Đề-đáp án HSG lớp 9- Vĩnh Phúc- môn Toán

 
  !"
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
%&'(')*+,


( )
( )
 
 
 
    
y x x
x y x xy y

= + +


− + = + −


%&'('*+, 
 !
n
"#$%& '(
a
)

a n≤
*+,*-

%&/'(/'*+,
 .,  *    / 
ABC
 0  +  12  3  4
O
5- 
6 6AD BE CF
')  7*  12  *,
( )
6 6D BC E CA F AB∈ ∈ ∈
-829
EF
:
BC
;
6G
129
AG
:';123
4 5O
;1<
M
-
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6 6 6A M E F
?> 0123-
- /
N
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ABC
-.=>
GH AN
⊥
%&0'(')*+, 
.=>

( )
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B
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4 54 54 5
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a b c abc
a b b c c a a b b c c a
abc
+ + +
+ + + ≥
+ + + + + +
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D&E$E1$FG*7H#%
C C×
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D+=&4E2FG45:HI +,-
Q+';=*G*$R!;
( )
( )
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6y x
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( )
( )

 
U      Vx x x x∆ = + − + − =
C-
W1"61I
 6  y x y x= − = +
C-
X Y+
y x= −
6*$),=61I
C6 6 x x x= = − = −

C-
• Q%
Cx =

 y x= − =
• Q%
x = −

 y x= − =
• Q%
x = −

 Vy x= − =
C-
X Y+
 y x= +
6*$),=61I
C6 6 Vx x x= = − =
C-
• Q%
Cx =

  y x= + =
• Q%
x = −

  y x= + = −
• Q%
Vx =


  VVy x= + =
C-
QS6G*')
( ) ( ) ( ) ( ) ( ) ( )
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C-
%&'('*+,
D+=&4E2FG45:HI +,-
/
a
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-a n≤
[
( )
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n a< +
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Y+
Na
≥

6 6a a a− −
')%'(G*
-n
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06 
( ) ( )
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-^*

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   N   C N   Ca a a n a a a a a a a− − ≤ < + ⇒ − + − < ⇒ − + − <
-QE'\4!,
Na
≥
5-
C-
_,1"
a =
,L
Ba =
,L
a =
C-
X Y+
a =

{ }
 
 B 66B6666N6n n≤ < ⇒ ∈
C-
X Y+
Ba =

{ }
 
B  V66666n n≤ < ⇒ ∈
4!,

6B]n
5 C-
X Y+
a =

{ }
 
 N BC6n n≤ < ⇒ ∈
4!,
6B6]n
5 C-
QS !
n
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D+=&4E2FG45:HI +,-
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6 6 6A M E F
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- -PA PB PC PD=

X c!dSef ,=
AMBC
0+6*1I
GM GA GB GC
× = ×
 /0 123/45 )%
C-

X c!d,=
BFEC
0+6*1I
GB GC GF GE× = ×
C-
X ^*
GF GE GM GA× = ×
C-
X _,1"6=
AMFE
0+-
C-
670
N
'/ 8
BC
4
H
'9 : 
ABC
&'(
')
;+<25 =>'6?6
GH AN⊥
X g,H+JO6$)=`hPi0+*
M
> 12312H#
AH
6
!,1"

HM MA⊥
-
C-
X *
MH
:';123
4 5O
;
K
6H1"!,
VCAMK∠ =
o
 
AK
')12H#G*
4 5O
-
C-
X W1"*
6KC CA KB BA⊥ ⊥
-^*
]] 6 ]]KC BH KB CH
6!,1"
BHCK
')7)-
^*
KH
1J*
N
C-

X [1"
6 6M H N
9)- C-
X ,*
GAN
"*12*,
6AD NM
:*;
6H
 
H
')@AG**

GAN
-^*
GH AN⊥
C-
%&0'(')*+, '
N
D
K
M
G
F
E
H
O
B
C
A

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( )

B
B
   
4 54 54 5

a b b c c a a b c abc
a b b c c a
abc
 
+ + + + + + ≥ + + +
 ÷
+ + +
 
C-
.=
  
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C-
c!d719=.*Xa*,HX^j*k
( )
( )
  
B

B


B
   
4 5 4 5 4 5 

   
- - -  -

c a b a b c b c a abc
a b b c c a
abc
c a b a b c b c a abc
a b b c c a abc
c a b abc
 
+ + + + + + + + + ≥
 ÷
+ + +
 
 
≥ + + + + + +
 ÷
 ÷
+ + +
 
= + + +
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
4 5 4 5 4 5  -c a b a b c b c a abc abc a b c+ = + = + = ⇔ = =
C-
C-

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D+=&4E2FG45:HI +,-
X  &$E,6H#%
 ×
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C-
X pH#77q$E6H#%
 ×
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+,B-Q$S6r')0,66N-
X W1"6g, '\_'g6"0e#N'O-
C-
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;+<25 =>'@?6