TNG HI TUN

DIN N VT L PH THNG

GII BT NG THC
THI HSG QUC GIA MN TON 2015

http://vatliphothong.vn

Cho a, b, c l cc s thc khng m. Chng minh rng

3(a2 + b2 + c2 ) (a + b + c)( ab + bc + ca) + (a b)2 + (b c)2 + (c a)2 (a + b + c)2 .

Li gii

H
iT
u

1 u tin, ta s chng minh

(Trch thi chn HSG quc gia mn Ton 2015)

3(a2 + b2 + c2 ) (a + b + c)( ab + bc + ca) + (a b)2 + (b c)2 + (c a)2 .

Bt ng thc trn tng ng

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

hay

a+b+c

Tht vy n lun ng v

ab + bc + ca.

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

a + b + c ( ab + bc + ca) =
0.
2
ng thc xy ra khi v ch khi a = b = c. 

T
ng

2 Tip theo, ta s chng minh

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

Cch 1:
Bt ng thc ny tng ng

[
(

)]
(a b) + (b c) + (c a) (a + b + c) a + b + c
ab + bc + ca 0,
2

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

[
2

2]
1
(a + b + c) ( a b) + ( b c) + ( c a) 0,
2

Sc (a b)2 + Sa (b c)2 + Sb (c a)2 0,

trong

a+b+c

Sa = 1

2( b + c)

a+b+c

Sb = 1

2( c + a)2

a+b+c

2
Sc = 1
2( a + b)

Gi s a b c, ta thy

a + c + 4 ac b
a+b+c

S =1
2 =

2 0

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

a
+
b
+
c
a
+
b
+
4
ab c

2 =
2 0
Sc = 1

2( a + b)
2( a + b)
Mt khc, ta c

ac
a
, do
bc
b

H
iT
u

Sc (a b)2 + Sa (b c)2 + Sb (c a)2 Sa (b c)2 + Sb (a c)2

( (
)
)2
ac
= (b c)2 Sb
+ Sa
bc
(
)
a2
2
(b c) Sb 2 + Sa
b
2
a Sb + b 2 Sa
= (b c)2
.
b2
Nh vy, ta ch cn chng minh a2 Sb + b2 Sa 0, tc l chng minh

)
(
a+b+c
a+b+c
2
a2 1
2 + b 1 (
)2 0,
2( c + a)
2 b+ c

b2
a2

a2 + b2 (a + b + c)
)2 .
2 + (

2( c + a)
2 b+ c

T
ng

Tht vy, v c 0 nn ( a + c)2 = a + c + 2 ac a + c, tng t ( b + c)2 b + c. Do

)
(
2
2
a
b
a2
b2

(a + b + c)
2 + (
)2 (a + b + c) 2 (c + a) + 2 (b + c)
2( c + a)
2 b+ c
a2 + b2
a2 b
b2 a
+
+
2
2 (c + a) 2 (b + c)
2
2
a2 b b2 a
a +b

+
+
2
2a
2b
a2 + b2
=
+ ab
2
a2 + b2 a2 + b2

+
2
2
= a 2 + b2 .
=

Bi ton c chng minh xong.

ng thc xy ra khi v ch khi a = b = c hoc a = b, c = 0. 
Cch 2:
i bin a, b, c bi a2 , b2 , c2 ta cn chng minh
(a2 + b2 + c2 )(ab + bc + ca) + (a2 b2 )2 + (b2 c2 )2 + (c2 a2 )2 (a2 + b2 + c2 )2 ,
tng ng vi
a4 + b4 + c4 + (a2 + b2 + c2 )(ab + bc + ca) 4(a2 b2 + b2 c2 + c2 a2 ),
2

a4 + b4 + c4 + abc(a + b + c) + ab(a2 + b2 ) + bc(b2 + c2 ) + ca(c2 + a2 ) 4(a2 b2 + b2 c2 + c2 a2 ).

Theo bt ng thc Schur bc 4, ta c
a4 + b4 + c4 + abc(a + b + c) ab(a2 + b2 ) + bc(b2 + c2 ) + ca(c2 + a2 ).
Do ta cn chng minh
ab(a2 + b2 ) + bc(b2 + c2 ) + ca(c2 + a2 ) 2(a2 b2 + b2 c2 + c2 a2 ).
Bt ng thc ny lun ng v theo AM GM th a2 + b2 2ab, b2 + c2 2bc, c2 + a2 2ca.
Bi ton c chng minh xong.
ng thc xy ra khi v ch khi a = b = c hoc a = b, c = 0 hoc cc hon v ca n. 

Ta c,

H
iT
u

Cch 3:
Bt ng thc cn chng minh tng ng

(a + b + c)( ab + bc + ca) + a2 + b2 + c2 4(ab + bc + ca).

ab( a b)2
0.
a+b

2ab
2bc
2ca
T suy ra ab
. Tng t ta cng c bc
, ca
.
a+b
b+c
c+a
S dng cc bt ng thc ny v bt ng thc Cauchy Schwarz ta c

(a + b + c)( ab + bc + ca) + a2 + b2 + c2
)
(
ab
bc
ca
+
+
+ a2 + b2 + c2
2(a + b + c)
a+b b+c c+a
(
)
1
1
1
= 2(ab + bc + ca) + 2abc
+
+
+ a2 + b2 + c2
a+b b+c c+a
(
)
9
2(ab + bc + ca) + 2abc
+ a2 + b2 + c2
a+b+b+c+c+a
9abc
+ a2 + b2 + c2 .
= 2(ab + bc + ca) +
a+b+c

T
ng

2ab
ab
=
a+b

Mt khc, theo bt ng thc Schur bc 3 ta c

9abc
+ a2 + b2 + c2 2(ab + bc + ca).
a+b+c

(a + b + c)( ab + bc + ca) + a2 + b2 + c2 2(ab + bc + ca) + 2(ab + bc + ca)

= 4(ab + bc + ca).

Bi ton c chng minh xong.

ng thc xy ra khi v ch khi a = b = c hoc a = b, c = 0 hoc cc hon v ca n. 
Cch 4: Bt ng thc cn chng minh tng ng

(a + b + c)( ab + bc + ca) + a2 + b2 + c2 4(ab + bc + ca).

Ta c,

2ab
ab
=
a+b

ab( a b)2
0.
a+b
3


2ab
2bc
2ca
. Tng t ta cng c bc
, ca
.
a+b
b+c
c+a
S dng cc bt ng thc ny ta c

(a + b + c)( ab + bc + ca) + a2 + b2 + c2
(
)
ab
bc
ca
2(a + b + c)
+
+
+ a2 + b2 + c2
a+b b+c c+a
(
)
1
1
1
= 2(ab + bc + ca) + 2abc
+
+
+ a2 + b2 + c2 .
a+b b+c c+a

T suy ra

ab

(
2abc

rng

1
1
1
+
+
a+b b+c c+a
2abc
+ 2a2 = 2a
b+c

Do ta cn phi chng minh

2 (ab + bc + ca)
hay l chng minh

(
)
+ 2 a2 + b2 + c2 (a + b + c)2 .

H
iT
u

tng ng

Chng minh hon tt nu ta ch ra c

)
(
1
1
1
+ a2 + b2 + c2 4(ab + bc + ca),
2(ab + bc + ca) + 2abc
+
+
a+b b+c c+a

bc
+a
b+c

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

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

(a + b + c)2 ,

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

.
b+c c+a a+b
2ab + 2bc + 2ca
Bt ng thc ny hin nhin ng theo Cauchy Schwarz

T
ng

b
c
a2
b2
c2
a
+
+
=
+
+
b+c c+a a+b
ab + ac bc + ba ca + cb
(a + b + c)2

.
2ab + 2bc + 2ca

Bi ton c chng minh xong.

ng thc xy ra khi v ch khi a = b = c hoc a = b, c = 0 hoc cc hon v ca n. 

Cch 5:
D thy rng khi abc = 0 th bt ng thc hin nhin ng.
Xt abc > 0.
V bt ng thc hon ton thun nht, nn ta c th chun ha abc = 1.
Bt ng thc cn chng minh tng ng

(a + b + c)( ab + bc + ca) + a2 + b2 + c2 4(ab + bc + ca).

S dng bt ng thc AM GM , ta c

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

)
a+ b+ c
= ab (a + b) + bc (b + c) + ca (c + a) + abc

3
2ab + 2bc + 2ca + abc 3
abc
= 2 (ab + bc + ca) + 3
= 2 (ab + bc + ca) + 2abc + 1.
4

Do , ta cn chng minh
a2 + b2 + c2 + 2abc + 1 2(ab + bc + ca).
Theo nguyn l Dirichlet th trong 3 s dng a, b, c lun tn ti hai s nm cng pha so vi 1.
Gi s hai s l a v b. Khi ta c 2c(a 1)(b 1) 0 hay tng ng
2abc + 2c 2(bc + ca).
T , chng minh s hon tt nu ta ch ra c
a2 + b2 + c2 + 2abc + 1 2ab + 2abc + 2c,
(a b)2 + (c 1)2 0.

H
iT
u

Bt ng thc cui lun ng nn bi ton c chng minh xong.

ng thc xy ra khi a = b = c = 1. 

hay

T
ng

Tng Hi Tun
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