Những Viên Kim Cương Trong Bất Đẳng Thức Toán Học
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TRAN PHU’O’NG C{)n_g téc vién: Trn Tuin Anh l\guyén Anh Cuimg; Bi1iVie_“:tAnh
i NHU’NG
VIEN KIM CU’O’NG
TRONG BAT DANG THU’C TO/§\N HQC
(Tdi bdn lcin thzi nhd’t)
NHA XUi\T BAN TRI THU’C
MUC
ugc
CHU’O’NG I: NHU’NG VlEN KIM CU’O’NG TRONG BAT DANG THLPC CO DIEN
§1. Bit ding thirc AM - GM vi cic R9 thuit chqn aiém ro’i §1.1. Bit ding thirc AM — GM §1.2. NhCrng sic miu diim rdi trong bit ding thirc AM GM §2. Bit ding thirc Cauchy - Bunhiakowski — Schwarz vi ky thuit chgn dié'm rdi §2.1. Bit ding thfrc Cauchy — Bunhiakowski Schwarz Ky thuit chqn diim rdi trong bit ding thfrc Cauchy ' ' BunhiaC6pski —Schwarz §3. Bit ding thlfrc Holder vi ky thuit chgn diim ro’i §3.1. Bit ding mac Holder §3.2. Ky thuit su’ dung bit ding thllrc Holder —~
-—
§4. §4.1.
Bit ding mac Minkowski vi
kv
thuit sir dung
Bit ding thirc Minkowski §4.2. Ky thuit su’ dung bit ding thirc Minkowski §5. Bit ding thirc Chebyshev vi ky thuit sir dung §s.1. Bit ding thirc Chebyshev §5.2. Ky thuit sir dung bit ding thirc Chebyshev CHUUNG "3
Nl-‘|l~)’NG
VIEN KIM CU’O’NG TRONG BAT DANG THUC CAN DAI
§6. Bit ding thirc hoin vi vi K9 thuit su’ dung §6.1. Gidi thiéu v'é bit ding thirc hoin vi §6.2. Ky thuit sir dung bit ding thirc hoin vi §7. §7.1.
§7.2. §7.3.
Bit ding thirc Schur vi ky thuit sir dung Gidi thiéu vé bit ding thu’c Schur Ky thuit sir dung bit ding thllrc Schur Ung dung bit ding thin: Schur trong chirng minh bit ding thirc -.. .1. .
xu’ng ba blen
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§s. Djnh ly Muirhead vi bit ding thirc d6i xu’ng §s.1. Gié'i thiéu dlnh I9 Muirhead §8.2. Ky CHU’O’NG
Ill:
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§9. thvirc §9.1. Gidi thléu dinh 19 Fermat §9.2. U'ng dung dlnh ly Fermat §10. Djnh ly Lagrange vi cic (rng dung trong bit ding thirc §10.1. Dlnh ly Lagrange cho him mét bié'n vi cac u’ng dung '
§10:2. Cu'c trl cua him nhi'éu biin vi phu’dng phip nhin tu’ Lagrange §10.2.1. Cgrc trj khéng c6 dléu kién ring buéc
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28
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1
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O
10. Cho
.
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nhc'>
I
thzir AM — GM
Y
nhét cua biéu thirc
S
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2
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a,b,c20. Tim gié
P:A{/
a
+§/
b+c,
mg
+S=i/a+b+\‘/b+¢'+\3/¢-+as =§
:>MaxS=§3-
Nguyén nhén sai lm:
>
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c.-+u=l
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121
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b,
C
nén Max s am tgi cnéu kién
1
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2
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3
Va bét dang
3
thtrc
khi gap bét G
trong cn
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thira
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lain nhét cua S
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O
a
Tim gié U;
'
a+ b +c=3
lc'>‘n
mat cila biéu
1/a(b + 2c) + Q/b(c +
S =
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1:1
mét biéu
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3 S
3
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3
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1 _
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+
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2
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a
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xét: Hai vé
sa dung
béit
121
caic
biéu mac
ding mac AM
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b2f1c
(-3) nén biéu mac
ta cé
ccfmg
thém cé béc (_3).
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2
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+
2
()
“
c+a
b—fL?+;€-z+?f:-52
va,b,¢
> 0
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Tacé" '
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c-+0
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b+c
:>
2 B£1i14.Cho a,b,c>O.Ch£rng minh: (a +b
2
+¢-
2
2(a3+b3+c3) )(-,-+—,+-,-)z3+_i—abc a“ b" c" 1
1
1
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ca
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I:
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Béi 15. [Hi Lap MO 2007] Cho a,b,c
151
dc}
clili ba canh cua mét tam giéc. Chimg minh
(a+b—c)4 2
(c+a—b)4
(b+c—a)4
a(a+b-c)+b(b+c'—a)+c(c+a—b)
/9
U
Z‘
'
+7c+w
Chlhzg minh
su dung
aéng thtrc
bém
,
AM - GM,
ta C6
.
Z_ ,7 (b+C-(1)4 +(l([l+b'_L)ZL.(b+(_(I) -2(a
3
3
+19
a(a+b——c)
_1
+( )+4(bc—ca—a
4
+b(b+c'—a)22(¢'+a—b)' =2(a2 +b3 +c3)+4(ca—ab—bc
+
b(b+c—a) 4
+c"(c'+a—b)22(c1+b—c‘)' :2(a2 +193 +c3)+4(ab—bc—ca
c(c+a—b)
-3
-1
-1
+(C+a_b) +(”+b—C) 25(a:+b3+c:)—4(ab+bc+c'a) :(b+‘_”) cz(a+b—c) b(b+c—a) c(c+a—b) Ta chi can Chirng minh 5(a1 a2
+111
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Ding thin: xéy ra
a
=b=c.
>0; 3§ne
Biai 16. Cho a],a2,...,a” A‘
a3 al - (13+(l3 + (l3"|"[l4
+...+
a
a
+ A, = + 3 ‘ a + aI a + a3 ‘
I1
N. Dal aII—|
+
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I
Cl
a /1-1
an
”"
+a
_
,
(ll+(12 + /1
a
”
a :1—] +
(1 I
Chirng minh ring: Max{Al.A2}2l—2Z Chzhzg minh
sa dung
bait
Al+=Z 2
ding thtrc AM a
——-—'
a,+a _
1_\¢
>2 _
a +a a +0, +—i‘ ——' +l2 =2l a,+a __,Z a,+a "
_
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3
_
3
3
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2
+L
al
an +
(12
,
,
(a2+a3)3
A1+i:Z -
3
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2
:
a
l
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+
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+
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ii
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n—l
(a+b+c)—3(n—I)
3
3
n—l
C
n
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i
1|
3
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' B5 dé:
,
Vx,_v,z>O;ne N
' /ip dgmg:
l
a"+b,1+Cn:Z:a"+b3”+b"2Z(a+‘g+b) ('_\'('
('_
‘('
_
w
~
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_
k&a>\1A,
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"@v*=-~-_: __ _‘
, ;~'— ¢.
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:(a—g2b) +(b-I520) +(c+32a)
Ch zrovzg
I: Nh 17'ng vién
Béi 20.
cm
kinz cu‘0’n g tron g brit (ing I/1 tic c5
(711511
@.b.(->0. Chtrng minh b2‘ud€ing1hL'rc sau: ((12
+Zbc)(b: +Zc'a)((':
+2411?)
2 abc(u+Zb)(b+Z(‘)(('+Za) (1)
Chzhzg minlz
y=Q,
Dim .\~=%,
('
;=£
{hi ta c6: .\-,\~.;>0 v xyz =
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1.
CI
[1+%j;(1+%j(1+%);(2+.\~)(2+;-)(2+;);{+2;%z2;.\'+Z,\;. 1..
Do d6 ta chi cn Chung minh céc
dang thfrc:
bén
su dung bit ding thtrc AM - GM _2
.
Z-‘:22; \ r\a*,b>./2".(‘>¢" /i_-4 *1 14’? *5 11 i
a2+8bC
Véy
4
.
1
a?+b*‘+c-‘
a
-1
.
ab’
+
\/a'+8bc Being thtrc xay ra
\/b‘+8ca
a
b_+8¢'@
a7‘+b"‘+c'3 1
+
we
\/a“+8c1b
=b=c>O.
{+8515
4
2al+bi+c:4:1 (,3
.4.
+1;-‘+53
a‘+b-‘+c~‘
B fit (Hing thzic AA/I
42
—
GA/I
lBi22_Chtmg minh: b(a+z')+c(b+d)+d(a+c)+u(/2+d)>4_ V I a_J~C~d>0(l) ('(c1+lJ) 5/(b+c') a(c+d) b(a'+a)_ Chzivzg minh
_
».
Ta
CO.
d
b
_
-
vT(l)—(a+C)[c(a+b)+ a(c+d)]+(b+d)[d(b£+c)+b(c1a+a)]
=(abc+abd +acd +bcd)[
(a+b+¢+d béim
bc1(b+c)(d+a)
]
abcdbcda
(1—+-1N1~-+—1)+P-A+~~1)P~+»W
ding thirc AM - GM la c6:
l+l a
b+d
+
l+l bd
L+l ac
:ll_1__l_) Sir dung
a"+C
ac(a+b)(c+d)
4(-1»+l)
>
C
a
1+--1» b d
c
4(1+1) b
>
cl
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b
d
c
4(l+l_) " ‘
=>vT(1)z(l+l+l_+l) V
‘
b
Q
d
c
12
a
4(i+é?)
,+
Bi 23. Chirng minh rang; Va, b.
3+3 a a +c +b 3+3 1
1
1
L161
'
2 0 ta c6: 243
1
1
32 g(a+[,+¢-+4) 3+3+?»+3 +d 3+3 c +d b 1
+41
=4 :>(dpcm)
b
D2'§ngthL'rcxéyraa=c>0,b:a'>O.
3
,
(1+i+1+iy (1+l+;+ly b c d c d a
d
a
a
”1_)2
Z1
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1
[9
a1+b1 +0 b c a
2
a
c
b
Ch lhzg nzinh
Bit ding
thfrc
dz?
cho tLro'ng duong véri
§+i+é+2(Q+Q+_Q)2§((z+b+b+c+c'+a) };-
0-
5-
Dzfal
.\'=%,
_\'
c
=Q_
1
+_v2
+13
b
/9
=£,b§td§1ngthL'1‘c cn chtrng minhu'O‘LhZ1nh
+2(l+l+l)>§£.\'+\*+"+l+l+i) X \‘ -2 ” X I 1
2(.x3
(l
c
2
a
¢
xi
a
'
Y
+y1 +¢1)+[%+i+l)z3(.\-+_»~+;) (1). TheoAM-GMta cé -A
b
VT(1)=(x2
+.\-3
.Y
+l)+()*2
Dang thirc xéy ra
X
Z
+_\'2
+lj+(:2 +13 +l)23(x+.\"+:). I Y
a :12 = c >
O.
(G
pcm)
44
B51
1v.
some sgxc BAT BANG
a.b.c>O B:Z1'l.Cho
a+b+[-:3
I
ling thl?'c AA/I
‘
.cr'-
'11;
-
(b+c)(b+a)
¢-~
+
(c+cz)(c+b)
Clz lhzg milzh SL1"
dung gia thiél a +
u
'
I
+c
/7
mu-C
dua véb§1dé11gth(1'c
3 dé
_‘
+
b
(a+b)(a+c) su dung bét ding
3
AM - GM,
ta
3
(l2+c)(b+a)
8
:>
2
a
8
+
b
1
?
a_|_b+C:3
Clzzhzg
Sir dung gié thiét H
3
b(2c+a)
+
(1
+b+¢- =
I93
c(2a+b)
+
3 aé
C3
dua 2
a(2b+c)
u+b+c =23 (dpcm)
2
(c+a)(c‘+b)
bém
0+1) +
c
"
4
3
8
_»
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4
3
_c+a_c+b:3c
a,b,c>O B£1i2.Cho
3
C3
c
4
8
A12+“
a(2b+c)
3
-
a3
b
-
GM.
La
c(2a+b)
c
3
9[b(2c+a)+c(2a+b)+ a(2b+c) 1
:>
3
‘I
b(2c+a)
3
+
+
b
c(Za
+12)
‘
_3
a(2b+c)
2
u(2b+c)
aéng béc bgic
b(2c+a)
c(2a+lJ)
‘
21
minh
L+3b+(2c‘+a)23-3 i~3b(2c+u):9a b 2c+a) +
hai vé:
0"
1
C61
\J(b+(')(b+u)
8
bzfac
2a+b+c
0‘
\/(a+b)(a+c‘)
_
3
béc
déwng
(c+u)(c+b)
(b+c)(b+c1)
L+i+1iO
\.
B2113. Cho
b17't
thlic c5
ziring
:1
q
~
-
~
"
C
b}
1
—”—+——+‘i2— l7+Zc (“+211 a+Zb
, ; .Chu'nO mmh rang: .
La: +1): +62
45
(711511
I
1
3
Ch tin g minh SL1‘
dung gié thiét (1: + I73 + cg = l dé du'a vé bél ding thfrc déng bgic
+
:13
.
+
b3
b+2c
c+2a
2a3+b2+c2
c3
a+2b
3
AM - GM,
Sir clung béit ding thtrc
L3 b+2C+a(
ta
+ 2-()2
1,
cc’):
2/L
(1( b
1-: 6a +4.)
c+2a
9-L‘ /9-L‘ a_£2b+c(a+2l2)22 ;m'C((I+2b)—6L
3 1
3
3
_;
3
9(27f7C+C_{[%a+gj+3(ub+b('+ca)26(a:
b
3
1
1
c‘
2
2
:>9(b+'2¢-+¢~+2a+a+2b)23(" H’
Bi1i4.Ch0
3
i+b(c+2a)22,li-b(c'+2a)=6b3 c+2a
+
a’
2 6' hai vé:
b:_?1c
(">0
{(1.19
+‘
"b+b¢'+“'=1
1
a‘
)3
.ChL'rng minh
Q
7
_2
+192
1
b‘
7
7
a"
c‘
+c3)
+19"
b+2c+c+2a+a+2b2
r§1n(dpcm). -
c1.b.c>O 3 a'+b"+ -": '1
'1
C-7
.
Chirng minh:
£i+@+%z3 C‘
Chlhzg minh
Bit ding (
thtrc d cho tucmg ducmg vdi
ab + be + ca ¢
0
Hazb‘ , )29-,3
17
2
¢
+ bzc,
+
cqa ,
b
a
2 blag czaz azbz +b 2 \:»*—,—+—,—+-—q—2a b‘ a“ c‘
St! dung b§td?1ngth(1'c
_
2
+c.
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2
ta c6:
2
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Ch ll'0'Il g I : Nhfrng viérz kim Clt’0'l1g tron g bfit rin g thzkc 05 (Tién (
ab '7
7
'7
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v
2
~
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bc '7
a
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~
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+
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7
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7
b
'\
2
2
2b:
b2
'
2
'
2
2
(
2
b
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2
7
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c"
b" 2
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21: _a3 :20-
2
2 -
5
'7
a
2b: ' ccj +(1w 22 b‘ c‘ 2
(1
bc _ca
'7
+011 22
2
47
W
:> a—;—+——%~+i2z1‘ +b“+c“ :> (dpcm) c’ a“ b“ Ba18. Cho
a.b.c'>O a
'7
.
.
+b+c=l
.ChL'rng mmh réng:
S
=
In
+
+
uz
\/a+bc
si
ab
\/c+ab
\/b+c¢lz
2
Ch tin g minh bc
bc
\/4l+bC
\/a(a_+b+c)+bc
C61
C61
ca
\/c‘+ab
\/c'(a+b+c)+ab
\/(c+a)(('+b)
a
1
+
(1+b)(1+c)
b
1
—
2
(
>
1
“+51 + "+19 )
:u+1>+¢-zi
2(c+b)
[IMO Shortlist 1998] Cho a,b,c > O
1
)(1+¢~)
8
8
8
8
4
>a+b+@_;>3-~/“abc_;=;
‘
2
4‘
2
4
4
Bfit ming
48
AM - GM
ml:-C
Biii 10. [IMO 1995] Cho a,b.c >0 thba min abc=l. Chimg minh ring: I 3 + ., > ,+ b‘(¢‘+a) c'ia+b) 2 u“(b+r) 1
(1)
1
§
|
|
Clz lhz g min lz
su dung gia thiér abc = +
abc
.
.
+
a/2c
bu-+a>
a*‘(b+c)
x=i,y=l.:=l b a
bién
1
S6 aé
2(a
—+i
)+~Z2
2
X
'_
1*"
:+x
+.1‘+_\‘>2
:3
4
Bili
_-0
x _Z
1
+
"‘
"
+
~
\'+; :+x x+v 11.
ta cé:
~
+.\:) -
4
:3 .\'+_\" 4 x+_\'
—
2
.\"+_\'
--—
'7
.
4
+—*'-Zl(.\"+y+;).
4
3
Iv" :+x
+M+.\22
x+y :>
_\‘+
-
.
1"
_\_
+
b
a '7
~
A
4
'7
+
a '>
--
1
_l::‘Y
,2
1
_\'+3
c
c
\+,f : +1J
AM - GM.
-
b \
dung b§1td5.ngth1'1'c
N
2l(.\-+_\-+g)_-
=3
3 §/A3";
-
~
2
[IMO Shortlist 1996] Cho ab a"‘ +1)“
+ab
(1,1). c
>
O
théa min abc =1. ChL'1'ng minh réng:
bc
+
+
_
b‘
+0“
(dpcm)
,
+120
an
ca _
+u” +ca
S1
Ch zhzg minh °
B5
2x3)‘: + x2_\'3
x5 + ys
(T5:
' Ap dung: Str dung bi; J5 ab
0
ta c6:
abc
1
a2b2(a+b)+ab
l
abc
bc(b+c)+l
bc(b+c")+abc
s
ab
ca
s
bc
Ding thirc xéy ra
ca(c+a)+1 ‘
i c‘+a'+ca a =b =c=l. s
5
+lJc
+
“
1
1/1+5‘ 1
:2 Z33
2
'1_ (1+“)+(1"4'+”)
’
,l1+a3
1
1
1
.
49
dién
Chung m1nh ring: —i—;:+%——,i_+——}~2l \/1+u" \/1+1)" \/1+(“‘
.
1:
:>
th12'c 1:5
22 (
,+
1
\2+a“
,+
1
1
J
,
2+c'
2+b'
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Chung minh
(,1.
8+2(a: su-
dung
Bili
+c2)_%a2b2c3 a3 +b3 +02 212
+192
bit déng G
13. Cho
thtrc
AM- GM
,b,'>O Ch, unc
m cé (H +111
.h
C
m1n
.
2
(b+¢) +
b
3
=12.
((+0)
+
L
_:
1
mg + 2c1\/Z
b\/I; + 2c\/Z
{(11% =1
I
a
+6 z3\~‘/611%-1
(1) (u+b) >2 _
a\/Z + b\/E
Ch /rn g minh
a3(b+c) bJB+2@\/Z
2
1f(¢-+11)
+
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b’.2JZZ
2/NZ c"\/Z+2a\/Z
21*
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a\/;+b\/Z—a\/;+b\/E
a\/;+b\/E
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b2(c+a)
b¢z+2¢-pi-@+2a¢;+u¢;+b¢1: 411+ p
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2
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2m
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y:4p+m—Z1z:> T:2[4n+p—2m+4p+m—2n+4n1+n-Qpj
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l m
9
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m
p
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M
p
9
9
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2(b+c)
>‘=b@>°
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3:“/;>0
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a
x=a\/;>O
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S=
Bat
2a\/21-
b\/5+2“/F " />\§+2¢~\/Z " .\*+2z
2
c\/Z+2a\/Z
3
a:.2\/1;
B+m+1 m
n
p
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n
p
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zing
111 111-
14111
v. PHOI HOP c/lxc BAT DANG THU’C DONG BAC NGU’O’C CHIEU
iu./>>() bi
1.
Cho
2
1
.Ch1'1'ns minh
1’
|Lu+/1:1
151112: V
‘_
S =
Z -10
‘
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11'
11:/7:
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rng ta cé hai ba‘l déng [hire
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3
') 84:1\‘;_—,z
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11g11'Q'c
(u+/1)
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1:
1
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w
+
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(u+/1)”
1*
v
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1h1'1c
xay ra
=12 =
:1
u
1
B:Z1i2. ChL'1‘ngminh1'ang:
(—12
b Z u + /2 +—) +1-l\"'_ —[_i__—_—.:2 I a
N/U
10.
»
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V u./2 >0
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bit ding
thtrc
ngu'Q"c
chiéu suu Ciily
+’lz2; 415-"-"I-"_§4f-f=s [7
\/c1:+b:
U
B§td§11gthL'1‘cIL1‘o'ng du'o'ng %+
4\[:_,7Clb
\./(13
\/11’
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1’°\[5Ub
+b)] 24\/51112.
2)+(u+l)) ta 06:
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I: Nh 1711 g vién kim
Ch u'0‘n g
troll g
cu‘!/n g
17111
(hing
Vu./>_¢'>(J
3.
Choj
.
1” +1, +
.ChL'1'ngm1nh ring:
:1
(-
~
1
5
C6 (Ii @511
H1121"
1?“? l
+0‘
:1“ +1J'
5
l
l
I
S=
1
_f
ab
-'
“ + 2‘
2
(1/7
+
be
v
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11'
9
'\"/u1>./2c.c'c1
fl/?+/’('1“,c+cu)+(u12+11c+'c
chiéu sau dy
+ u :12
bit
Bi1i5.Ch(rn@c minh
+/73c:
AM - aw
AM - GM w cé
Sir dung b§1Id§1ng\hL'1'c 11‘.
+u: )2-H1121):
+04 +(1l7(a3 +123)-¥-l7('(/>2 +c'(a +11 + z-)(u3
+123
11
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F
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_}
+123
+c3)S(a+12+c)$
.
.
+9(‘f”+”,‘+‘f’)z12
“bf
‘
Suy ra bét ding thirc
SL1‘
_
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thtrc AM— GM ta c6
ab+l1c+cu
27(u3
'
(a+;)+(.)»‘
bit
+123
$6.
27(”+b)(b+C)((‘+”) >16
ta cé hai
ab+b('+c'a Ap dung
r//2+/J('+ (dpcm)
=b = c.
Ta cflng c6 min $6 két qua tu‘0'ng
al+b2+c'2+
1
Lu"
suu
\'c'>'i
moi
41.12,
020.
>2
Saba"
(a+l2)(b+c)(c+a)_ a
GM
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-H11‘:
kim z'u"0'ng rrong
.\'h17'ng vién
~‘+b*+w xay
D11gll11'1‘c
Jzilzg
I1 ('1!
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_
9(u+/1)(/>+(‘)(‘('+z1)2%(u+/7+” (U +/,+(.)» _»
_
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15112:
b
U
/9+6
h
+
+
/7
("+11
+.1f(
(I
U/H’
\/11+/JH/2+(')((‘+(1
¢1+/>
Clz 1212 g min I1
Nlzn xét: D5
}'"
rring
111
c6 hui
hit
dzing [hire
11g11'Q'c
chiéu sau
1151)’
1"1—1**
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l>+("
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1
~
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/{p dgmg: 11:11
0+7
3.13";
2
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2
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minh: Kl1611g1n?11tfnh16ng
VT(l)—VP(l)= .\'(.\'—
1
11+!)
Y8
Z
~
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Chzivzg
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SL'1'd1_111g
7+(
»1’)+4u/7(‘Z2(u+/2)(l7+(')((*+z1) (13
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my
ra
a
:[2:c
hoéc
((1—b): +(b—c)3 +(c—u): 211199
E)2f1tt:u‘b+ab‘ =c1b(a+b)Da3
ru
8
ab
-3
3I)S(Z) :16.
=a4b4(u+/Jr‘ +
4
16
(I/7
ab
I" (8-3!)
uh
+i+—L+—l—+i25-5)u4l24i~--J——~L-L=5 ab
(11;
(11;
bng xéy ra vdi Cl-l-17:2 I
Binh lufin: Néu dilt '
'
ab ab ab ab
(11;
a
151
=12:
l.
Zu 2/) .\'=i.\'=—— u +1) u +12
nén du \
lhl
'
bng xziy ra
151
\':\" + xx +
(.\‘+ Y):
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chit
+11} =1
\
vdi .\".\‘>0 va '
.\'v\"
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tozin hQc nhung czic bién
If do tai sao ta
Ngoili ra ta hoiln toim c6 thé chqn ab=l hay u} dén két qual
=b>0
P=;%+i'i
.\'+ y = 2. Ta thy biéu Lhirc P khéng thay déi vé ban
thi bi rng buéc bdi diéu kién .\‘+ y =2. D6
a
Q6
thé chun héa a
nhung nhu' lhé
sé
.\".
_\'
+b=2.
khé dé di
.\'lu7"ng
58
nzdu Jiéhz ru'1'
.s¢'
f(u+/v)(/v++l)('+¢'u):9
dgm c
.
+ ;\'
_\':
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((1./7.0)
(>1)
3’u+/1+z"23 :>
O. ChL'rngminh1"§1ng 1.
\/3(a3+b3+¢'3)>\/Z+\/l?+\/Z
2_
(a+b)(b+c)(("+u)>
_ ab+bc+cu
3ubc'
Saba
Clzfmg minh. 1.
Chun héu
Theo
41}
+12" + (:5
AM- GA/It
3:113
:
3.
+123
—
a+1;+¢~
\,/;,Z+\/E+v/Q
»
Ta cén chirng minh u/>+b(‘+ca 2c(\,G +\/B + +03 23ub().ChL'1'ng minh réng:
(U
+/+~)‘ ’
abr
6|
/-+-
SI
+27»1;"f+ H‘ ‘fl 254 a' +b“ + 0‘
‘
0 ,
,
3!
.
Jab + ac + bc
/ab + ac + bc
Suy ra
,
\a‘+b'+c'
2
_
.
‘
Chuén héa a + b + c
3(ab +
3(ab + ac + bc) _,
(a+b+c')“
=1. khi
.
La
Céch 2.(V5 Q1150 Bd
1
—— 2
abc
bit
V
9 81(ab + at +bc) 2 54. ———i+
-2121
1'15.
ta
du''c
9[3(ab +bc + ca)-112 2
(dimg)
a+b+c
4
.
1
\la'
(ab +bc + ca)“
_
+ b‘ + c‘
2 54
/
ta (.0
—
Z((z+b+c') \/Qrq/(ub+b(.+m)1(u1 +bI +01)
y su y ra
Q
.
+b' +c')
(a+b+c)'
,
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3(ab+bc+ca)
;;(a +b + cf)‘ 1
S
~
1
~
Z
2(a+b+c) \/§]i(a+b+C)(,
\27 hay
O
(ab + bc + ca):
(ab +bc +ca).(ab +bc + ca).(a“
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