Chuyen De Bat Dang Thuc
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Y%>%
x 2 + y 2 = ( x − y ) + 2 xy = ( x − y ) + 2 2
a
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2
2
+ y2
?8 a
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+ 4.( x − y ) + 4
/
5%
2
4
⇔
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2
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a %h) # >%
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2
(x − y ) + 4(x − y ) + 4 ≥ 8.(x − y ) (x − y )4 − 4(x − y )2 + 4 ≥ 0 [(x − y )2 − 2]2 ≥ 0 4
⇔
2
5%
1 1 2 + ≥ 2 2 1+ x 1+ y 1 + xy
Y%>% 1 1 2 + ≥ 2 2 1+ x 1+ y 1 + xy
a ⇔
1 1 1 1 ≥0 + − − 2 2 2 1+ x 1+ y 1 + y 1 + xy
xy − x 2 xy − y 2 + ≥0 1 + x 2 .(1 + xy ) 1 + y 2 .(1 + xy ) x ( y − x) y( x − y) + ≥0 2 1 + x .(1 + xy ) 1 + y 2 .(1 + xy )
⇔
(
)
(
)
⇔
(
)
(
)
⇔
( y − x )2 (xy − 1) ≥ 0 (1 + x 2 )(. 1 + y 2 ).(1 + xy )
) % 4
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0
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6
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(1.a + 1.b + 1.c ) ≤ (1 + 1 + 1).(a (a + b + c )2 ≤ 3.(a 2 + b 2 + c 2 ) 2
a ⇔ ⇔
a2 + b2 + c2 ≥
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1 1 1 + + ≥9 a b c
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2a 3 + 2b 3 + 2c 3 < 3 + a 2b + b 2c + c 2 a
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a b3 + c3 < 1 + b 2c a3 + c3 < 1 + c 2a
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11
= 255 < 256
b R $ 256 = 24.14 = ( 24 ) = 1614 < 1714 11 ^• K 7 14 O # 5 S 14
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6
a+b b+c c+d d +a + + + <3 a+b+c b+c+d c+d +a d +a+b
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a
a+b a+b a+b+d < < a+b+c+d a+b+c a+b+c+d b + +c b+c b+c+a < < a+b+c+d b+c+d a+b+c+d d +a d +a d +a+c < < a+b+c+d d +a+b a+b+c+d
H
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1 1 1 1 S + + ... + < 1.3 3.5 (2n − 1).(2n + 1) 2 1 1 1 S 1+ + + ... + <2 1.2 1.2.3 1.2.3.....n
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H 8
a
1 1 ( 2k + 1) − (2k − 1) 1 1 1 = . = − ( 2n − 1) . ( 2n + 1) 2 (2k − 1).(2k + 1) 2 2k − 1 2k + 1
= w
& R Jl ) a
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a
1 1 1 1 2 1 + + ... + = . 1− < 1.3 3.5 (2n − 1).(2n + 1) 2 2n + 1 2
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a 1 1 1 1 1 1 + + ... + < 1+ + + ..... + 1.2 1.2.3 1.2.3.....n 1.2 1.2.3 ( n − 1) .n
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9
O# 5S
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L5 34 F 34
%$ 6 k p P |Q |I |Q |I|Q |I |Q +|
Y%>% a |Q |I |Q +| P |Q |I |+ Q| ≥ |Q I+ Q| P ^ 4 x − 2 + x −3 = x − 2 + 3− x ≥ x − 2+ 3− x =1 ^; P |Q |I |Q |I|Q |I |Q +| ≥ I P + a w OS ? ) Q> 6 R % 1 ≤ x ≤ 4 OS ? ) Q> 6 R % 2 ≤ x ≤ 3 ^; a %$ 6 k 34 + R % 2 ≤ x ≤ 3 A5 %$ 6 3L p l P Q XJOQI SJO IXSJOXIQS CL%QW WX G < C4 QI IX P Y%>% ^ AQW WX G < W$# Hi % a 3 QI I X ≥ 3 xyz 3
xyz ≤
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( x + y ) .( y + z ) .( z + x ) ≥ 3 3 ( x + y ) .( y + z ) .( x + z ) 2 ≥ 3 3 ( x + y ).( y + z ).( z + x )
? )
Q> 6 R %QP PXP ^; l ≤
8 1 8 . = 27 27 729
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a
1 3
34
8 1 R %QP PXP 729 3
H 8 Q I XIXQ P p x4 + y4 + z 4
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(
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:
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a
( x 2 + y 2 + z 2 ) 2 ≤ (12 + 12 + 12 )( x 4 + y 4 + z 4 ) → ( x 2 + y 2 + z 2 ) 2 ≤ 3( x 4 + y 4 + z 4 )
w O S C4 O S
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1 ≤ 3( x 4 + y 4 + z 4 ) 1 x4 + y4 + z 4 ≤ 3
x 4 + y 4 + z 4 a %$ 6
68
5
k
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=
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1 2
a l P . ( x + y ) .h = a.h = a. h 2 = a. xy ^A R i '%5 4 QI P ^ ; l 3L R %QJ 3L ^ ; 68 $ 5 %$ a
⇔ x= y
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6A
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4 3x + 6 x + 19 + 5 x 2 + 10 x + 14 = 4 − 2 x − x 2 2
Y%>% a 3x 2 + 6 x + 19 = 3.( x 2 + 2 x + 1) + 16 = 3.( x + 1) 2 + 16 ≥ 16 5 x 2 + 10 x + 14 = 5. ( x + 1) + 9 ≥ 9 2
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R %Q P
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x + 2 − x = 4 y2 + 4 y + 3 2
Y%>% $# -
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a
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x + 2 − x ≤ 1 + 1 . x + 2 − x 2 ≤ 2. 2 = 2 2
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2
2
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x + 2 − x2 = 4 y2 + 4 y + 3 = 2
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R %Q P C4 P
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x =1
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6A
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y=−
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x + y + z =1 x + y 4 + z 4 = xyz 4
Y%>% $# -
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a
x4 + y 4 y 4 + z 4 z 4 + x4 + + 2 2 2 2 2 2 2 2 2 ≥x y +y z +z x x4 + y4 + z4 =
≥
x2 y 2 + y 2 z 2 z 2 y 2 + z 2 z 2 x2 z 2 + y 2 x2 + + 2 2 2
LV
≥ y 2 xz + z 2 xy + x 2 yz ≥ xyz.( x + y + z )
^ AQI IX P S g E x 4 + y 4 + z 4 ≥ xyz ? ) OPS Q> 6 R % Q P P X P ^;
x + y + z =1 x + y + z = xyz 4
4
4
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Y%>% D # !" xy − 4 = 8 − y xy = 2 + x 2
w # !" w # !"
6A 6A
1 3
2
6A
1 3
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OS OS
y ≤ 8
x2 + 2 = x . y ≤ 2 2 x
x 2 − 2 2 x + 22 ≤ 0 ( x − 2)2 ≤ 0 x = 2 x=± 2
g &) Q P 2 A P g &) Q P 2 A P ^;
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2
Y%>% ^ AQW WX 34 $
) E
E
x + y + z ≤ xy + 3 y + 2 z − 3 2
2
2
⇔ x 2 + y 2 + z 2 − xy − 3 y − 2 z + 3 ≤ 0 y2 3 y2 + − 3y + 3 + z2 − 2z +1 ≤ 0 4 4
(
⇔ x 2 − xy + y ⇔ x− 2
2
y 2
2
x−
4
y + 3 −1 2
2
y −1 2
2
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)
+ ( z − 1) ≤ 0 2
+ ( z − 1) ≥ 0 2
O{S ∀x, y ∈ R
2
y y 2 + 3 − 1 + ( z − 1) = 0 2 2 y x− =0 2 x =1 y ⇔ −1 = 0 ⇔ y = 2 2 z =1 z −1 = 0 ⇔ x−
x =1
QW WX # >% A5 34 y = 2
H$
z =1
A5
%D5
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p # !"
6A
1 1 1 + + =2 x y z
Y%>% f i
5 1 x
1 y
1 z
a 2= + + ≤ 4 X
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x≥ y≥z
BY
X P C48 # !"
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g E P 8b ^ L% P R i
1 1 + =1 x y 1 1 1 P + ≤ y≤2 5 4 x y y
!j
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6A
x+ x = y
Y%>% O{S ^ L%Q K < W K < A# !" 6A O{S ^ L%Q G < W G < a x + x = y ⇔ x + x = y2
R i
a
⇔ x = y2 − x > 0
b a g !
x = k OR ) E k .(k + 1) = y 2
k 2 < k ( k + 1) < ( k + 1) k < y < k +1
4 %z R C4 RI 34 !" 48 > gE R i a b# ^ ; # !"
6A
a
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