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200904


2009 4-

12009

m 2 4mn + 4n 2 = m 2 4n 2


1 __________ 5x 2 x 5
2

2 (2009 ) 5 x 2 3x 5 = 0 5 x 2 2 x

x+ y x2 y2 3(2009 ) 1 _______ x 3 y x 2 6 xy + 9 y 2
2 1 = _________ 3x x + 1 x +1 5(2009 ) 0 x ______________ x 1 1 62009 y = x x 3 1 72009 x______ ׳ x +1
4(2009 ) 1



a 2 + 2a 8 (2009 ) a
a + 2
92009 x



1 ׳ x

x0
102009 a > b > 0 a + b 6ab = 0
2 2

a+b ba



2 112009 0 2
122009 2x 1 x + = x +1 x +1

x2 x 2 0 x x2 + 2x + 1




1 29

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1
2x 1 x + = x +1 x +1

132009



1
142009 x =



1 ׳ x2

2 15(2009 ) 240 1 2 ( )

40 a 1 x ____________ x2

162009

2
17(2009 )

3 1 = 2x x + 3



x = 9
1809 x=



2 x3

׳

3
192009 13 xy xy 0 m =

x y + x y




202009

x2 + 4 x + 4 x = 2 x 4 x2





2 x2
2 29

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21 2009 15ab ab=1 P= P Q = 222009 1 + 12009 A x 1 B

a b 1 1 + Q= + a +1 b +1 a +1 b +1



1 a = a 1 a 1
2

1 x x +1 B x 1 C x 0

D x > 1

2(2009 )8 (a A a b B a + b

b2 a ) a ab


1 a+b



C

1 ab

D

B 3(2009 )4 A4 B4 A 42009 y

a 4 a2 a a a 2 a + 2
C2a D2a





1 1 x x y
C



A

y x

B

x y

x y

D

y x

D 52009

1 1 + a + 1 a (a + 1)
C



A

1 a +1

B

a a +1

1 a

D

a +1 a

C

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3 29

=

a +1 1 a +1 1 = C + = a (a + 1) a (a + 1) a (a + 1) a

62009 A



x2 4 2 x x + 2 x 4x + 4 x + 2 x 2
C



8 x2

B

8 x2

8 x+2

D

8 x+2

D

x2 4 2 x x ( x + 2 )( x 2 ) x 2 x 2 x 2 = x + 2 ( x 2 ) + = 2 2 x x+2 x x x ( x + 2) ( x 2) x 4x + 4 x + 2 x 2
2

( x + 2) ( x 2) = x ( x + 2)
2

2

=

8 D x+2
(ab)2 a 2b Db
2

72009 Aa Bb C1



( ab )
a 2b

=

a 2b2 = b B a 2b

B

8 2009 A
b a

a 2 b2 B a 2 + ab ab a+b B C a a

D b

92009 A

x x+2

xy 2 y x 4x + 4 x y B C x2 x+2
2

D

y x2

D 102009 A
x+3 2 x 2 6x + 9 2x 6


x2 9 2

B

x2 + 9 2

C

D

x3 2

D
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112009 =

x+3 2 x + x + 2 x2 4

( x + 3)( x 2) x 2 x 2 + x 6 x 2 x 2 8 2 = = 2 x2 4 x 4 x2 4 x 4

= ( x + 3)( x 2) + (2 x) = x 2 + x 6 + 2 x = x 2 4 = A B C

1 x+3 x2 x+3 x + 3 1 = = = 1 x + 2 ( x + 2)( x 2) x + 2 x + 2 x+2
C D

122009

a b a b = a b a
C



A

a+b b

B

a b b

a b a

D

a+b a

A 132009 3

a b a b = a b a
C



A

a+b b

B

a b b

a b a

D

a+b a

A

14 2009 Ax1 Bx>1

2 x x 1
C x=1 Dx<1

A 15(2009 )

a b a b = a b a
C



A

a+b b

B

a b b

a b a

D

a+b a

A

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5 29

162009 y = A x 2 B 172009 B x 2

1 x x2
C x 2 D x 2



1 1 + a + 1 a (a + 1)
C

C

A

1 a +1

B

a a +1

1 a

D

a +1 a

182009 A



x2 4 2 x x + D 2 x 4x + 4 x + 2 x 2

8 8 8 C D x2 x+2 x+2 x 3 192009 7 x x+3 A3 B 3 C 3
B A

8 x2

D0

b2 4a 2 202009 + 2a b b 2a
A 2a b B b 2a A 21(2009 ) A3 A C 2a b D b + 2a

x 3 x x+3 B 3 C 3

D0

12009

x+3 3 + x = 1 2 x 9 x 3
=2

x = 1 = 1 22009

4 x3

4 2 x + x = 3 x 16 x 4 x + 4
2

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6 29

=

4 x4 x 2 x x+2 + = + = ( x + 4)( x 4) 2 x+4 x+4 x+4 x+4 5 x = 3 = 7

x2 4 x2 ) 32009 ( x+2 x+2 x


42009 =

x2 4 x+2 x x2 2 2 x 4x + 4 x +1 x 2

= =

( x + 2 )( x 2 ) i x + 1 x 2 x+2 x2 ( x 2)

x +1 x x2 x2 1 = x2
x = 2 2 =

1 1 2 = = . 2 2 2 2 2 1 x2 2 x + 1 i x = 2 x 1 x +1

52009

1 ( x 1) x 1 = = x 1 x +1 x +1
2

x=2 =

x 1 2 1 1 = = x +1 2 +1 3

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7 29

62009







=

3

=

4

=

5

=

6

(x 02 ) y x=1 7

=

=0 8

72009 =

a b + +1 a- b b- a

a b +1 a- b a- b a- b = +1 a- b

=1+1 =2

82009 :

y 3 5 (y + 2 ) 4y 8 y2
8 29

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= y 3 ( y + 2)( y 2) 5 4y 8 y2 y 2

2 y2 = y 3 y 9 = y 3 4 y 8 y 2 4( y 2) ( y 3)( y + 3)

=

1 4( y + 3)

9 2009 6

1 1 x+ y ( x2 y 2 + ) 2x x + y 2x

x =

2y = 3

=

1 1 1 x+ y ( x + y )( x y ) 2x x + y x + y 2x

=

1 1 x y ) 2x 2x

= ( x y )
= yx
x=

2y = 33 2

10(2009 )18 1 = 1 x = 2 = 1



1 x2 1 x = 2 x+2 x+2



1 x2 1 1 = x + 2 x + 2 x 1

(2009 )19 =

2a + 2 a2 1 (a + 1) 2 a 1 a 2a + 1

2(a + 1) 1 (a + 1)(a 1) a 1 a +1 (a 1)2
9 29

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=

2 a +1 a 1 a 1 2 a 1 = a 1 = 1

112009

x2 4 1 1 x = 2 4 x 2 x 2 x + 2x

x2 4 1 2 = x 2 x + 2x
=

( x + 2)( x 2) 1 x2 x( x + 2) 1 x 1 1 = = 4 1 4 4

=

x=

x2 4 x + 4 12 (2009 ) ( x + 2) x = 5 2x 4


( x 2 ) x + 2 (3') = x 2 4 ( x + 2 )( x 2 ) 6 ' = ( ) ( ) 2 ( x 2) 2 2
2

x x = 5

2

2

4 ( 5) =

2

4

2

1 = (8') 2

13(2009 ) 1 _____ _____ 2 100 ֧ z_____ 3

1 8

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10 29

15020 3 2 10

20 x 1 3 1000 50 + 800 30 + 20 x = 8
x 530 x =530 530 142009 =

1 1 4 + 2 x 4 x+2 x2

4 x2 1 2 + ( x + 2 )( x 2 ) ( x + 2 )( x 2 ) x 2
=

x+2 x 2 ) 4 ( ( x + 2 )( x 2 )

=1

5

152009 

8 a2 a+2 + 2 2 a a 2a 4 a

=

a+2 a 8 2 i a ( a 2 ) ( a + 2 )( a 2 ) a 2

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11 29

( a + 2 ) 8a i a 3 = a ( a + 2 )( a 2 ) a 2
2

( a 2) a = i a ( a + 2 )( a 2 ) a 2
2

4

=

1 5 a+2
x+ y x y 1 )i ( 1 ) x y x + y x2 y 2

162009 x=2 3 y=2 3 ( (

x+ y x y 1 ( x + y ) 2 ( x y )2 y 2 x 2 i 2 2 )i( 2 12 ) x y x+ y x y x2 y 2 x y
4xy i

1 4 4 xy x2 y 2
x+ y x y 1 )i( 1 ) 4 6 x y x + y x2 y 2

x2 3 y2 3 (

x x2 + 2x + 1 x 2 1 17 2009 x+2 x+2 x 1 x = 3 2


x ( x + 1) 2 x 1 = x+2 x + 2 ( x + 1)( x 1)
= x =

x x +1 1 = x+2 x+2 x+2 1 3 = 3 32+2

3 2 =

182009 a



1 a 2 2a + 1 a a

a 2 1 (a 1) 2 (a + 1)(a 1) a a +1 = = = 2 a a a (a 1) a 1
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19 2009 1 + =



4 a+2 a a 4
2

a2 a+2 a = (a + 2)(a 2) a a2

a 2 b2 2ab + b 2 a + 202009 2 b = 1 a a ab a
=

(a + b)(a b) a 2 + 2ab + b 2 a ( a b) a

=

a+b a a ( a + b) 2 1 a+b 1 1 x+ y ( x2 y 2 + ) 2x x + y 2x

=

212009 20 6

x =

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

20 =

=

1 1 x y ) 3 2x 2x

= ( x y ) 4
= y x 5
x=

2y = 33 2 6

x2 2 x 2x 4 222009 2 x 2 x = 2 + 2 x 4 x+2


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13 29

=

x( x 2) ( x 2)( x + 2) 2( x 2) [ ] ( x + 2)( x 2) x+2 x+2

=

x( x 2) x( x 2) ( x + 2)( x 2) x+2 x( x 2) x+2 ( x + 2)( x 2) x( x 2)
1 x2

=

=

x = 2+ 2

1 2 = 2 2

232009

x-4 4 x + 2 x= 2 x-2 x -4x +4 x-2
=

x4 x2 4 + 2 x 2 ( x 2) x x4 4 + x 2 x( x 2)

=

x2 4 x + 4 = x( x 2)
=

( x 2)2 x( x 2)
x2 x 2 2 2

=

x= 2 =

=1- 2 242009
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1 1 ( x 2 ) x = 2 1 + 2 x 1 x 1
1 +



1 1 ( x 2) 2 x 1 x 1

=

x ( x + 1)( x 1) x + 2 3 x 1 1

= x 2 + 2 4
x=

2 =

( 2)

2

+ 2 5

= 4 6 1 1 x ) 2 252009 ( 2,1, 1 x 1 x + 1 2x 2
x 25= x-1 x +1 =

1

2 x-1 x +1 x

4 x 4 2 =2 2

x= 2 =

262009 =

a3 5 ( a 2)a = 3 3 2a 4 a 2

5 (a + 2)(a 2) a 3 2 2( a 2) ( a 2)

=

a 3 a2 3 2( a 2) 9 a 2 a 3 a2 4 2(a 2) (3 + a )(3 a )

=

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15 29

=

1 5 2(a + 3)

272009 a =

3 3 =

1 2( 3 3 + 3)

=

3 6

x 2x 3x x = 3 2 x2 x+2 x 4


3 x ( x + 2 ) x ( x 2 ) ( x 2 )( x + 2 ) 2x x 3x = 3 2 2x ( x 2 )( x + 2 ) x2 x+2 x 4
=x+4 5

x =3 =3+4 =7 7

28(2009 )

a 1 2a 1 a a a a

a 1 a 2 2a + 1 a 1 a 1 = = = a 0 1 2 a a a ( a 1) a 1
292009 17 ( x + y )( x y ) 2x + y ( x y) xy2xy x2y x3y2 341 302009

x2 y2 2 x + y x =3 y =2 x y

x2 6x + 9 2x 6 2 9 x2 x + 3x

=

( x 3)2 x( x + 3) (3 + x)(3 x) 2( x 3)
16 29

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=

( x 3) x x = ( x 3) 2 2
312009 =

x2 y2 1 2( x + y ) x = 3, y = x+ y 3

( x + y )( x y ) 2( x + y ) x+ y

= x y 2x 2 y = x 3y x = 3, y = = 3 3 ( ) = 2 322009 a ( 4 = 4a 1 +

1 3

1 3

1 a 1 1 )+ a = a 2a a2 3

= 4a 1 a = 3a 1 1 a = 3 1 = 3 1 = 1 1 = 0 3

a a2 2a 1

332009 =

x y x2 y 2 2y 2 2 x + 3 y x + 6 xy + 9 y x+ y

x y ( x + y )( x y ) 2y 2 x + 3y ( x + 3 y) x+ y

x y ( x + 3 y)2 2y = x + 3 y ( x + y )( x y ) x + y

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17 29

=

x + 3y 2y x+ y x+ y x + 3y 2 y x+ y
3 x )( x + 2) x = x+2 2

=

=1 3422009 (3 =

3x + 6 x x + 3 2 x + 2) = x + 2) = 2 x + 6 x+2 x+2 3 3 x = ,= 2 6 =3 + 2 2

352009 1



1 x2 + 2x + 1 x = 3 x+2 x2 4

x + 2 1 ( x + 1) 2 = x+2 ( x + 2)( x 2)
= = x + 1 ( x + 2)( x 2) i x+2 ( x + 1) 2 x2 x +1

x = 3 =

3 2 5 = 3 + 1 2

a a2 1 3a 36 a 7 1 0 . 01 a a +1 a +1

B 2a a2 1 a +1 a 2a ( a + 1) (a 1) a +1 a 2a1 a 7 1 2a1 2 7 11 2 7 5 . 29
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x2 + 4 x + 4 x + 2 372009 x x = 1 x2 4 x2


( x + 2 ) x 2 x 6 = ( x + 2 )( x 2 ) x + 2
2

= 1 x 7 x = 1 = 1 1 = 0 8 x + 4 x + 4 = ( x + 2 ) x 4 = ( x + 2 )( x 2 )
2 2 2

x+2 x2 = ? 2 x2 x+2

382009 k 2009 2 1 13%K 2 40000 15000 65 1E x 40 000 13% 13%

x

15 000

2 1 1 5

2x

40 000 15 000

13% 13%

40 000 13% 5200
15 00013% 1950

x

40000 13% 5200 2600 2x 2x x 15000 13% 1950 x x

2:

x = 10 8 x = 10 9

40000 13% 15000 13% = 65 7 2x x

2 x = 20
: 20 10 10

39(2009 )17
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1 1 yx x = 2y = 3 2 x y 3x

19 29



1 1 yx 2 x y 3x

=

yx yx xy 3x 2

=

y x 3x 2 i xy y x
3x 3 2 x = 2y = 3 = =2 y 3

=

40(2009 )19 6 ܧԧ 40 3 1 y 2 y MN 6 x 5 4 3 2 1 A A (30 B 0) O 10 2030 B x 1O x 3x

6 6 = + 40 x 3x

x=

1 10

x =

1 1 2 (50 0) 10 10 1 x 41(2009 )18 + = 2 x 3 3 x
1 x = 2( x 3) 3 x = 7

x=

7 3 7 3

x =

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20 29

x =

7 3

422009 20 x = 2008y = 2009

x y 2 xy y 2 x x x

x y 2 xy y 2 x y x 2 2 xy + y 2 x = x x x x
= x y x x ( x y)2 1 x y 1 1 = = 1 x y 2008 2009

=

x = 2008y = 2009 =

43(2009 ) =

x 1 x2 2 x + 1 1 + 2 x+2 x 4 x 1

x 1 x2 4 1 2 + x + 2 x 2x + 1 x 1

=

x 1 ( x + 2)( x 2) 1 + 2 x+2 x 1 ( x 1)

x2 1 + x 1 x 1 x 2 +1 = x 1 x 1 = x 1 =1 =
442009 (
2 a +1 + a+2 a ) atan60 2 a 1 a 1

2sin30 a
www.xuetongedu.com 21 29

=

2(a 1) + (a + 2) a 1 3 i = (a + 1)(a 1) a a +1
3 2 1 3 = 3 = 3 1 = 2 3 1 +1

atan602sin30 =

452009

a 1 2a 1 (a ) a a a

=

a 1 a 2 2a 1 a 1 a 1 i = = 2 a a a (a 1) a 1

462009
2 1 x 1 1+ x x


2 x 1 1 x 1 x + 1 ( x 1)( x + 1) x + 1 1 + = = = x x x x ( x 1)( x + 1) x 1 x

x = 2 =

1 = 1 2 1

472009 =

x2 y2 1 2( x + y ) x = 3, y = x+ y 3

( x + y )( x y ) 2( x + y ) x+ y

= x y 2x 2 y = x 3y x = 3, y = = 3 3 ( ) = 2 482009 a ( 4
www.xuetongedu.com 22 29

1 3

1 3

1 a 1 1 )+ a = a 2a a2 3

= 4a 1 +

= 4a 1 a = 3a 1 1 a = 3 1 = 3 1 = 1 1 = 0 3

a a2 2a 1

492009

x+3 3 + x = 1 2 x 9 x 3


=2

4 x3

x = 1 = 1 502009

4 2 x + x = 3 x 16 x 4 x + 4
2

=

4 x4 x + ( x + 4)( x 4) 2 x+4 2 x = + x+4 x+4 x+2 = x+4 5 x = 3 = 7
a 2 b2 1 a 2 ab a

51(2009 ) a = 2 b = 1 1 + = 1 +

(a + b)(a b) a a ( a b)

=1 + a + b a = 2 b = 1 = 2 52 (2009 ) x



x 1 1 + 2 3 < x < 3 x + 1 x 1
23 29

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x

109

a 2 ab 0 ( a b ) b tan 60 a = 1b = 3 ab

a 2 ab 0 ( a b ) b tan 60 ab
a ( a b) 1 b 3 ab

=

= a b 3

a = 1b = 3 = 1 3 3 = 2
5309
1 x 2

(2

1 x 2

)


1 x 2

( 2

1 x 2

)

=

1 x 2

2

1 x 2

=2

2xy x2 + y 2 5409 M = 2 N = 2 + - M N x y2 x y2
M + NM NN M x y =52 M + N =

2 xy x2 + y 2 ( x + y)2 x+ y + 2 = = 2 2 2 x y x y ( x + y )( x y ) x y

5 y+ y 5 7 2 x y =52 x = y = = 5 2 y y 3 2
M N =

2 xy x2 + y2 ( x y ) 2 yx 2 = = 2 2 2 x y x y ( x + y )( x y ) x + y
24 29

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5 y y 5 2 =3 x y =52 x = y = 5 2 7 y+ y 2
N M =

x2 + y 2 2 xy ( x y)2 x y 2 = = 2 2 2 x y x y ( x + y )( x y ) x + y

5 y y 5 3 2 x y =52 x = y = = 5 2 y+ y 7 2


3 x 2 > x + 2 x + 1 x2 1 552009 1 2 2 1 3 x x 2 x 1 7 2 x.


x +1 x2 1 = i x ( x + 1)( x 1)
= x 4 x 1

3 x 2 > x + 2 2 1 3 2 x 1 7 2 x
x > 2 x 4 2 < x 4 4 562009 16

1 1 xy + 2 2 x y x+ y x y

1 1 xy x + y + x y x2 y 2 2 + 2 = = 2 ( x y )( x + y ) xy y x y x+ y x y

572009

x2 + 2x 2 2 x 4 x2


www.xuetongedu.com 25 29

=

( x + 2 )( x 2 )

x ( x + 2)



2 x2

=

x 2 x2 x2

=1

582009 M

2 xy x2 + y2 N 2 x2 y2 x y2

+ MN,

M+NM-NN-M xy=5 2 M + N =

2 xy x2 + y 2 ( x + y)2 x+ y + 2 = = 2 2 2 x y x y ( x + y )( x y ) x y

5 y+ y 5 7 2 x y =52 x = y = = 5 2 y y 3 2

2 xy x2 + y2 ( x y ) 2 yx M N = 2 2 = = 2 2 x y x y ( x + y )( x y ) x + y
5 y y 5 2 =3 x y =52 x = y = 5 2 7 y+ y 2
N M =

x2 + y 2 2 xy ( x y)2 x y 2 = = 2 2 2 x y x y ( x + y )( x y ) x + y

5 y y 5 3 x y =52 x = y = 2 = 5 2 y+ y 7 2

1

1 592009 | 3 2 | +2009 + 3 tan 30 3
0

=

2 3 + 1+ 3 + 3

=6
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3 3

26 29

36 a 2 6a a+5 602009 2 i a = 2 2 a + 10a + 25 2a + 10 a 2 + 6a
=

(6 + a )(6 a ) 2(a + 5) a + 5 i i (a + 5) 2 6 a a (a + 6)

=

2 a

a = 2 2 =

2 2
a2 b2 2ab + b 2 a + a 2 ab a b = 1 a

612009 =

(a + b)(a b) a 2 + 2ab + b 2 ( )..............................1 a ( a b) a

= =

a+b a ......................................................1 a ( a + b) 2

1 .....................................................................1 a+b a a 0 a 1 1 , 1
62(2009 ) x = 2008y = 2009

x y 2 xy y 2 x x x

x y 2 xy y 2 x y x 2 2 xy + y 2 x 2 = x x x x
= x y x 4 x ( x y)2 1 5 x y 1 1 = = 1 7 x y 2008 2009

=

x = 2008y = 2009 =

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27 29

63

x y x2 y 2 2y 2 2 x + 3 y x + 6 xy + 9 y x+ y

=

x y x 2 + 6 xy + 9 y 2 2y 2 2 x + 3y x y x+ y
2

( x + 3y) 2 y x y x + 3 y ( x + y )( x y ) x + y x + 3y 2y = x+ y x+ y
= =

x+ y =1 x+ y

64 ( x = [

2x x2 2 x + 1 ) x +1 x2 1

x( x + 1) 2 x ( x 1) 2 ] x +1 x + 1 ( x + 1)( x 1)

=

x( x 1) ( x + 1)( x 1) ( x 1) 2 x +1

=x
652009 x 2 = 0
2

( x 1) 2 x2 + x2 1 x + 1

=

( x 1) 2 x2 + ( x 1)( x + 1) x + 1

=

x 1 x2 + x +1 x +1 x2 + x 1 x +1

=

x2 2 = 0 x2 = 2
www.xuetongedu.com 28 29

=

=1

2 + x 1 x +1

662009 x 2 = 0
2

( x 1) 2 x2 + x2 1 x + 1



( x 1) 2 x2 = + ( x 1)( x + 1) x + 1
x 1 x2 + = x +1 x +1
=

x2 + x 1 x +1

x2 2 = 0

x2 = 2 = =1 2 + x 1 x +1

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