100 đề thi học sinh giỏi toán lớp 8 có lời giải

x2 7 x 6
x4 2008x 2 2007 x 2008

x 2 3x 2
8 x

1
x

2

x 1
4 x2

0

1
x2

2

4 x2

1
x2

x

1
x

2

x 4

2

64

4

x 2 x 4 x 6 x 8

x2 10 x 21

m

6

AB
GB
BC

HD
AH HC

2008


S

hai c

can, m

c v

k

N

b

M

th

th nh

s

th nh

ph n

bao nhi u k .

nh

ph
t

c

b

.

minh r

ph n

1
s k
11

(4

c

chia h

l

H

4 ch gi

1

1- 6 m

v chia

tr

1c

m

k

,b

sao cho AD = BC. T

b

1

.

1 x

2
1 x2

cho m

ng

ph n

cu

l

thi

nh

ph

th m

2c k
c

4
1 x4

ni n

c

1
s k
11

v

tr n c bao nhi u

v s d

s t nhi n n.

8
1 x8

t

m

s k .S

m

ng

nh sau:

th m

1
s k
11

trong ph n
k

3 ch s gi

ng kh ng
m

th n nh

ABC c n t A, c g

BDC

s g

ch s th th

th hai nh

ni n n

g

im

s rn

trong tay ch c

9 l v kh ng can n

nhau chia cho m

v

l .

Cho tam gi

can c dung t

xu c

nh th

thi

t s ng v n

n3 n chia h cho 6 v

1 x

Nh n ng

l . C ti

th m

s g

ng l 16 v s d l m

Ch

B

?

6l n

4l ,m

s b chia v s chia

(4

n

em

can c dung t

200. T

nguy n t

c

th n

chia dung t

nhau th
gi

L

-2009

c

l
nc

.

b

l . Sau khi b

k

v

vi n v m
A = 200. Tr n AB l

l

vi n nh
i

D


IL C

THI H C SINH GI I L

1 : (5

a)

2015 2014
2015 2014

x

(5

y

2

y

1

z

2

b)
c)

(
a) X4 + 6y2 -7 = 0
1
2011x 1

z

1

x

2

2014 2
2014 2

: (x2 8)2 + 36
ng minh :

l

am ts h ut.

ng th c sau :

1

2012 x 2

c2
a2

c : x19 + x5

1
2013x 4

1
2014 x 5

a
b

Cho tam gi

A

2B

b
c

x1995

G
i AE. Ax c t CD t i F. Trung tuy n AI c
ng th ng qua E song song v i AB c t AI G.

a) Ch
2
b) Ch ng minh : AEF ~
= FK.FC.
c)
i.
12cm.

(4

2015 2
2015 2

a2 b2
b2 c2
2
nh nh t c a A =
6x 5 9x 2

a) Ch ng minh b

b)

: H Th Song

m i bi u th

2

b)

c) Cho ba s h u t
1

Th

H

NGH

c 2013-2014)

c
a

c x2 -1

Qua E k tia Ax
t CD K.

t AC = 9cm, BC =


- 2014

150

a 3 4a 2 a 4
a 3 7a 2 14a 8

x

1969

1971

2

1
9 x 20

x

2

1
11x 30

x

2

1
13x 42

2 1970

(

I
c) Tam

HA'
AA'

HB'
BB'

HC'
CC'

IM, IN

(AB BC CA ) 2
AA' 2 BB' 2 CC' 2

1
18


2014)

8

1)
2)

: (3

: (4

x 2
2010

nh:
:(

a.

5: (7

b)
c)

3x

x2
27 3x2

:

1

x 3

)

2)

a)

x

2

3

1

2 : (2
1)
2)
1)

1
3

A

)

1
x 2 yz
2

y

2

x 3
2009

1

2 zx

x 4
2008
1
z 2 xy

x 2010 x
2
1 1
x y
0

2

2x 6
x 3 27

3x 2 8x 6
x 2 2x 1

D = CB.CK

2

t

.

-3

2009
3
1
0.
z

x 2008
4


- 2014

7

1

b)

b)

1 x

: x(x + 4)(x + 6)(x + 10) + 128

+ x5 + x3 + 1 cho x2 1

3 - 4x

x2 1
2
5 x 1 2x
:
x 1 1 x2 x2 1
a
ab + a + 2

b1) n2 n + 2

1
b
2c
bc + b + 1 ac + 2c + 2

b2) n5 n + 2

a) C
Cho ABC
OA'
AH

OB'
BK

OC'
CI

1

---------------------------

AB).
------------------------------


- 2014)

.
b

5x2 + 8x

3

A = 10x2

c)

y

x

3

1

x

7x

3

y

1

3.
2 x

. Gi
a) (x + x)2 + 4(x2 + x) = 12
b
c) x 2 x 3 x 4 x 5

x y
2

2

y

3

0

2

2007 2006

2005 2004

.

4
Ox),

2

.
.

AOB =

8a 2
3

CA
DB

OC 2
OB 2

; DB theo a.

+10x+21.


-2014

:(

(a2 + 1)(b2 + 1)(c2
A (1

:(

1
1
1
1
)(1
)(1
)...(1
)
2
2
2
x
( x 1)
( x 2)
( x 9) 2

)

P( x)

2x2 2x 3
x2 x 2

f(x) = x1994 + x1993 +1 cho g(x) = x2 1
:(

:(

n

)

x 2
x m

x 1
x 1

a
b c a

b
a c b

15n 1 225

- |x - 3| =14
c
a b c

3

u 5: (
CA
DB

OA2
OB 2
8a 2
3


N

GV:

1)Cho

(6

):
:

P

2
3x

2
x 2y
x 2y
x 2y
3x

(4

b)

):

:

x 2005 4 x 8038
9
18
x

2

1

:

x 2y
x

c )2 = a2 + b2 + c2 + 2ab 2ac 2bc.
2
+ y2 4 ( x y )

b) Cho

a)

2013-2014

x

x

2

:

1
3x 2

x

2

2 x 4004 3x 6022
24
20

1
5x 6

1

x 3

1
2013

a)
b)
4
BC = 3BM; AC = 3AN.
2

c BPQ?
C
(2
2)

= IN.IC

):
2

----------------

----------------

.

8


- 2014)
:
(3

x4

b/ Cho

)

30x2 + 31x

30

(a - b)2012 + (b - c)2013 + (c - a)2014
(4

)

n

A = n + 4n - 20n - 48
36
8
7
6
A = n + 4n + 6n + 4n5 + n4
3

(5

2

)

x m
x 1

:

M

x2

x 2
x 1
2x 2014
x2

x

0

0

FED.
5: (5,5

a/

b/ CH.CD = CB.CK

c/ AB.AH + AD.AK = AC2.

?

16

n


3-2014
-

)

1 x3
1 x

1 x2
x :
1 x x 2 x3

2
1 .
3

)

Cho a b

2

b c

c a

2

2

4. a 2 b2 c2 ab ac bc .

a b c.

)

)

a 4 2a3 3a 2 4a 5 .

)

0

a,
)
N.

SABCD.

AOB=

20132

1
AB

1
CD

2
.
MN

COD=

20142


3-2014)
.
1
2.5
3

1
1
........
5.8
(3n 1)(3n 2)

+ 6n2 -19n

x2 2y2 -2xy +6y

3

A= (x+1)(x+3)(x+5)(x+7) +2028

cho x + 8x +12
2

4,5

a/

2

c/
:

- 3b2

4

x 1
2012
2a
3a

x 2
2011
b 5b
b 3a

2

x 3
2010
a
b

b2

+ 2013x2 + 2012x + 2013
x 4
2009

x 5
2008

x 6
2007

0

x2y + y2z + z2x +zx2 +yz2 + xy2 x3 y3 z3 > 0
)

KM
b/
KN

DM
DN

2

.

c/ AB.AE + AD.AF = AC2.

2

2


_________
180
_
___
QUANG TRUNG

: (4
1.
2.

x2 7 x 6
x4

2014 x 2 2013x 2014

: (4
2
1. x 3x 2

1
2. 8 x
x

: (4
(a+b+c)( 1
3.

a

2

1
b

x 1

4 x
1
)
c

2

1
x2

2.
3.

4 x

2

1
x2

1
x
x

2

x 4

2

9

x 2 x 4 x 6 x 8

: (8
AH (H
1.

2

0

m

x 2 10 x 21 .

2008

AB .
GB
BC

HD
.
AH HC


-2014

a4

8a 3 14a 2

8a 15

10 n 18n 1

Khi ta
A= a

4

a/

2a 3

3a 2

4a 5

3x
x 2

x

5 x

3x
x 2 x 5

0.

S A0 B

S ABCD

20082

:

1
AB

1
CD

2
MN

SC 0 D

2009 2

0


-2014)
.
1
2.5
3

1
1
........
5.8
(3n 1)(3n 2)

+ 6n2 -19n

x2

3

A= (x+1)(x+3)(x+5)(x+7) +2028

cho x + 8x +12
2

4,5

a/

2

c/
:

- 3b2

2y2 -2xy +6y

4

x 1
2012
2a
3a

x 2
2011
b 5b
b 3a

2

x 3
2010
a
b

b2

+ 2013x2 + 2012x + 2013
x 4
2009

x 5
2008

x 6
2007

0

x2y + y2z + z2x +zx2 +yz2 + xy2 x3 y3 z3 > 0
)

KM
b/
KN

DM
DN

2

.

c/ AB.AE + AD.AF = AC2.

2

2


-

3-2014)

(3

s

a/
b/

2: (2
(3

- 1) - ( x + 2)2 = 3

Gi

a/ x2 3x 2 x 1 0
1
b/ 8 x
x

2

4 x

2

:

1
x2

:
2

4 x

2

1
x2

1
x
x

2

x 4

2

(2
P(x) = x3 + ax2 +bx+c chia cho x-1; x-2; x5: (6
a)
b)

)

K

6: (4
a)
b)

2

.
h BC.


a, a3 + b3 + c3 3abc

b, (x-y)3 +(y-z)3 + (z-x)3

1 x3

x(1 x 2 )2
Cho A =
: (
1 x
1 x2

1 x3
x)(
1 x

a, Cho x+

x)

1
2
2

x
( x 10) 2
a

a, Cho a,b,c > 0, CMR:

1<

b, Cho x,y

x2 y 2
+

y 2 x2

0 CMR:

Cho ABC
b, CMR: AM

AB

a b

+

b

b c

x y
+
y x

+

c

c a

+ y2 + z2

<2

ACM
MNP


A

a 1 a 3 a 5 a 7

x a x 10

B( x )

P

A

15

1
x4 3x3 ax b

x 2 3x 4

1
22

1
32

1
44

...

1
1002

1

a 1 a 3 a 5 a 7

a 2 8a 7 a 2 8a 15
a 2 8a

22 a 2 8a

2

a 2 8a 11

2

1

15

15

120

a 2 8a 12 a 2 8a 10

a 2 a 6 a 2 8a 10

x a x 10
x2

1

x m x n ;(m, n Z )

a 10 x 10a 1 x 2

m n a 10
m.n 10 a 1

mn 10m 10n 100 1
m(n 10) 10n 10) 1

m n x mn


1

a) Cho x 2

2xy 2y2
A

2

3

A

a b
c

2x 6y 13 0
a3

b c
a

b3
c a
b

3x 2 y 1
4xy

N

c3 3abc
c

a b

a

b c

b

c a

9

4
.
5

x6

3x 2

1 y4


a(b c) 2 (b c) b(c a) 2 (c a) c(a
1 1
a b
1
N
2
a 2bc b 2
M

x2

y2

b) 2 (a b)
1
0
c
1
1
2
2ca c 2ab

xy x y 1
( y 4,5) 4 ( y 5,5) 4 1 0

3x 2

5y 2

345


C u1.

a. Ph n t ch c

b. Gi i ph
c. Cho
C u2.

Cho bi

a th

sau ra th

s :

ng tr nh:

. Ch ng minh r ng:

u th c:

a. R t g n bi u th c A.

b. T nh gi tr c a A , Bi t x =

c. T m gi tr c a x A < 0.
d. T m c c gi tr nguy n c a x

.
A c gi tr nguy n.

C u 3. Cho h nh vu ng ABCD, M l m t i m tu
n
ch BD. K ME
MF AD.
a. Ch ng minh:
b. Ch ng minh ba

g th : DE, BF, CM ng quy.
c. X c
v tr c a i m M di n t ch t gi c AEMF l n nh t.
C u 4.

a. Cho 3 s d
b.

AB,

ng a, b, c c t ng b ng 1. Ch ng minh r ng:

Tinh: a

2011

H

+b

2000

2011

+ b2000 = a2001 + b2001 = a2002 + b2002
i

a. x4 + 4 = x4 + 4x2 + 4 - 4x2
= (x4 + 4x2 + 4) - (2x)2
= (x2 + 2 + 2x)(x2 + 2 - 2x)

(6
b.

( x + 2)( x + 3)( x + 4)( x + 5) - 24
= (x2 + 7x + 11 - 1)( x2 + 7x + 11 + 1) - 24
= [(x2 + 7x + 11)2 - 1] - 24
= (x2 + 7x + 11)2 - 52
= (x2 + 7x + 6)( x2 + 7x + 16)
= (x + 1)(x + 6) )( x2 + 7x + 16)
<=>

(*)

(2
(2


a 3 4a 2 a 4
a 3 7a 2 14a 8

3.
P=(x-

x

A=

2

a
b c a

1
9 x 20

x

b
a c b

2

1
11x 30
c
a b c

x

2

1
13x 42

1
18

3
0

minh :

BC 2
a) BD.CE=
4

a) (1,5) a3 - 4a2 - a + 4 = a( a2 - 1 ) - 4(a2 - 1 ) =( a2 - 1)(a-4)
=(a-1)(a+1)(a-4)

0,5

=( a -2 )(a2 - 5a + 4) = (a-2)(a-1)(a-4)

0,5

a3 -7a2 + 14a - 8 =( a3 -8 ) - 7a( a-2 ) =( a -2 )(a2 + 2a + 4) - 7a( a-2 )

quay


(3
a) x2 4x + 4 = 25
b)

x 17
1990

x 21 x 1
1986 1004

4

c) 4x 12.2x + 32 = 0

A

HA'
AA'

I

HB'
BB'

yz
x 2 yz
2

HC'
CC'

xz
y 2xz
2

1
x

1 1

0.
y z
xy
2
z 2xy

IM, IN

(AB BC CA) 2
AA' 2 BB' 2 CC' 2

c) 4x 12.2x +32 = 0
2x.2x 4.2x 8.2x
2x(2x 4) 8(2x 4) = 0 (2x 8)(2x
(2x 23)(2x 22) = 0 2x 23
x
2x = 2 3
= 22
1
x

1
y

1
z

0

xy

yz xz
xyz

0

xy

yz xz

x

22

0

yz = xy


)

1 x3
1 x

1 x2
x :
1 x x 2 x3

2
1 .

3

)

Cho a b

b c

2

2

g

)

(

c a

2

4. a 2 b2 c2 ab ac bc .

a b c.

)

a 4 2a3 3a 2 4a 5 .

)

0

b, Cho AB
)
N.
AOB=

ng

20082

1
AB

1
CD

2
.
MN

COD=

)
A=

1 x

-

x x2
(1 x)(1 x)
:
1 x
(1 x)(1 x x 2 ) x(1 x)

3

20092

ABCD.


B i 1:

Cho x =

a 2 (b c) 2
b2 c 2 a 2
;y=
(b c) 2 a 2
2bc

B i 2:
a,

1
1 1 1

= + +
a b x
a b x

(b c)(1 a)2
(c a)(1 b)2
(a b)(1 c)2
b,
+
+
=0
x a2
x b2
x c2

B i 3:
(3 x 1)
a
b
=
+
3
3
( x 1)
( x 1) ( x 1) 2

B i 4:
2x2
B i 5:
Cho

ABC; AB = 3AC