng ta s lm cht thm nh gi ca mnh

bng cch nng bc ln.

V tri c vit li :

2 2 2 2a b c d

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

2 2 2 2a b c d

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

2(a b c d)

4(ab bc cd da ac bd)

23(a b c d) 8(ab bc cd da ac bd)

2(a b c d) 4(ab bc cd da)

a,b,c abc 1

3 3 3

1 1 1 3

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

2

2 2 2

1 1 11 1 1

ab bc ca 3a b ca b c

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

abc 1

yz zx xya ;b ;c

x y z

a;b;c

2 2 2

2 2 2

(a b) (b c) (c a)6

c ab a bc b ac

2

2 2 2

4 a b cVT 6

a b c ab bc ca



p dng bt ng thc CAUCHY-SWARCH, ta a v chng minh bt ng thc:

Khai trin, ta c:

Ta c nh gi sau:

Vy ta cn chng minh:

Vy ta cn chng minh:

y l mt bt ng thc quen thuc cp phn trc.

V d 6: Cho l cc s thc dng tha mn .Chng minh rng:

Li gii

t

Khi , ta vit li P di dng:

p dng bt ng thc CAUCHY-SWARCH, ta c:

Vy bt ng thc c chng minh.

V d 7: Cho l cc s thc dng tha mn v tha mn

Chng minh rng:

Li gii

t tng t vi .Nh vy ta c .

Ta nhn thy lun tn ti 2 s cng ln hoc cng b hn 1. Gi s l .Vy ta c:


4

2 2

a bVT

a b c ab

22 2 2a b 6 a b c ab

4 4 4 2 2 2 2 2 2 2 2 2 2 2 22 a b c ab a b bc b c ca c a 2abc a b c 6 a b b c c a

2 2 2 2bc(b c ) 2b c

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

2 2 2 2 2 2 4 4 42(a b b c c a ) (a b c ) (a b c)(a b c)(b c a)(c a b)

abc (a b c)(b c a)(c a b)

a,b,c abc 1

a b cP 1

2 b a 2 c b 2 a c

x y za ; b ; c

y z x

22 2

2 2 2 2 2 2

xy(xz) (yz)P

(xy) 2xyz (yz) 2x yz (zx) 2y zx

2 2 2 2

2 2 2 2 2 22 2 2

xz xy yz xy yz zxP 1

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

x, y, z x, y, z (0;1)

xyz (1 x)(1 y)(1 z)

2 4 2 4 2 4x y y z z x 15

y z x 8

1 xa

xb,c abc 1

a, b

a 1 b 1 01 c

a b 1 abc



Nh vy ta c:

By gi, li s dng bt ng thc CAUCHY-SWARCH, ta c:

Mt khc, ta cng c:

Vy nn:

Vy

Vy ta c iu phi chng minh, php chng minh c hon tt.

V d 8: Tm gi tr nh nht ca biu thc trong l cc s

thc dng tha mn .

Li gii

p dng Bt ng thc CAUCHY-SWARCH, ta c:

Du ng thc xy ra khi:

V d 9: Cho cc s thc tha mn .Chng minh rng:

Li gii

V nn tn ti sao cho

Khi , bt ng thc c vit li di dng:

2 2

1 1 1 1 1 c

a b 1 ab c 11 a 1 b 1 ab 1 1 ab 1b a

2

2 2 2

2 2 2 2 2

c 11 1 1 c 1 3 3x y z

1 c 4 41 a 1 b 1 c 1 c 4 1 c

2 22 2 2 2 2 22 2 2

2 2 22 2 2 2 2 2 2 2 2

x y z x y zx y z

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

22 2 2

2 2 2

2 2 2

2 2 2

x y z 33 x y z

2x y zx y z

3

3 3 3 2 2 2 2

2 2 2 2

(x y z )(x y z) (x y z )

3(x y z ) (x y z)

3

3 3 3 2 2 2 2 3 33(x y z ) (x y z )4

3 3 3 3x y z8

1 1 1P

2 4a 3 9b 6 36ca,b,c

a b c 1

9 4 1 36 1P

2 36a 3 36b 6 36c 72 2

3 2 1 1

2 36a 3 36b 6 36c 12

1 1 1a ;b ;c

2 3 6

x, y, z 1 xyz 1

22 2x y z

1x 1 y 1 z 1

xyz 1 a,b,c2 2 2

bc ca abx ; y ;z

a b c




rng: nn ta c pcm.

Mt iu kh c bit bi ton ny l ng thc xy ra ti v s im.

Ta c th gii bi ton ny theo mt hng khc nh sau:

t .

Bt ng thc cho tr thnh:

p dng CAUCHY-SWARCH, ta c:

Ta ch l vi

Ta c:

Li . Vy ta c iu phi chng minh.

Ta th kho st mt vi v d v cc bt ng thc khng i xng xem phng php ny c sc cng

ph khng ? a phn, cc bi ton ny l tng ca nhiu s hng v c 2 s hng l i xng nhan v s

hng kia khng i xng, dn bin, ta phi nh gi 2 s hng i xng bng bt ng thc

CAUCHY-SWARCH. Ta hy theo di.

V d 10: Cho cc s thc khng m tha mn .Tm gi tr nh nht ca biu

thc:

.

Li gii

Trc ht, ta on du ng thc xy ra khi no, da vo iu kin ta s on nhng b s nguyn dng

v tha mn . D on c .

Thay ci im ri ny vo th ta thy ngay iu c bit , 2 biu

thc ny bng nhau. Li mt php tnh na l bc 3 cng bc 1 em chia cho 2 n s c bc 2 ( ng

bng bc ca gi thit ).Nh vy tng s dng AM-GM kh l thin.

Theo bt ng thc AM-GM, ta c:

By gi, ta cn chng minh:

tng s dng CAUCHY-SWARCH kh r rng, ta c:

Vy , ta c ngay iu phi chng minh.

Bnh lun: Cc bn hy th xt hiu: xem thu c

iu g nh ! l mt hng ng thc bnh phng rt p mt.

4 4 4

2 2 2 2 2 2

a b c1

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

2 2 2 2

2 2 2 2 2 2

(a b c )VT

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

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

a b cx ; y ;z

b c a2

a1

a b

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

xy yz zx 0 x (a b)(a c); y (b c)(b a);z (c a)(c b)

2 2 2 2 2 2(a b) (a c) x y z (x y z)

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

a,b,c ab 2bc 3ca 6

(a b)(b c)(c a) 4a b c

ab 2bc 3ca 6 (a,b,c) (1,0,2)

(a b)(b c)(c a) 4a b c 6

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

2(a b)(b c)(c a)(4a b c) 36 (ab 2bc 3ca)

2 2 2 2M a(b c) b(c a) c(a b) (4a b c) (2a(b c) b(c a) c(a b))

2M (ab 2bc 3ca)

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



V d 11: Vi cc s thc dng tha mn iu kin

Tm gi tr nh nht ca biu thc:

Li gii

Hnh thc ca bi ton khin ta ngh ngay n phng php hm s . Vn l ta s xc nh dn v bin

no ? Nhn thy c s i xng ca 2 bin nn ta khng th dn theo 2 bin ny c. Vy ta s

dn . By gi bin ta s khng ng chm g n, gi nguyn. Ta s c a 2 bin cn li v .

iu kin, vic a v ng ngha vi vic rt ra cc biu thc dng hoc cc biu thc c th

nh gi c vi .

s hng cui c cu hnh nn ta c ngay nh gi theo AM-GM:

By gi ta kh khn vic x l 2 s hng u tin, c mt phng php rt mnh x l biu thc

phn thc, l p dng bt ng thc CAUCHY-SWARCH:

Vy suy ra:

Kho st hm s trn ta c t ti

V d 12: Cho cc s thc dng a,b,c. Tm gi tr nh nht ca biu thc

Li gii

Nhn thy rng s xut hin c bit ca xut hin 2 ln nn s rt OK nu nh chng ta dn v

bin . Hai s hng u tin l i xng theo nn ta cn nh gi chng a v

. Ta s p dng CAUCHY-SWARCH:

Ta c:

x, y, z x y 1 z

3 3 3x y z 14P

x yz y zx z xy (z 1) (x 1)(y 1)

x, y

z z z

z (x y)

(x y)

ab

x y 2(x 1)(y 1)

22 2(x y 2) (z 1)

z xy x y 1 xy (x 1)(y 1)4 4

3 3 4 4

2 2

2 2 2

2 2

2 2 2 2 2

2 2 2 2

2 2

x y x y

x yz y zx x xyz y xyz

(x y )

x y 2xyz

(x y ) x y

x y (x y )z z 1

(x y) (z 1)

2(z 1) 2(z 1)

2 3

2 2

(z 1) 4z 28P

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

z 053

MinP8

1 5x y , z .

3 3

2a b c c

Pb c c a a b a b

c

a b

ct

a ba, b

ct

a b



Vy nu t th ta c: .

n y, ta c th kho st hm tm gi tr nh nht.

Bi ton tng t: Cho cc s thc dng . Tm gi tr nh nht ca

V d 13:[Poland 1991] Cho x,y,z l cc s thc tha mn: . Chng minh:

Li gii

BT

u tin vi u bng ti ta tch mi s hng trong v tri ra 2 nhn t du bng xy ra ng

thi tn dng gi thit thng qua BT Cauchy Schwarz v a v mt n. Khi thng hiu dng ny

cc bn s thy khng thng th khng phi quan tm du bng:

Ta c:

.

t Ta chng minh:

Ta c: .


V d 14: Cho x,y,z l cc s thc tha mn: . Chng minh:

Li gii

Ta c:

.

t :Ta chng minh:

D c iu v: .

Vy ta c iu phi chng minh

ng thc xy ra khi v ch khi v cc hon v ca n.

2 2 2 2

2

2 2

22

2

a b a b (a b) 2(a b)

b c c a ab ac bc ba 2ab c(a b) 4ab 2c(a b)

(a b)2

2(a b) c

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

c c

a bt

c

2

2 2

2t 1 1P

t 2t t t

a,b,ca b c

Pb c c a a b

2 2 2x y z 2

x y z xyz 2

x y z xyz 2

0;1;1

2 2 22 2VT x. 1 yz y z .1 x y z 1 yz 1

22 2 2 2 2x y z 2yz 1 yz 1 2 2yz y z 2yz 2

yz t 2 3 22 2t t 2t 2 4 t t 0 t 1

2 2 2 2 22 x y z x 2yz x 2t 2t 1 t

2 2 2x y z 8

4 x y z xyz 16

2 2 22 2VT x. 4 yz y z .4 x y z 2 yz 16

22 2 2 2 2x y z 2yz 4 yz 16 8 2yz y z 8yz 32

yz t 22t 8 t 8t 32 256 t 4

2 2 2 2 28 x y z y z 2yz 2t

x, y,z 0;2;2



V d 15:[JBMO 2002 Shortlish]Cho a,b,c l cc s thc dng. Chng minh rng:

Li gii

p dng BT Cauchy ta c:

M . Suy ra pcm!

Cch 1:

p dng BT Cauchy Schwarz ta c:

M

Ta c: ng!

Vy ta c pcm! ng thc xy ra khi v ch khi

Tuy nhin, bi ton ny li c mt cch gii khc, chnh l nh gi i din hay rng hn l phng

php tip tuyn. Nhng dng ny ta ch cn mt cht k thut trong phng php nh gi i din nh

sau:

Cch 2: Ta s tm x,y sao cho BDDT sau lun ng:

BT

Li c BT lun ng nn ta s chn x,y, sao cho:

th BT trn lun ng!

Bi gii: Ta c: . Tht vy:

BT (lun ng)

Tng t ta c:

Cng v theo v cc BT trn suy ra:

ng thc xy ra khi v ch khi

nh gi thng qua s dng BT ph s c s dng nhiu sau ny.

3 3 3 2 2 2

2 2 2

a b c a b c

b c a b c a

3 2

2

a aa 2

b b2 2a a

b 2 ab b

3 2

2

a a2

b b

3 3 32 2 2 2 2 2

2 2 2 2 2 2

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

a ab b b bc c c ca a2

2 2 2a b c

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

3 3 3 2 2 2

2 2 2 2 2 2

a b c a b c

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

2 2 222 2 2a b c a b c 3 a b c a b c

a b c 3

a b c

3

2 2

aax by

a ab b

3 2 2 3 3 2 2a ax by a ab b 1 x a y b x y a b b a

23 3 2 2a b a b b a a b a b 0

2 11 x y x y x ; y

3 33

2 2

a 2 1a b

a ab b 3 3

3 2 2 3 3 2 23a a ab b 2a b a b a b ab

3 3

2 2 2 2

b 2 1 c 2 1b c; c a

b bc c 3 3 c ca a 3 3

3 3 3

2 2 2 2 2 2

a b c a b c

a ab b b bc c c ca a 3

a b c



Cch 3: (Cauchy ngc du Kt qu khi a ra BT ph c th ging nhau nhng t tng th khc):

Ta c:

V li ging nh trn.

V d 16:[USAMO 2003] Cho x,y,z l cc s dng. Chng minh rng:

Li gii

Bi ton th v c nhiu cch gii v sau y l cch s dng bin i kh hay:

Ta c:

Suy ra ta cn chng minh:

(lun ng)


Bi ton tng chng n gin trn li c mt s ng dng tt trong vic nh hng em n t tin khi

i cc bi ton khc a v n:

V d 17:[H Khi A nm 2007] Cho x,y,z l cc s thc dng thay i tha mn: . Tm gi

tr nh nht ca:

Li gii


Suy ra

t . Ta c:

quay tr li v d 16

Vy ti

Dng ca bi ton trn li m ra mt hng i nhanh chng cho bi ton sau:

V d 18: Cho a,b,c l cc s thc dng. Chng minh rng:

3 2 2

2 2 2 2

ab a ba a b ab a b 2 1a a a a b

a ab b a ab b 3ab 3 3 3

2 2 2

2 2 22 2 2

2a b c a 2b c a b 2c8

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

2 2 2 2

2 2 2 2 22 2 22 2

2a b c 2 b c a 2 b c a b c a3 3 3

a b c2 a b c2a b c 2a b c

2 2 2

2 2 2 2 2 2

2 2 2

a b c b c a c a b1 a b c b c a c a b a b c

a b c2 2 2a b c ab bc ca

a b c

xyz 1

2 2 2x y z y z x z x yP

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

2 2x y z x 2 yz 2x x

2y y2x x 2z zP

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

x x a; y y b; z z c

a b cP 2

b 2c c 2a a 2b

P min 2 x y z 1

2

2 2 2

a b c2a 2b 2c

a b b c c a a b c



Li gii

nguyn nh vy nh gi kh lng em li kt qu g!


Tng t cho 2 bin cn li ta c:

; .

Cng v theo v cc BT trn ta c:

.

p dng BT Cauchy Schqarz ta c:

M

Suy ra



2a 2a 4a 4a

a b 2a a b 3a b2a. a b

2b 4b

b c 3b c

2c 4c

c a 3c a

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

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

24a 4b 4ca 3a b b 3b c c 3c a 4 a b c

3a b 3b c 3c a

2 2 2 2 2 2a 3a b b 3b c c 3c a 3 a b c ab bc ca 4 a b c

2

2 2 2

a b c4a 4b 4c

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

a b c

chinh ph c b t ng th c trong thi i h c

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