30 đề thi học sinh giỏi toán lớp 9

    


Thời gian làm bài 150 phút không kể thời gian giao đề
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$%!&'() *
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    A = + + − −

( ) ( )
!   " #  #B = − + −
$
#%
 
  & C = − − −
'(
 
$
  & 
D =
− + −
$
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$%'&'() *
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( ) ( )
$ $# #$4 $$$ $ f x x x= + + + + +
$5
( )
f x =
6789:.

2 2 2
x y z 6
xy yz zx 1
x y z 14
+ + =


+ − = −


+ + =

$%+&'() *
%;.9<.2=%>?@A.2=
( )
B
'(C+=
( )
C B
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( )
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+
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( )

+
KL$
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( )

+
L2
$%,&+() *
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.2W.M%GC1$6'(X.Y%.3.ZT(E'*AJ2V5?-G
C>SCBT(EJ25?.<X1$
20?%A[>$
\]-@'@.J2V.2-C>A+8.TF^.$
\3HIV.2WS.<T*L2=.?@$
$%-&!() *
%-0?
  
S2 S$$$S2 a
_5-@.Y%*.302

=
+ + +
n
2
a
(2n 1)( n n 1)
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
<
1 2 2009
2008
a + a + . . . + a

2010


Thời gian làm bài 150 phút không kể thời gian giao đề
 !"#
$%!&'() *
1
§Ò 1
§Ò 2
    
1) Rót gän biÓu thøc:
( ) ( ) ( ) ( ) ( ) ( )
   
x y x
P
x y y x y x x y
= − −
+ − + + + −
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#bc.K.-.@\.3
$$# $#$4 #$4$" $$$ &$$Q = + + + +
$%'&'() *
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    
    
C` d d d d
5 5 #  5 " ! 5  5  x x x x x+ + + + + + + + +
5
"
4""

A =
%89:.
   
x y a b
x y a b
+ = +


+ = +

3H'F'F0?M+:.2A
n n n n
x y a b
+ = +
#%5SSe≥'(
5dde #≤
$
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  
  
x y z
x y z
+ +
+ + +
$%+&'() *
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C>SCBT(EJ25?.<X1$
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#Vb)V.2WS.<T*L2=.?@$
$%-&!() *
%0? .ZMf  '(d0?M+:2

S 2

S$$$S2
d 
.%7/UI8

≤
2

g2

g$$$g2
d

≤
#$
3HT*.K.G20?2

S2
h

( )

 h  d≤ ≤ ≤
02%%g2

i2
h
g


Thời gian làm bài 150 phút không kể thời gian giao đề
 !"#
$%!&'() *
1) Rót gän biÓu thøc:
( ) ( ) ( ) ( ) ( ) ( )
   
x y x
P
x y y x y x x y
= − −
+ − + + + −
 5ST(-0?9:X`
#%
 
  & C = − − −
'(
 
$
  & 
D =
− + −
$

2
§Ò 3
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Đề thi học sinh giỏi vòng 2 lớp 9
năm học 2008-2009
Đề thi môn: Toán
(Thời gian làm bài: 150 phút )
Câu 1: (1,5điểm)
2 Cho các số thực dơng x, y thoả mãn: x

100
+ y
100
= x
101
+ y
101
= x
102
+ y
102
Hãy tìm giá trị của biểu thức: A = x
2008
+ y
2008
.
\ Giải hệ phơng trình:
2 2 2
x y z 6
xy yz zx 1
x y z 14
+ + =


+ =


+ + =

Cõu 2:Q

5
Đề 6
Đề 7
    
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g$$$g2
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Câu 3:Q
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n
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=
+ + +
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2007
2009
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Câu 4: QS"
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Câu 5:Q4
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(Thời gian lm bi 120 phút không kể thời gian giao đề)
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+ + − = − + − + −
AB+&,() *
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P
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+ −
= +
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#
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x
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= +
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Q a b
a b
= + + +
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#*
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J2T(2I-F.'(I*97T(%J2>$p.%\2CS>S
2T(I-F.'F2I2{
>(#Q 
%.qQN.PNS\-IbS2IC>'(1'*A'F2$~
.qQN
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
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.2-N>1'(.E95c.%'F.qQN$~.qQN
#
.E95c'FGN>'(
NJ2.2-N>'(.E95c.%'F.qQN$~.qQN
4
.E95c'F.2
C'(1'(.E95c%('F.qQN

$\-IJ2-.qQN

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QN
#
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4
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§Ò 18
®Ò kh¶o s¸t chÊt lîng HSG líp 9 M«n: To¸n
C©u 1:
2 TÝnh gi¸ trÞ cña biÓu thøc:
#""#$#""#$#"4$
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#
++−+++++
+
=A
\ %0?`I
4
dI
#
i!I
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E.%!$
$%'
%\.3
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


 zy
zy
zx
zx

M
−
+
+
−
+
=
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UI8


zyx
=+
$
p\.3VA.n-.@p^.\H\2%M{
$%+
%9:.k5S 
5
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d#

d5i5i4d&`
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2 ~G.-.@TF^.B
14
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\ ~G.-.@p^.$
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+ xx
a
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axxxx
a
a

$%-
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-C>1T(S+8.J2-.2-CN>'(N1TxT_.T(
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
SSS +=
$
(§Ò 19)
®Ò kh¶o s¸t chÊt lîng HSG líp 9 M«n: To¸n
Bµi 1: (2 ®iÓm):
2 Chøng minh r»ng:
( )
( ) ( )
44

! −++=++ xxxx
'F5I*PB
\ 6789:.
( ) ( )





=−++
−=+


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44

xx
xxx
 679:.
$##"

=+−−−− xxxx
AB'(2 điểm):
2 \20?M.?TM.E92S\S02%%

cba ++
ˆT(0?M.?{
\ %20?5S.%7/
#4

−−=−+ xxyxyyx
$/.-.@J2\.3
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−
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−
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yy
y
xxx
x
A
{
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( )
4#
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=−++− axax

A8M{
AB+(2,5 điểm):
2 679:.8M5
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\ 679:.
$

+−=++−+−+ xxxxxx
AB,(1 điểm):%5fBf.%7/


≤+
y

x
$
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x
y
y
x
P !! +=
{
AB-(2,5 điểm):%.qQNI>`b'(C.2W.MQNQC
I*.,'F>S$~9P-.%ACJ2.2-C>y.QN.G)Q)I-
C$GC'*A'F>$
2 ~;.C`5$+8.J2.2-C).Y%b'(5$502%%G.-.@
TF^.$
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0?%A>J2.2-C>\E.
"
#
=
HK
AH
$
§Ò 20
®Ò kh¶o s¸t chÊt lîng HSG líp 9 M«n: To¸n
C©u1:
Chøng minh r»ng



++= nA
T(0?9:I
NA ∈
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Nn ∈
$
15
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$%'
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{ }
BB ++∈ kkkx
'F
Zk ∈
$3
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Zm
∈∀
$
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2679:.
( )( )
# −=−+−+− xxxx
\6789:.






++=+
−=−
ayxyx
yxyx
!
#
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A$6=+(VCSV>SVSV1TxT_.T(2S\SS+$
3H2
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4
$%-
%2S\ST(-0?.Z+:$-.@p^.J2
`







+
b
a
#


$






+
c
b
#


$






+

d
c
#


$






+
a
d
#


§Ò 21
®Ò kh¶o s¸t chÊt lîng HSG líp 9 M«n: To¸n
C©u 1:
2 Gi¶i ph¬ng tr×nh
#### yx −=−
.M.n9_9-0?m.O$
\ 6789:.
( )( )



=++

=+++
4
4

xyyx
yxxyx
$%'
DE.9:..<L2CQ4B#Sy..s..GA.=T(
=.0?M+:Sy..s%(.GA%(=T(0?M.?$
$%+
2 %5`'(5f$3


≥
−
+
yx
yx
\ %2S\ST(=+(\2GJ2=..2-.%7/2d\d`$3


<+++ abccba
$%,&"M('#NEOEB*
%.qQNI>`b'(C.2W.MQNQCI*.,'F
>S$~9P-ACJ2.2-C>y.QN.G)Q)I-C$2C'*A
'F>$
2 ~;.C`5$+8.J2.2-C).Y%b'(5$502%%G.
-.@TF^.$
\ 3HIC.2WS.WC


d)

T*T(=.GT_I*W$
0?%A>J2.2-C>\E.
"
#
=
HK
AH
$
§Ò 22
®Ò kh¶o s¸t chÊt lîng HSG líp 9 M«n: To¸n
1-a, T×m tÊt c¶ c¸c sè tù nhiªn cã 3 ch÷ sè
abc
02%%
16
    

( )





−=
−=





ncba
nabc
'FT(0?MTA:$
\S-0?5SSeM+:.%7/<.3
Qde`5Qe
S%0?M.?9f#$>E.HA0?.ZM02%%.%-'E..n99P
J20?9

Acm0?$3H
%m0?(A.^.#m0??2$
$ %5T(0?.Z.%7/
#
≤≤
x
-.@TF^.'(-.@p^.J2X`5
x−"
dQ#5
+x
$ 2S679:.
x
xx
x
xx
x
=
−−
−
+
++
+

#
#
#
#




\S6789:.





=++
−=−+
=++
4

!

zyx
zxyzxy
zyx
#$ %.2-UC>G2%G.E9.qQ$V=..E9.EJ2.q
y.GC>SC.3.ZR1S[$~;.C1`5BC[`B1[`e$
3H2S

zxyyx =−+
\S

EC
AE
DB
AD
+
I*WI.E9.E1[.2W$
§Ò 23
®Ò kh¶o s¸t chÊt lîng HSG líp 9 M«n: To¸n
C©u1: 1) Cho hÖ ph¬ng tr×nh



=−
=+
byax
ayx#
Q2S\T(.20?
]-@\8T*A8Q5B'F2$
DE.9:..<L2VQB#Sy..s%('(.s..G
-CQ2B'(>QB\02%%2S\T(-0?M.?$
$%'%.q.PN=.E9.2-C>S.E95c'F-G>SCSC>Tx
T_..G1S[Sz$Du1'*A'F[zQ
∈
[z$
3H
HCAHBA
‹
‹
=
$

$%+3HzQ`4

d"iT*2E.%'FM+:$
.^.7-;90?MQ5B.%7/5
#
`
#
d

d$
$%,%2S\S
−≥
'(2
#
d\
#
d
#
`!$
-.@TF^.J2\.3C`2

d\

d

$
§Ò 24
®Ò kh¶o s¸t chÊt lîng HSG líp 9 M«n: To¸n
C©u 1: a) Cho a, b > 0, c
≠

$3H
cbcaba
cba
+++=+⇔=++ 

\6789:.





+=−+−
=+





xyyx
yx
17

$%'2%9

"T(0?M.?02%%9dT(0?M.?$3H
9d2E.%!'(9

dI*97T(0?M.?$
\679:.M5
#

`
#
d5d&
$%+2.^.7-0?M\^.<.3
Q

i5g#
#
i4

ddc'F0?M+:5$
\%5SSeT(-0?.ZI*P'(5dde`$-.@TF^.J2X`
5dede5$
$%,%'*C>1.PN$6)STxT_.T(.J2C>S>'(zT(
.J2$wCI}.<0%0%'F)zy.1.G6$3z6T(
.E9.EJ2.q.PN=.E9.%'*$
Đề 25
đề khảo sát chất lợng HSG lớp 9 Môn: Toán
Bài 1 (2 điểm):
a) Chứng minh rằng nếu b là số nguyên tố lớn hơn 3 và 10b+1 cũng là số nguyên tố thì
5b+1 chia hết cho 6.
b) Tìm nghiệm nguyên của phơng trình x
2
+x+13=y
2
.
Bài 2 (1 điểm):
Cho xyz = 1, chứng minh rằng :

=
++
+
++
+
++ xzzyzyxyx
$
AB+(3 im):bc.\.3
x
x
x
xx
x
x


$

+
+

+
AB,(2 im):-.@TF^.J2\.3
44
# xxxxA ++=
'F
x
$
AB-(2 im): %'*C>1AG:'@$MG>'(1TxT_.T^V
'(02%%VddV`$-%G.<CVSCTxT_.y.r%1>.GtS)$
3-%G.<>tS)S)1Tn9.(\2GJ2=..2-'*$
Đề 26
đề khảo sát chất lợng HSG lớp 9 Môn: Toán
Câu 1: a) Chứng minh rằng với mọi số tự nhiên


n
.0?

#
+
n
2E.%#
d

I*2E.%#
d
$
\.^.7-0?M5S\E.5ff.%7/5
#
d `
#
d 5
$%'679:.
axxxx =+ 444#
Q2T(.20?$
$%+wXH%(.q.PN\-IbI}2.E9.EXCSX>'FCS>
T(-.E9$6T(P'*AG.wCEI>$
2 3HXy.C.G.J2CB
\ C.Y%b'(XN`+$
$%,%2S\ST(-0?.ZI*P.%7/2d\d`$-.@TF^.J2\
.3
( )
++= cbcabP
$
Đề 27
đề khảo sát chất lợng HSG lớp 9 Môn: Toán
Câu 1: a) Chứng minh rằng với mọi số tự nhiên

n
.0?

#
+
n
2E.%#
d

II*2E.%#
d
$
\.^.7-0?M5S\E.5ff.%7/5
#
d `
#
d 5
$%'679:.
axxxx =+ 444#
Q2T(.20?$
$%+wXH%(.q.PN\-IbI}2.E9.EXCSX>'FCS>
T(-.E9$6T(P'*AG.wCEI>$
18

3HXy.C.G.J2CB
+ C.Y%b'(XN`+$
$%,%2S\ST(-0?.ZI*P.%7/2d\d`$-.@TF^.J2\
.3
( )

++= cbcabP
$
Đề 28
đề khảo sát chất lợng HSG lớp 9 Môn: Toán
Câu 1: a) Giải phơng trình:
( )
( )


4$4

#

#
=

+
+
+
y
yy
x
x
$
\6789:.

+=
=+

&!&


xxy
yxyyx
$%'
2 M.<&5i#d!`S/.-A.%G=MHm22
.<5`i'(5`"$
\ 0?.ZMS\E.HI\p\2m0?.n,\M97J2A._0?
FA-.@\H
#
n
$
$%+%-0?.Z5fS
y
.%7/5
#
d
#
`5$-.@TF^.J2\.3
X`5

d

$
$%,%.2-t1%G.E9.q.PN$)}.<C>.E95c'FQN
.GV'(0%0%'F1QC.=t1S>.=t$)}IVJ2.qQN$

62%J2t'FC>T(V

$3H
2 CV

$`>V

$1B
\ CV

`>V$
Đề 29
đề khảo sát chất lợng HSG lớp 9 Môn: Toán
Câu 1:
2 Rút gọn biểu thức:
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bbaaba
ba
bab
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+
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'F2fS\f'(
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$
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Đề 30
đề khảo sát chất lợng HSG lớp 9 Môn: Toán
Câu 1: (2,0 điểm)
2 Rút gọn biểu thức:






+









+


+
+
=
aa
a
aa
P

'F2f'(2

$
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AB'(3,0 im)
2 ?%2GA'*J2=..2-'*T(8J29:.\n2
Qi5

iQi5d`$/5-@-.@J20?%2%3'F
GUJ2.2-T(
"

$
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H.%mn.FG\R-.<.MI*A
M(%.=.<#5d"` $Q~VQ5S_T(ME
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AB+(2,0 im)
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Đề 31
đề khảo sát chất lợng HSG lớp 9 Môn: Toán
Phần 1 (Mỗi câu 5 điểm): HS chỉ cần trình bày vắn tắt cách giải và ghi đáp số.
$ Tìm số lớn nhất trong hai số


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§Ò 32
®Ò kh¶o s¸t chÊt lîng HSG líp 9 M«n: To¸n
C©u 1: (1,5 ®iÓm) Cho biÓu thøc
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§Ò 33
®Ò kh¶o s¸t chÊt lîng HSG líp 9 M«n: To¸n
C©u 1: (2,5 ®Óm)
2 Gi¶i hÖ ph¬ng tr×nh:
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.P.q=.E9.2-C>$3H


$BMAN
S
ABC
=
∆
$Q
ABC
S
∆
T(I‚8
+8..2-C>$
$%,(2,0 điểm)%.2-C>S-2%>1S[B..ECV$‰2C
I}-.<0%0%'F['(>1y.>1'([.3.Z.GX'(‰$6)T(2%
J2CV'(X‰$3CX1).3-=.E9$
$%- (1 điểm) >E.H20?.Z,+^'(.%7/<.3
( )
( )
44#
##
=++++++ yxyxyx
$
/.-.@TF^.J2\.3
yx
Q

+=
$
§Ò 34
®Ò kh¶o s¸t chÊt lîng HSG líp 9 M«n: To¸n

C©u 1. (2 ®iÓm) Cho
aaa
aaa
aaa
aaa
P
4
4
4
4




−+
−−
−
−−
−+
=
2 UI8J22\.3XA•2'(c.X$
\ 2
"<P
$
22

$%'(1,5 im)%2.3Q`
"
i"
#

d4'FM+:$3HQ
2E.%'F-.@M+:J2$
$%+(1,5 im)679:.
#!

+=++ xxxx
$
$%,(1,5 im)%2S\ST(\20?.Z+:$3H
cba
aca
ca
cbc
bc
bab
ab
++
+

+
+

+
+


##

##

##

#
"
#
"
#
"
$
$%-(1,5 im)%.%C>1A
o
A

=
$2C5.G%'F.2C>A>C5\H"
%

'(y.G>.GVSy..<1.G$
3

#
4
ABANAM
=+
$
$%F(1,5 im)670l.3-C>1A.qIC>.E95c'A
.<1$3HEC1>..qI1.E95c'FC>$
Đề 35
đề khảo sát chất lợng HSG lớp 9 Môn: Toán
Bài 1: (2 điểm)
a) Chứng minh rằng nếu p là số nguyên tố lớn hơn 3 thì ( p 1)( p + 1) chia hết cho 24.
b) Tìm nghiệm nguyên dơng của phơng trình: xy 2x 3y + 1 = 0.

Bài 2: (2 điểm) Cho các số a, b, c khác không và đôi một khác nhau, thoả mãn điều kiện
a
3
+b
3
+c
3
= 3abc. Tính:







+

+








+

+

ba
c
ac
b
cb
a
c
ba
b
ac
a
cb
$
AB+(2 im)
2 29:.
## =+ aaxx
A8+^.$
\ %.2.3\n2Q5`25

d\5d.%7/UI8
Q xf
'F
[ ]
Bx
$
-.@TF^.J2\.342

d#\

$

AB,(1,5 im)%A5N'(2CS>TxT_..M2.2N5SNS.%7/NCi
N>`QT(=+(%.3:$3H.<L2..P6J2.2
-C>N'('*A'FC>T*L2=.?@$
AB-(2,5 im)%.2-C>$6
2
S
\
S

TxT_.T(-2%'(
2
S
\
S


TxT_.T(-..EJ2-G>SCSC>Bb'(TxT_.T(\-IJ2
-.q%G.E9'(=.E9J2.2-C>$3H
r
rR
h
m
h
m
h
m
c
c
b
b

a
a
+
++
$
Đề 36
ST Toỏn
Bài 1.
Giải hệ phơng trình





=+
=+
##
##


xy
yx
65

S5

T(8J29:.5

d!5d`'(5
#

S5
4
T(8J2
9:.5

d 5d`$-.@J2\.3C`Q5

d5
#
Q5

d5
#
Q5

i
5
4
Q5

i5
4
i!$
AB'%.qQt=.E9.2-C>.E95c'F-G>SCSC>TxT_.
.GVSSX$6T(P'*AI}.wV5?X$3H
VT(.29P-J2A>$
>?>SS[Sz,.==..q$Q[Sz.3.ZT(2%J2>tSt
'FX$
23
    

AB+%2S\ST(#0?M+:SM.?,2'(.%7/
cba

=+
$3
H.W2d\T(0?9:$
AB,%5SSeS.T(-0?.Z.%7/5

d

g$3H
( )
( )( )



−+−+≥−+ tzyxytxz
$
24