bất Đẳng thức-sinh hoạt cụm-võ văn nhân
TRANSCRIPT
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
1/24
Chuyn : Mt s phng php chng minh bt ng thc
Chuyn : MT S !"#$% !& C!'$% M($! )*T +,$% T!'C T- T.n / T!T $0i Th1nh
23 $!4$% )*T +,$% T!'C C5$ )6$ T!"7$% %8:
93 )t ng thc 2M/%M 2;ithm
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
2/24
Chuyn : Mt s phng php chng minh bt ng thc
2 2
nn na b a b+ +
946& h%& 5A# hu h7n s thBc:
1 2 3
1 2 3
...
...n
n
a a a a
b b b b
h>Oc
1 2 3
1 2 3
...
...n
n
a a a a
b b b b
Ph& /@ t% c@:
1 1 2 2 1 2 1 2... ... ....n n n na b a b a b a a a b b b
n n n
+ + + + + + + + + Du ! xA# $% kh&: 1 2
1 2
...
...n
n
a a a
b b b
= = = = = =
Et /0ng thc 'CFEGHCFI ch> h%& 5A# /Jn /&Ku ng(*c ch&Mu:
'h> h%& 5A# hu h7n s thBc:1 2 3
1 2 3
...
...n
n
a a a a
b b b b
h>Oc1 2 3
1 2 3
...
...
n
n
a a a a
b b b b
Ph& /@ t% c@: 1 1 2 2 1 2 1 2... ... ...
.n n n na b a b a b a a a b b b
n n n
+ + + + + + + + + Du ! xA# $% kh&: 1 2
1 2
...
...
n
n
a a a
b b b
= = =
= = =^3)t ng thc !_&$ ]`:1Et /0ng thc !_&$ ] ch> h%& 5A# /Jn /&Ku cLng ch&Mu:
'h> h%& 5A# hu h7n s thBc:1 2 3
1 2 3
...
...n
n
a a a a
b b b b
h>Oc1 2 3
1 2 3
...
...n
n
a a a a
b b b b
QR& ( )1 2, ,..., nt t t l mSt h>n 4T tuU V c% ( )1 2, ,..., nb b b
Ph& /@ t% c@: 1 1 2 2 1 1 2 2... ...n n n na b a b a b a t a t a t + + + + + +2Et /0ng thc !_&$ ] ch> h%& 5A# /Jn /&Ku ng(*c ch&Mu:
'h> h%& 5A# hu h7n s thBc:1 2 3
1 2 3
...
...
n
n
a a a a
b b b b
h>Oc
1 2 3
1 2 3
...
...
n
n
a a a a
b b b b
QR& ( )1 2, ,..., nt t t l mSt h>n 4T tuU V c% ( )1 2, ,..., nb b b
Ph& /@ t% c@: 1 1 2 2 1 1 2 2... ...n n n na b a b a b a t a t a t + + + + + +a3)t ng thc SC!: 'h> %,,c khng m 4 k l s thBc t kU, t% lun c@:
9 9 9 9 9 9
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
3/24
Chuyn : Mt s phng php chng minh bt ng thc
5X2 2 2 2
2 2 2 9 : 9 : 9 : 9 :
3
a b c a b b c c aa b c
+ + + + + + + =
YX 3 3 3 2 2 29 :9 : 3a b c a b c a b c ab bc ca abc+ + = + + + + +
)3 MT S j$% T!"7$% %8 T_$% C!'$% M($! )*T +,$% T!'C:
j$% 9: T!kM \l T !)1i 9: 'h> %??c l 3 s 5(Jng . 'hng m&nh $Zng:
=3 ( )3 3 3 1
9 :9 : 9 :9 : 9 :9 : 8
a b ca b c
a b b c b c c a c a a b+ + + +
+ + + + + + Cch9: + 5[ng )+T C Sit% c@
3 3
9 9 \ \ 8
a a b b ca
a b b c
+ ++ +
+ +91
3 3
9 9 \ \ 8
b b c c ab
b c c a
+ ++ +
+ +92
3 3
9 9 \ \ 8
c c a a bcc a a b
+ ++ + + + 93 .
'Sng 3 E] t$n suy ;= c/&Mu c^n chng m&nh.9]&Mu _u%n t$Rng t$>ng t$()ng h*+ n# l /` V 5u Zng x"# $%CchG: + 5[ng )+T C Sit% c@
3\
9 9 a9 9
aa b b c a
a b b c + + + +
+ +91
3\9 9 a
9 9
bb c c a b
b c c a + + + +
+ +92
3\9 9 a
9 9
cc a a b c
c a a b + + + +
+ +93
'Sng 3 E] t$n suy ;= c/&Mu c^n chng m&nh
b33 3 3 1
9 9 9 9 2
a b ca b c
b c a c a b a b c+ + + +
+ + +Cch9: + 5[ng )+T C Sit% c@
38
2 9 a9
ab c a a
b c a + + +
+91
38
2 9 a9
bc a b b
c a b + + +
+92
3
8 2 9 a9
c a b c ca b c
+ + + +
93
'Sng 3 E] t$n suy ;= c/&Mu c^n chng m&nhCchG: b+ 5[ng )+T C Sit% c@
3 3
9 2 8 2
a b c aa
b c a
++ +
+91
3 3
9 2 8 2
b c a bb
c a b
++ +
+92
T- T.n > T;@ng T!T $0i Th1nh $%ng 3
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
4/24
Chuyn : Mt s phng php chng minh bt ng thc
3 3
9 2 8 2
c a b cc
a b c
++ +
+93
'Sng 3 E] t$n suy ;= c/&Mu c^n chng m&nh.I6& V t(ng t$n t% c@ th` g&"& t(Jng tB cc cu 5(6& /# :
c3d d d
3 3 32 2 2
a b ca b c
b c a+ + + + oX
d d d3 3 3a b c a b c
bc ca ab+ + + + 3 s 5(Jng39 ,
3 3+ a b c c
c4
1
3t% /(*c:
39 1
3 3 3
a b c ca b c
c
+ + + +
39 8 1
3 3 3
a b c ca b
c
+ + 91.
(Jng tB:39 8 1
3 3 3
+ +
b c a ab c
a 92?
39 8 1
3 3 3
+ +
c a b bc a
b93.
'Sng 91, 92 4 93 t% su# $% 1 m&n 1 =P P kh& 1= = =a b c .)1i tqp K:( Bi ton ny chng ti sng tc v ch nh gii theo th! y"# t$ %h&' t#y nhin c) th*gii theo cch khc nhanh h+n,
'h> x, #, th>" mAn 3x y -+ + . m g& t$T nhf nht c% &`u thc2 2 2x y -
.x y- y -x - xy
= + ++ + +
-% c@:2
2 2
y - x y -x y- x x y-
+ + ++ + +
2 22
2
x x
x y -x y-
+ ++91, (Jng tB:
2 22
2
y y
y - xy -x
+ ++92?
2 22
2
- -
- x y- xy
+ ++93
-# 91 cSng 92 cSng 93 4 thY> 4 t% /(*c:2 2 2
22 2 2
x y -.
x y - y - x - x y
+ + + + + + + +
e7& c@: ( )21a
2 \2
xx y - x
x y -+ + +
+ +91i? ( )
21a2 \
2
yy - x y
y - x+ + +
+ +92i?
( )21a
2 \2
-- x y -
- x y+ + +
+ +93i
-# 91i cSng 92i cSng 93i 4 thY> 4 t% /(*c:
T- T.n > T;@ng T!T $0i Th1nh $%ng 8
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
5/24
Chuyn : Mt s phng php chng minh bt ng thc
2 2 2
1a 89 2 2 2
x y -x y -
x y - y - x - x y
+ + + + + + + + + +
2 2 2 3
2 2 2 8 8x y - x y -
x y - y - x - x y
+ + + +
+ + + + + +3
2. . Ij#
3
2/in.= kh& x#1. 9 Nu 5Lng C=uchy/SchH=;I s nh%nh hJn
)1i tqp X:( Bi ton ny chng ti sng tc v ch nh gii theo th! y"# t$ %h&' t#y nhin c) th*gii theo cch khc nhanh h+n,'h> x, #, th>" mAn 3x y -+ + . m g& t$T nhf nht c% &`u thc
3 3 3x y -.
x y- y -x - xy= + +
+ + +
-% c@:2
2 2
y - x y -x y- x x y-
+ + ++ + +
3 32
2
x x
x y -x y-
+ ++91 ? (Jng tB:
3 32
2
y y
y - xy -x
+ ++92 ?
3 32
2
- -
- x y- xy
+ ++93
-# 91 cSng 92 cSng 93 4 thY> 4 t% /(*c:3 3 3
22 2 2
x y -.
x y - y - x - x y
+ +
+ + + + + + e7& c@:
( )3
21a 2 \2
xx x y - x
x y -+ + +
+ +91i
( )3
21a 2 \2
yy y - x y
y - x+ + +
+ +92i
( )3
21a 2 \2
-- - x y -
- x y+ + +
+ +93i
-# 91i cSng 92i cSng 93i 4 thY> 4 t% /(*c:2 2 2
1a 29 :2 2 2
x y -xy y- -x
x y - y - x - x y + + + + + + + + + + + 2 2 2a9 x y - + +
2 2 2
\ 9 :2 2 2
x y -xy y- -x
x y - y - x - x y
+ + + + + + + + + + +
2 2 239 x y - + + 9
-7& c@: 2 2 2x y - xy y- -x+ + + + 9# 9 cSng 9 4 thY> 4 t% /(*c:
( )2 2 2
2 2 2\ 2
2 2 2
x y -x y -
x y - y - x - x y
+ + + + + + + + + +
m
( )2 2
2 2 2 3
3 3
x y -
x y -
+ +
+ +
2 2 2 3
2 2 2 8
x y -
x y - y - x - x y + + + + + + + +3
2. . Ij#
3
2/in.= kh& x#1.
)1i tqp Z:'h> 3 s thBc %,,c;< 4 th>" % c 3%c+ + = .'hng m&nh $Zng:
( ) ( ) ( )3 3 3c c% %
1% 2 c 2c % c 2%
+ + + + + 9
T- T.n > T;@ng T!T $0i Th1nh $%ng d
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
6/24
Chuyn : Mt s phng php chng minh bt ng thc
j$% G: Sr $% )*T +,$% T!'C C2C!\/SC!293 ]$ $% C# )6$ T!"7$% %8:
)v( 9: 'h> %??c l 3 s ong th>": =bwbcwc= K=bc
Chng minh ;ng:8 8 8
12 2 2
a b b c c a
a b b c c a+ +
+ + + 9 T.n hAc tu-i t;x / ;= ng1y 9f393Ga
%ii: %ecec% 3%c 1 1 1 3a b c
+ + 9
]Ot :1 1 1
, ,x y -a b c
= = = ? Ph& /@ 9 /(*c 4&t l7& : z w y w I K zJyJI { ? 9?
T= c|:8 8 8
12 2 2
a b b c c a
a b b c c a+ +
+ + +
3 3 3
12 2 2
a b ca b b c c aab bc ca
+ + + + +
3 3 3
12 1 2 1 2 1
a b c
b a c b a c
+ + + + +
9
Ph& /@ 9 /(*c 4&t l7& :
33 311 1
1
2 2 2
yx -
y x - y x -
+ +
+ + +
22 2
2 2 2
11 1
1
2 2 2
yx -
xy x y- y -x -
+ +
+ + +eb+ 5[ng E] C=uchy/SchH=;I :
( )222 2
1 2 331 2
1 2 3 1 2 3
a a aaa a
b b b b b b
+ ++ +
+ +} Qi ?
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
7/24
Chuyn : Mt s phng php chng minh bt ng thc
% c@ :2
2 2 2 28 8 8 8
8 8 8 8
8 8 8 8 8 8 8 8
1 1 1 11 1 1 11 1 1 1
11 1 1 1 1 1 1 11 1 1 1
1 1 1 1 8
a b c 0 a b c 0
a b c 0
a b c 0 a b c 0
+ + + = + + + = + + +
+ + + + + + + + + + + +
Hu# $%:
2
8 8 8 8 2 2 2 2
1 1 1 1 1 1 1 18a b c 0 a b c 0
+ + + + + + + h%#
2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 1 1
2 a b a c a 0 b c b 0 c 0 + + + + + 91 thY> t /0ng thc -Q th
a2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1a . . . . .
a b a c a 0 b c b 0 c 0 a b a c a 0 b c b 0 c 0 + + + + +
h%# 2 2 2 2 2 2 2 2 2 2 2 21 1 1 1 1 1 a
a b a c a 0 b c b 0 c 0 abc0 + + + + + 92. o 91 4 92 su# $% /(*c 3abc0 .
]0ng thc x"# $% kh& 8 3a b c 0 = = = = .
)v( X: 'h> 3 s thBc %,,c khng m th>": %ecec%1. 'hng m&nh $Zng:3 1 3 1 3 1
8ab bc ca
a b b c c a
+ + ++ +
+ + +.
Et /0ng thc c^n chng m&nh t(Jng /(Jng 46&:3 % c c% 3 % c c% 3 % c c%
8ab bc ca
a b b c c a
+ + + + + + + + ++ +
+ + +8% 8c 8
8ca
a b ca b b c c a
+ + + + + + + +
]` V $Zng:9 : 9 : 9 :
9 :9 : 3
9 :9 :
ab a b bc b c ca c a
a b c ab bc ca abc
a b c ab bc ca a b c
+ + + + += + + + + + + + + = + +-Hp 5[ng E] C=uchy/SchH=;I t% c@:
2 2 2 2 2 2% c %
9 : 9 : 9 :
ca b c c a
a b b c c a ab a b bc b c ca c a+ + = + +
+ + + + + +
29 : 1
9 : 9 : 9 :
ab bc ca
ab a b bc b c ca c a a b c
+ +
+ + + + + + +
D> /@, t% c@: 8% 8c 8 8
9 :ca
a b c a b ca b b c c a a b c
+ + + + + + + ++ + + + +
:8 8
9 : 2 .9 : 8a b c a b ca b c a b c
+ + + + + =+ + + +
Hu# $% /&Mu +h"& chng m&nh. ]0ng thc x"# $% kh& 4 chq kh& mSt t$>ng 3 &n Zng < 4 h%& &n cn l7&Zng 1.
)v( Z: Ch. cc s thc ong =JbJc Chng minh :( ) ( ) ( )
2 2 2
2 2 22 2 2
1
3d d d
a b c
a b c b c a c a b+ +
+ + + + + +
% c@: ( )
( )
( ) ( )
22
2 2 2 2 22
2=
8 2d
a aa
a b c a bca b c
+= + + + ++ +
2 2
2 2 2 288 2a a
a b c a bc ++ + +
( ) ( ) ( )
2 2 2
2 2 2 2 22
2
= = 2d
a a a
a b c a bca b c +
+ + ++ +
Hu# $%:( ) ( ) ( )
2 2 2
2 2 22 2 2d d d
a b c
a b c b c a c a b+ +
+ + + + + +
2 2 2
2 2 2
1 2
= = 2 2 2
a b c
a bc b ca c ab
+ + + + + +
T- T.n > T;@ng T!T $0i Th1nh $%ng r
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
8/24
Chuyn : Mt s phng php chng minh bt ng thc
D> /@ t% c^n chng m&nh:2 2 2
2 2 2 1
2 2 2
a b c
a bc b ca c ab+ +
+ + +h=y
1 1 11
2 . 2 . 2 .b c c a a b
a a b b c c
+ + + + + D?
-]Ot , ,a b c
x y -b c a
= = = , t% c@ : zJyJI{ 1 zyI~9
Ph& /@ D? 4&t l7&:
1 1 11
2 2 2- x yx y -
+ + + + + 12 2 2x y -
x - y x - y + + + + +
1
2 2 2
x y -
x - y x - y + +
+ + + 3
y y -
- x y + + 9 ' H& nh nhng.
)v( ^: 'h> cc s thBc 5(Jng %,,c. 'hng m&nh $Zng2 2 2
2 2 2
a b c ab bc ca
ab bc ca a ab bc b bc ca c ca ab
+ + + +
+ + + + + + + +9
-% c@:2 2 2
2 2 2 1a b c
a b c ab bc caab bc ca
+ ++ + + +
+ +
D> /@ t% c^n chng m&nh: 2 2 21
ab bc ca
a ab bc b bc ca c ca ab + ++ + + + + +
2 2 23 2
ab bc ca
a ab bc b bc ca c ca ab + + +
+ + + + + + 2 2 21 1 1 2
ab bc ca
a ab bc b bc ca c ca ab + +
+ + + + + +2 2 2
2 2 2 2
a bc b ca c ab
a ab bc b bc ca c ca ab
+ + + + +
+ + + + + +2 2 2
2 2 2 2 2 2 2
a b c bc ca ab
a ab bc b bc ca c ca ab a ab bc b bc ca c ca ab + + + + +
+ + + + + + + + + + + +9
-% c@:2 2 2 2
2 2 2 2
9 :1
9 :
a b c a b c
a ab bc b bc ca c ca ab a b c
+ ++ + =
+ + + + + + + +91
2 2 2bc ca ab
a ab bc b bc ca c ca ab+ + =+ + + + + +
2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
b c c a a b
a bc ab c b c b ca bc a c a c ab ca b a b+ +
+ + + + + +
2
2
9 :1
9 :
ab bc ca
ab bc ca
+ + =
+ +92
o 91 4 92 su# $% /&Mu +h"& chng m&nh.
)v( a: Ch. cc s thc ong =JbJc Chng minh ;ng:2 2 2
1
2 2 2 3
a b c
a b b c c a
+ + + + + .
)v( d: Ch. cc s thc ong =JbJc th.: =bc ~9 Chng minh ;ng: 2 2 2 12 2 2
a b c
a b c+ +
+ + +
)v( f: 'h> cc s thBc 5(Jng %,,c. 'hng m&nh $Zng
2 2 2
32 2 2
9 :9 :9 : 8
3
a b b c c a a b c
abc a bc b ca c ab
+ + + + + + + +
9
T- T.n > T;@ng T!T $0i Th1nh $%ng \
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
9/24
Chuyn : Mt s phng php chng minh bt ng thc
G3 Sr $% \l T C T NHT :)1i tqp9:(;=ni=n (M_ Gf? 'h>% s 5(Jng %,,c thf% /&Mu k&Kn 3a b c+ + = . 'hng m&ng $Zng:
2 2 2 2 2 2
1 1 1 3
2 2 2 8a b b c c a+ +
+ + + + + +9
% c@ 9 t(Jng /(Jng 46& t /0ng thc s%u: 2 2 2 2 2 23 1 1 1 3
8 2 2 2 2a b b c c a+ + +
+ + + + + +
2 2 2 2 2 21 1 1 1 1 1 39 : 9 : 9 :2 2 2 2 2 2 8a b b c c a
+ + + + + + + +
2 2 2 2 2 2
2 2 2 2 2 2
3
2 2 2 2
a b b c c a
a b b c c a
+ + + + +
+ + + + + +DD?
SV oYng:2 2
2 2 9 : 9 :
2
x y x yx y
+ + + = , t% c@ DD? /(*c 4&t l7&:
2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
9 : 9 : 9 : 9 : 9 : 9 :3
2 2 2 2 2 2
a b b c c a a b b c c a
a b b c c a a b b c c a
+ + + + + + + +
+ + + + + + + + + + + +
% c@:2 2 2 2
2 2 2 2 2 2 2 2 2
9 : 9 : 9 : 89 :
2 2 2 29 : a
a b b c c a a b c
a b b c c a a b c
+ + + + ++ +
+ + + + + + + + +
9?
Nu 5Lng C=uchy/SchH=;I t$Bc t&+ t% c@:2 2 2 2
2 2 2 2 2 2 2 2 2
9 : 9 : 9 : 9 :, 4 h&`n nh&n2 2 2
2 2 2 2 2 2
9 : 9 : 9 :ng 98, 9d, 9a c@ mSt t /0ng thc /ng th & t>n c% t% /(*c chng m&nh x>ng.% c@: [ ] [ ] [ ]
2 2 29 :9 : 9 :9 : 9 :9 : 9 : 9 : 9 : /@ t$>ng 9 :9 :a b b c , 9 :9 :c a a b , 9 :9 :b c c a c| t nht mts khng m. c l t$>ng % t/0ng thc X?J Z?J ^? c| t nhtmSt t /0ng thc /ng. Ij# & t>n /(*c chng m&nh. Du ! x"# $%kh& %c1.)1i tqp G: 'h>% s 5(Jng %,,c thf% /&Mu k&Kn 2 2 2 3a b c+ + = . 'hng m&ng $Zng:
2 2 2 2 2 2
3a b b c c a
a b b c c a
+ + ++ + + + +
9
]` V $Zng:2 2 29 :
2 29 :
a b a b a b
a b a b
+ + = +
+ +, 5> /@ t /0ng thc /A ch> 4&t l7&:
2 2 29 : 9 : 9 :3
29 : 29 : 29 :
a b b c c aa b c
a b b c c a
+ + + + +
+ + +
b+5[ng t /0ng thc C=uchy/SchH=;I t% c@:2 2 2 29 : 9 : 9 : 9 :
29 : 29 : 29 :
a b b c a c a c
a b b c c a a b c
+ +
+ + + + +
E& t>n /(*c _u# 4M chng m&nh:
29 :
3
a c
a b c a b c
+ + + + +
% c@:2 29 : 9 : 9 :9 : 9 :9 :a c a c b a b c a b b c
a b c a b ca b c a b c a b c
+ + + + = + + + +
+ + + + + +2 2 2 239 : 9 : 9 :9 : = 9 :9 : 9 :9 :
329 : 2 29 :
a b c a b c a b b c a b c a b b c a b b ca b c
a b c a b c a b c a b c a b c
+ + + + + + = + + + + = + + +
+ + + + + + + + + +Nh( 4j# & t>n /(*c chng m&nh x>ng nu
9 :9 : /@ & t>n /(*c chng m&nh. Du Zng x"# $% kh& %c1.)1i tqp K: 'h>% s %,,c thf% /&Mu k&Kn 2 2 2 1a b c+ + = . 'hng m&ng $Zng:
1 1 11
d a d a d aab bc ca+ +
Et /0ng thc c^n chng m&nh /(*c 4&t l7&:
1 1 1 1 1 1 1 1 2 1 2 1 2 1
2 d a 2 d a 2 d a 2 d a d a d a 3
ab bc ca
ab bc ca ab bc ca
+ + + + D> 2 2 2 2 21 2 2 9 :ab a b c ab a b c = + + = + , t(Jng tB 2 2 2 21 2 9 : ,1 2 9 :bc b c a ca c a b = + = +D> /@ E] t$n 4&t l7&:
2 2 2 2 2 29 : 9 : 9 : 1d a d a d a d a d a d a 3
c a b a b b c c aab bc ca ab bc ca
+ + + + +
2 2 2 29 :
d a d a d a 1d a9 :
c a b a b c
ab bc ca ab bc ca
+ ++ +
+ +2 2 2 2
9 : 9 : 9 : 89 :
d a d a d a 1d a9 :
a b b c a c a c
ab bc ca ab bc ca
+ +
+ +
% c^n chng m&nh2 29 : 89 : 1
1d a9 : 1d a9 : 3
a b c a c
ab bc ca ab bc ca
+ + +
+ + + +
T- T.n > T;@ng T!T $0i Th1nh $%ng 1
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
11/24
Chuyn : Mt s phng php chng minh bt ng thc
2 29 : 89 : d 29 :a b c a c ab bc ca + + + + +2 2 2 2 29 : 89 : d9 : 29 :a b c a c a b c ab bc ca + + + + + + +
9 :9 : /@ & t>n /(*c chng m&nh. Du Zng x"# $% kh&1
3a b c= = = .
M ;ng: o & t>n n# t% c@ th` m $Sng $% & t>n s%u
'h>% s %,,c 5(Jng thf% /&Mu k&Kn 2 2 2 1a b c+ + = . 'hng m&ng $Zng: 1 1 1 =
1 1 1 2bc ca ab+ +
Tp ch C;uz M=thng 3 s %, 4 c t kU lun tn t7& h%& s m cng ln hn hoc bng mh%# cng nh hnhoc bng m 9 46& m l s thBc tuU V.Phng mt twnh tng _ut g&" sp : a b c .-Nu b ! th a ! 4 b ! .-Nu b !> th ;m 4 c;m .
9/&Mu +h"& chng m&nh.
)- G: Nu x !
y !
h>Ocx !
y !
th t% c@: 29 xy ! x y ! +
9 5y 5ng /(*c su# $% to E]: 9 :9 : T;@ng T!T $0i Th1nh $%ng 11
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
12/24
Chuyn : Mt s phng php chng minh bt ng thc
)1i tqp 9:'h> 3 s thBc %,,c 5(Jng th>" : 2 2 2 8a b c abc+ + + = 9'hng m&nh $Zng: < 2ab bc ca abc + + .
S2M_/G?/o g&" tht su# $% c@ wt nht mSt t$>ng % s %,,c khng l6n hJn 1.Q&" sp s /@ l c, kh& /@ t% c@:
91 : 9 : ng 3 s %, 4 c t kU lun tn t7& h%& s mcng ln hn hoc bng mh%# cng nh hn hoc bng m 9 46& m l s thBc tuU V!. kh& /@ t% c@:
1
1
a
b
h>Oc 1
1
a
b
9 1:9 1:
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
13/24
Chuyn : Mt s phng php chng minh bt ng thc
2 2
9 9 9 9 9 9
2 1 1
9 9 9
1 1 1 1
xy y-
x y x - y - y - y x - x
-x xy
x y x - y -- x - y x y
y- -x
y - y x - x - x - y x y
++ + + + + +
+ + + + + ++ + +
+ + +
+ + + + + +
( ) ( ) ( )
81
x # # g g x
xy-= +
+ + +h%# 4> /(*c /&Mu c^n chng m&nh.)1i tqpG:'h> 3 s thBc %,,c 5(Jng th>": 8ab bc ca abc+ + + = .'hng m&nh $Zng: a b c ab bc ca+ + + + .
](T $2M/G?/Cch 9:-Phng mt twnh tng _ut, g&" sp h%& s % 4 th>": $>ng 3 s %, 4 c t kU lun tn t7& h%& s mcng ln hn hoc bng mh%# cng nh hn hoc bng m 9 46& m l s thBc tuU V!., kh& /@ t% c@:
1
1
a
b
h>Oc
1
1
a
b
9 1:9 1: Oc% 1
1
. Hu# $% 9% 1:9 1: T;@ng T!T $0i Th1nh $%ng 13
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
14/24
Chuyn : Mt s phng php chng minh bt ng thc
2 2 2
2 2 2 2 21 1 1 1 1 12c % 29% c: 3 % 2% 2 3 29% 1:9 1: 9% 1: 2 s thBc x 4 # khc < 4 thf% : ( ) 2 2xy x y x y xy+ = + 9
m g& t$T l6n nht c% &`u thc 3 31 1
3x y
= + hi 2/G^?
E@i gii:]Ot : H xe# , { x# . Ph& /@ t% c@ : ( )
2 2
xy x y x y xy+ = + 4&t l7& :229 3:
3
;P ; ; P
;+ = =
+% c@:
3 3 2 2
3 3 3 3 3 3
22 2
3 3 2 2
1 1 9 9
9 9 9 3
9
x y x y x xy y3
x y x y x y
x y xy x y x y ; ;
x y xy P ;
+ + += + = =
+ + + + = = = =
CJn n% : 9xe#
2
8x# su# $% :
22 2 388 < /@2 2
2
1 1?
1
t t t t y x ty
t t t
+ += = =+ +
. o /@22 2
2
1 1 2 1
1
t t3
x y t t
+ += + = + .
zt hm s( )
2 2
22 2
2 1 3 39 W9
1 1
t t tf t f t
t t t t
+ + += =
+ +.
j+ "ng &n th&n t% tm QN c% l: 1a /7t /(*c kh&1
2x y= = .
)1i Z:'h> cc s thBc x, #, l cc s thBc +hn &Kt 4 khng m. m g& t$T nhf nht c%
T- T.n > T;@ng T!T $0i Th1nh $%ng 1r
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
18/24
Chuyn : Mt s phng php chng minh bt ng thc
2 2 2 2 2 21 1 1
9 : 9 : 9 :9 : 9 : 9 :
: x y y - - xx y y - - x
= + + + + + + +
)1i ^: 'h> h%& s thBc x 4 #. m g& t$T nhf nht c% &`u thc ( ) ( )2 22 21 1 2. x y x y y= + + + + + .
hi )/G^?)1i a: 'h> x, # l h%& s thBc khng m th%# /&. m g& t$T l6n nht 4 g& t$T nhf nht c% &`u thc
2 2
9x #:91 x#:{
91 x: 91 #: =
+ + hi /Gd?
)1i d: 'h> cc s thBc x, # th%# /& 4 th>" mAn 9x e #3e 8x# 2.m g& t$T nhf nht c% &`u thc : 39x8e #8e x2#2 29x2e #2 e 1. hi )/Gf?)1i f: 'h> cc s thBc khng m %, , c thf% mAn: % e e c 1. m g& t$T nhf nht c% &`u thc
39%22e2c2ec2%2 e 39% e c e c% e 2 2 22 a b c+ + .hi )/G9?
)1i 9: 'h> x, #, l 3 s thBc thuSc />7n [ ]1? 8 4 , .x y x - m g& t$T nhf nht c% &`u thc
.2 3
x y -P
x y y - - x= + +
+ + + hi 2/G99?
)1i 99: 'h> % 4 l cc s thBc 5(Jng thf% mAn 29%2e 2 e % 9% e 9% e 2. m g& t$T nhf nht
c% &`u thc { 3 3 2 2
3 3 2 28 =
a b a b
b a b a
+ +
. hi )/G99?
)1i 9G: 'h> 2 s thBc x,# ;< th>" x # 1+ = .m g& t$T nhf nht c% :x #
1 x 1 #
= + .
)1i 9K: 'h> x,# 4 x, # ; 1. m g& t$T nhf nht c%( ) ( )3 3 2 2
9 19 1
x y x yP
x y
+ +=
)1i 9X: 'h>x 4 y l cc s thBc thf% mAn: 21 9 y x x y = .
m g& t$T l6n nht 4 nhf nht c% &`u thc:
a a
3 3
1x yP x y xy+ = +
)1i 9Z: 'h> x,#, l % s 5(Jng th>" m"n /&Mu k&Kn: 21 1 1 3 1
= 19 : \1.x y - x y - xy-
+ + + +
.
m g& t$T nhf nht c% &`u thc : ( ) ( )3 3 3 3 9 3:91 :
2 28
- -P x - y -
+ = + + + +
)1i 9^: 'h> x, #, " mAn xe#e ; cc s thBc khng m -yx ,, th>" mAn 3222 =++ -yx .m g& t$T l6n nht c%
&`u thc-yx
-xy-xy3++
+++= d
.
)1i 9f: 'h> h%& s thBc x,# th%# /& s%> ch>: 29x2e #2 - x# 1. m QNN 4 QN c% &`u thc :8 8x e #
{ 2x# e 1
.
T- T.n > T;@ng T!T $0i Th1nh $%ng 1\
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
19/24
Chuyn : Mt s phng php chng minh bt ng thc
)1i G: 'h> 2 s thBc x,# ;< th>"d
x #8
+ = . m g& t$T nhf nht c% :8 1
x 8#
= + .
)1i G9: C+ hi 2J ) > Gd ? 'h> ,x y l s thBc thf% mAn 2 2 2x y+ = . m g& t$T l6n nht, g& t$Tnhf nht c% &`u thc 3 329 3P x y xy= + .
)1i GG: 'h> % s thBc
1
%, , c ?33
. m g& t$T l6n nht c% &`u thc:a b c
P a b b c c a= + ++ + +)1i GK:'h> , ,a b c l % s thBc thf% mAn /&Mu k&Kn abc a c b+ + = . m g& t$T l6n nht c% &`u thc :
2 2 2
2 2 3
1 1 1P
a b c= +
+ + +)1i GX:'h> % s 5(Jng , ,a b c thf% mAn /&Mu k&Kn 21 2 \ 12ab bc ca+ + . m g& t$T nhf nht c% &`u
thc:1 2 3
Pa b c
= + + .
)1i GZ:'h> %, , c l /S 5& % c7nh c% t%m g&c c@ chu 4& Zng 3. m g& t$T nhf nht c% &`u thc:2 2 239 82 a b c abc= + + +
)1i G^: 'h> cc s thBc 5(Jng , ,x y - thf% mAn /&Mu k&Kn8
3
x y -
xy-+ + = = . 'hng m&nh $Zng:
8 8 81\3 1ad d 1\x y x + + )1i Ga:'h> cc s thBc x, #, thf% mAn /&Mu k&Kn x e# e h%& s thBc 5(Jng x 4 # thf%:8 8 1
2x y xyxy+ + = + .
m g& t$T l6n nht c% & thc 2 22 2 3
1 1 1 2:
x y xy= +
+ + +.
)1i K:'h> % s thBc x,#, thf%:1
1? 1? 18
x yx xy- = . m g& t$T nhf nht c% &`u thc
1 1 1
1 1 1:
x y -= + +
+ + +)1i K9:'h> % s thBc x,#, thf%: 1 , , =, ,x y - x y x - . m g& t$T nhf nht c% &`u thc
3 8x y -:
x y y - - x= + +
+ + +)1i KG:I6& x 4 # l h%& s thBc l6n hJn 1. m QNN c% &`u thc:
3 3 2 22 2x # x #{ 29x # : 1a x#
9x 1:9# 1:
+ = + +
)1i KK:'h> % s thBc 5(Jng %,,c thf% /&Mu k&Kn: 29 9 8a c b c c+ + = .
m g& t$T nhf nht c% &`u thc:3 3 2 2
3 3
32 32
9 3 9 3
a b a b:
b c a c c
+= +
+ +
T- T.n > T;@ng T!T $0i Th1nh $%ng 1=
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
20/24
Chuyn : Mt s phng php chng minh bt ng thc
hi 2 /G9K?)1i KX:'h> cc s thBc 5(Jng %,,c. m g& t$T nhf nht c% &`u thc s%u:
3
2 3{ .
% % %c % c=
+ + + +)1i KZ:'h> cc s thBc 5(Jng x,#, thf% mAn 2# xg 4 2g x# . m g& t$T nhf nht c% &`u thc :
x # 3g{ x # # g g x= + ++ + + .
)1i K^:'h> cc s thBc 5(Jng %, , thf% /&Mu k&Kn8% 2c c
1 1 a % %
+ + + =
. m g& t$T nhf nht c% &`u
thc:c %c %
{ 2%9 2c: 9c %: c92% :
= + + + + +
)1i Ka:'h> cc s thBc %,,c khng /ng th)& Zng < thf% mAn: 2 2 2 29 : 29 :a b c a b c+ + = + + . m g& t$T l6n nht 4
g& t$T nhf nht c% &`u thc:3 3 3
9 :9 :
a b cP
a b c ab bc ca
+ +=
+ + + +
j$% Z: $% !"#$% !& $ )(l$(3+`$! E ] $ )(l$: Q&" sp ( )1 2 3, , ?..., nf x x x x l hm s l&n t[c 4 /& xng 46& tt c" cc
&n 1 2 3, , ?..., nx x x x xc /Tnh t$n mSt m&Mn l&n thng thf% mAn /&Mu k&Kn:
( ) 1 2 1 21 2 3 3, , ,..., , , ,...,2 2
n n
x x x xf x x x x f x x
+ +
9?
Ph& /@ t /0ng thc s%u /(*c thf% m"n ( ) ( )1 2 3, , ,..., , , ,...,nf x x x x f x x x x 46&
1 2 3 ...
nx x x x
xn
+ + + +=
DEu F:9?c@ th` th%# mSt s 57ng khc s%u: ( ) ( )1 2 3 1 2 1 2 3, , ,..., , , ,...,n nf x x x x f x x x x x x ,
( )2 2 2 2
1 2 1 21 2 3 3, , ,..., , , ,...,
2 2n n
x x x xf x x x x f x x
+ +
,
e Nu chq xt t$n , m&Mn l&n thng th()ng xt: [ ] ( ] [ )9 ? :, ? , ? , ? , 9 ? :, 9 ? :,...a b a b a b a b a b +((3 )*T +,$% T!'C )2 )(l$ ]( CC T` +jT Tj( %(& T` )(l$ +( '$%
=3hng php:
Q&" sp t /0ng thc c^n chng m&nh c@ 57ng: ( ), ,
-
7/25/2019 Bt ng Thc-sinh Hot Cm-V Vn Nhn
21/24
Chuyn : Mt s phng php chng minh bt ng thc
Lhng ;1ng buc cc bin bng mt ng thc h=y bt ng thc? ]& 46& l>7& n# th 5n &n thY> cc /7& l(*ng t$ung nh ch0ng h7n nh(
2 2
, ,2 2
x y x yt t t xy
+ += = = l nhng k thujt chwnh 5Lng /` 5n h%& &n Zng nh%u.
)1i 9:'h> % s thBc khng m x,#, chng m&nh $Zng 33.x y - xy-+ +
DCch 9: 3 33. 3. t ch#n h)a theo tCng $i v6i bDt Eng thFc ?ng b>c,
Nu th t% c@: 3 32r
3. 1 4 lw. Ij# 22 ,a b t t ab+ l nhng s khng m. Hu# $% 9 /ng.C%# 9 , , : 9 , , :f a b c f t t cE# g&) t% /& chng m&nh: 9 , , :