V Tin Trnh Trng PTNK 1

BT NG THC (Nng cao cho lp 9)

Phn I. Chng minh bt ng thc bng php bin i tng ng.

PP. Dng php bin i tng ng bin i bt ng thc cn chng minh v cc bt ng thc khc m ta d dng nhn thy ng.

Bi 1. Cho , ,a b c . Chng minh 2 2 2a b c ab bc ca .

Ta c: 2 2 2 2 2 22 2 0a b c ab bc ca a b c ab bc ca

2 2 2 0a b b c c a .

Bt ng thc cui cng ng nn ta c iu phi chng minh.

Bi 2. Cho ,a b tha 2a b . Chng minh 2 2 2ab a b

Ta c: 42 2 2 22 8 16ab a b ab a b a b

3 3 4 3 2 2 3 48 8 4 6 4a b ab a a b a b ab b

4 3 2 2 3 44 6 4 0a a b a b ab b

4 0a b

Bt ng thc cui cng ng nn ta c iu phi chng minh.

Bi 3. Cho , ,a b c tha 3a b c . Chng minh

2 2 2 6a b c ab bc ca .

Ta c: 22 2 2 6 6a b c ab bc ca a b c ab bc ca

223 3 3 3ab bc ca ab bc ca a b c ab bc ca

2 2 2a b c ab ba ca

V Tin Trnh Trng PTNK 2

2 2 2 0a b b c c a

Bt ng thc cui cng ng nn ta c iu phi chng minh.

Bi 4. Cho ,a b tha 1ab . Chng minh 2 21 1 2

11 1 aba b

Ta c: 2 2 2 21 1 2 1 1 1 1

1 1 11 1 1 1ab ab aba b a b

2 2

2 21 1 1 1ab a b aba ab b ab

2 2 01 1 1b a a b

ab a b

2 2

2 20

1 1 1b a a ab b a b

ab a b

2

2 2

10

1 1 1b a abab a b

Bt ng thc cui cng ng nn ta c iu phi chng minh.

Bi 5. Cho , , 1;4a b c tha 2a b c . Chng minh 2 2 2 18a b c

Ta c: 2 21 4 4 1 0 3 4 0 3 4a a a a a a a

Tng t 2 23 4, 3 4b b c c

Do : 2 2 2 3 12 18a b c a b c .

Bi tp tng t.

Bi 6. Chng minh cc bt ng thc sau.

a) , 0a b . Chng minh 3 3 2 2a b a b ab b) ,a b tha 2a b . Chng minh 3 3 2 2a b a b v 3 3 2a b

V Tin Trnh Trng PTNK 3

c) Cho , ,a b c tha 3a b c . Chng minh 4 4 4 3 3 3a b c a b c d) Cho ,a b tha 0a b . Chng minh

2 2 3 3 6 64a b a b a b a b e) Cho ,a b . Chng minh 10 10 2 2 8 8 4 4a b a b a b a b

f) Cho 3 36a v 1abc . Chng minh 2

2 2

3a b c ab bc ca

g) Cho ,a b tha 0a b . Chng minh 2

2 2 1 2aba ba b

h) Cho , ,a b c . Chng minh 2 3ab bc ca abc a b c v 4 4 4a b c abc a b c

i) Cho , ,a b c . Chng minh 4 4 2 21 2 1a b c a ab a c

j) Cho , ,a b c . Chng minh 2

2 2 24a b c ab ac bc

k) Cho , , 0a b c . Chng minh a b c ab bc ca

Phn II. p dng mt vi bt ng thc n gin chng minh bt ng thc khc.

Dng 1. p dng bt ng thc 2 2 2 1a b c ab bc ca vi , ,a b c

Theo hng ng thc: 2 2 2 2 2a b c a b c ab bc ca v 1 ta c:

2 3a b c ab bc ca Du " " xy ra a b c

2 2 2 23a b c a b c Du " " xy ra a b c

Cc bt ng thc trn rt c bn v n gin nhng nu c p dng hp l th c th gii c nhiu bi ton bt ng thc khc.

V d 1.(Canada 2002): Cho ba s , , 0a b c Chng minh

3 3 3a b c a b c

bc ca ab

V Tin Trnh Trng PTNK 4

Gii: 3 3 3

4 4 4a b c a b c a b c abc a b cbc ca ab

p dng BT 2 2 2, , :a b c R a b c ab bc ca hai ln ta c:

4 4 4 2 2 2 2 2 2a b c a b b c c a abbc bcca caab abc a b c

Du " " xy ra a b c .

+ Nu cho 1abc ta c bt ng thc:

Cho ba s thc , ,a b c tha: 1abc . Chng minh: 4 4 4a b c a b c

+ Nu cho 1a b c ta c bt ng thc:

Cho ba s thc , ,a b c tha: 1a b c . Chng minh: 4 4 4a b c abc

V d 2. (BMO 2001)

Cho ba s , , 0 :a b c a b c abc Chng minh: 2 2 2 3a b c abc

Gii: 2 22 2 2 2 2 23 3a b c abc a b c abc

Ta c: 22 2 2 2 2 2 2 2 23 3a b c a b b c c a abbc bcca caab

23 3abc a b c abc

Du " " xy ra 3a b c

V d 3. (Bearus 1996) Cho , , 0 :a b c a b c abc . Chng minh:

9ab bc ca a b c

Gii: 2 3 3ab bc ca abbc bcca caab abc a b c

33 233 3 3 81 81a b c abc abc a b c

Vy 9ab bc ca a b c

Du " " xy ra 9a b c

V Tin Trnh Trng PTNK 5

Bi tp.

Bi 1. Cho , ,a b c l cc s thc. Chng minh

a) 2 2 2a b c ab bc ca

b) 22 2 23 3a b c a b c ab bc ca

Bi 2.

a)Cho , , 0a b c . Chng minh 3 3 3a b c a b c

bc ca ab

b)Cho , , 0a b c tha a b c abc . Chng minh 1 1 13a b ca b c

c)Cho , , 0a b c tha a b c abc . Chng minh 2 2 2 3a b c abc

d)Cho , , 0a b c tha a b c abc . Chng minh 9ab bc ca a b c

Bi 3. Cho , , 0a b c tha 1a b c . Chng minh

a) 1ab bc cac a b

b)2 2 2

1a b cb c a

c) 4 4 4a b c abc d) 2 2 21 1 12a b b c c aa b cc a b

Bi 4. Cho , , 0 : 1a b c a b c . Chng minh

4 1 4 1 4 1 21a b c

Dng 2 .Cho , 0a b .Chng minh: 1 1 4 2a ba b

Chng minh: 2 21 1 4 4 0a b a b ab a ba b

(ng)

Du " " xy ra a b

V Tin Trnh Trng PTNK 6

Bt ng thc 2 thng c s dng di dng: 1 1 4 3a b a b

y l mt bt ng thc n gin nhng dng n ta s gii quyt nhiu bt ng thc khc kh hn.

V d 4: Cho ba s dng , ,a b c . Chng minh: 3 3 1 12 2a b a b a b

Gii: Dng bt ng thc 3

3 3 3 1 1 3 2 2 1 3

3 2 3 22 4 4 3 2 2 22 2 2 2a a b a ba b a a b a a ba

Tng t

3 1 32 2 2a b b a b

.Du " " xy ra a b

Cng v theo v ta c iu phi chng minh.

V d 5. Cho , , 0a b c . Chng minh: 1 1 1 1 1 1 12 2 2 4a b c a b c a b c a b c

Dng BT (3)

Ta c:

1 1 1 1 1 1 2 1 12 4 16a b c a b a c a b a c a b c

Du " " xy ra a b c

Tng t ta c: 12a b c

1 1 2 116 a b c

Du " " xy ra a b c

12a b c

1 1 1 216 a b c

Du " " xy ra a b c

Cng v theo v ta c:

1 1 1 1 4 4 4 1 1 1 12 2 2 16 4a b c a b c a b c a b c a b c

V Tin Trnh Trng PTNK 7

Du " " xy ra a b c

V d 6. Cho , , 0a b c . Chng minh:

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

Gii:

Ta c: 1 1 4 23 2 3 2 2a b a b c a b a b c a b c

1 1 4 23 2 3 2 2b c a b c b c a b c a b c

1 1 4 23 2 3 2 2c a a b c c a a b c a b c

Cng v theo v ta c iu phi chng minh.

Du " " xy ra a b c

V d 7. Cho , 0a b . Chng minh: 22 2

1 1 2a ba b

a b a b

Gii:

Ta c: 2 2 2 21 1 1 1 1 121 1 1 1 1 1

a b a b a ba b a b a b

M 1 1 41 1 2a b a b

Du " " xy ra a b

Do ta c: 22 2 42

1 1 2 2a ba b a b

a b a b a b

Du " " xy ra a b

+ Nu , 0 : 1a b a b ta c bi ton: Chng minh: 2 2 11 1 3

a ba b

(Hungary 1996)

V Tin Trnh Trng PTNK 8

+ Nu , 0 : 2a b a b ta c bi ton: Chng minh: 2 2

11 1

a ba b

Bi tp tng t:

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

1 1 1 92b c c a a b

Bi 3: Cho ba s dng , , : 1a b c a b c . Chng minh:

2 2 21 1 1 92 2 2a bc b ca c ab

Bi 4: (Nesbitt) Cho , , 0a b c .Chng minh:

32

a b cb c c a a b

Bi 5:(IMO 1995) Cho 1, , , 0abc a b c Chng minh:

3 3 3

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

Bi 6: (IMO Short List 1993) Cho , , 0a b c . Chng minh:

22 3 2 3 2 3 2 3 3

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

Bi 7.

a) Cho , 0 : 1a b a b . Tm GTNN ca 2 21 1

2A

a b ab

b) Cho , 0 : 1a b a b . Tm GTNN ca 2 21 1

1 2B

a b ab

c)Cho , 0 : 1a b a b . Tm GTNN ca 2 21 1 4C ab

a b ab

d)Cho , 0 : 1a b a b . Tm GTNN ca 3 3 2 21 1 1D

a b a b ab