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81 phương trình, hệ phương trình hay và khó

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Sưu tầm:
Hồ Văn Diên – THPT Thái Lão


1
PHƯƠNG TRÌNH, BẤT PHƯƠNG TRÌNH VÀ HỆ PHƯƠNG TRÌNH

1.
( )
2 2
1 2 3 1x x x x+ − + = + 2.
2 2
1
1 1 2
2
x x x− − = −
3.
2 2
2 1
2 1
1 1
sin cos
cos
sin
n n
m
m
x x
x
x
+


+
+ = + với ,m n là các số nguyên dương
4.
1 3
1 0
4 2
x
x x
+
− =
+ +
5.
3
3 2x x x− = +

6.
4 3 2
2006 1006009 2 2007 1004 0x x x x x+ + + − + + =

7.
( )
( )
( )
( )
( )
( )
2
2 2 2 2
2
2 2 2 2

2
2 2 2 2
3 1
4 1
5 1
x y z x x y z
y z x y y z x
z x y z z x y

+ = + +


+ = + +


+ = + +


8.
( )
( )
( )
2 2
2 2
2 2
6 1 9
6 1 9
6 1 9
x y x
y z y

z x z

= +


= +


= +



9.
3
3
6 1 8 4 1x x x+ = − −
10.
4
6
2
cos2x
3 1 4tan 7
cos
x
x
 
+ + =
 
 


11.
( )( )
2 2 2
4
3 2007 2005 4 4 30 1 2006x x x x x x x x− + + − − = + − +

12.
( )
( )
2
2 2
1
2z
8
3
3 4 z 2
4
0
0 ,
x xy y
x y xy x yz
x y z
x y z

+ + =




+ + + + =



+ + =


< <

13.
( )( )
2 3 2 3
2
6 6 5 2 6 4
2 2
1
x x x x x x
x
x x

− + = + − +


+ ≥ +




14.
3
3 3 0x x− + =
15.

2006 2005
2005 2006 1x x− + − =
16.
( )
4
4
16cos 3 2048cos 768x x+ = −
17.
3 2
3
2 1 1 3 2x x x x+ + = + +

18.
2 2 2
3 3 3
7
37
1
x y z
x y z
x y z
+ − =


+ − =


+ − =

19.

3
3
2 3
1 3
82
y x
x y

− + =


+ =



20.

6 5 4 3 2
16 16 20 20 5 2 7 0x x x x x x− − + + + − =

21.
( )
2 2 2
11 14 9 11 2 3 17 2 3 2 2 4x x x x x x x− + + − + + + + = +
22.
2 2 2
45
2 5 4 40 5
4
x x x x x x+ + − − + = + +


23.
2
3
2 11 21 3 4 4 0x x x− + − − =
24.
2
2 3 3x x x− − = +

25.
( )
( )
( )
2
2
2
4 8
4 8
4 8
x y y
y z z
z x x

− =


− =


− =



26.
2 2
2
4 6
2 8 3 7
x y xy
x y x
+ + =


+ = +


27.
( )
2 2
2 1 2 1 2 3 0x x x x x x+ + + + + + + <

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Sưu tầm:
Hồ Văn Diên – THPT Thái Lão


2
28.
2
35
12

1
x
x
x
+ >

29.
2
2
2
2
6 5
x x
x x
+ >
− + −

30.
2 2
1 13 1 13
16 16
97
36
0
0
y x y x
x x
x y
x
y


+ + + − = + +



+ =


<


>


31.
1
1
1
1
1
1
y
x
x
z
y
y
x
z
z


+
= +



+

= +



+

= +



32.
2 2
2008 2007
2008 2008
2007
x
x
x
+ − − =


33.

( ) ( ) ( )
2
2 7 2 7
log log 3 log 2log 3
2
x
x x x x x
 
+ + = + +
 
 

34.
( )
( )
2
2 2
2
3 2
9
2 6ln
9
2 1
y y
x y x y xy
x x
x x y

 
+ +


 
− + + − =

 

+ +
 

− + =



35.
4 4
2 2
16 1
8
2 8
x y
x y
x xy y

− −
=



− + =


36.
cos cos
2
1
1
1 2
x y
x
e
y
y x x

+

=

+


= − −


36.

( )
3
1 log 3
1
1
x

x x
+ +
>
+

37.
cos5 cos sin3 cosx x x x+ = −

38.

(
)
( )
(
)
2 2
3 2 9 3 4 2 1 1 0x x x x x+ + + + + + + =

39.

( ) ( )
2
8 8
4 4
2 6 6 2
log 4 .log 8log .log
2
x
x x x x x x x
 

+ + = + +
 
 

40.

3 2 1
0
x y x y
x y x y

+ − + = −


+ + − =



41.

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

− = −



− = −



42.

( )
2
2 2
3 5 2 2 3 .2 -3 5 2 2 .3
x x
x x x x x x x− − + + > − + +

43.

( )
3
2cos3 6cos 1 162cos 27x x x+ + = −

44.

2
2 1 3 1 0x x x− + − + =

45.

2
2 3 1 3 2 2 5 3 16x x x x x+ + + = + + + −

47.


3
7 1x x+ = +

48.

( )
2
3 3
1 log 4 log 16 0x x x x+ + − =

49.

( )
( )
( )
( )
2 2
2 2
3
15
x y x y
x y x y

− − =


+ + =




50.

( )
( )
2
2
2 3 18
5 9 0
x x x y
x x y

+ + =


+ + − =



51.

( )
1 ln 1
x
e x= + +

52.

( )
3

3
1 1 2x x+ + = +

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Sưu tầm:
Hồ Văn Diên – THPT Thái Lão


3
53.

2
2 2
6 3 1
1
x xy x y
x y

− + + =


+ =



54.

2
2
18 12

9
3
xy x
y
xy

− = −


= +



55.
( )
4 12 2 11 0
x x
x x+ − + − =

56.
1 2 1
4
4 3.4 2
3 2 log 3
x y y
x y
+ − −

+ =


+ = −


57.

2
tan
cos 2
x
e x+ =
trên ;
2 2
π π
 

 
 

57.

1 1 1
1
x
x
x x x

− − − >

58.
2 2 2

3 2
2 0
2 3 6 12 13 0
x y x y
x x y x

− + =


+ + − + =



59.

( )
( )
( )
2
3
2
3
2
3
2
30
16
x x y z
y y z x
z z x y


+ − =


+ − =


+ − =



60.

5 3
15 11 28 1 3x x x+ + = −

61.
4 3 10 3 2x x− − = −
62.

( ) ( )( )
( )( ) ( )( )
2 2 4
2 7 3 3 2 7
x y x y
y x y x
− = + −




+ − = + −



63.

4
4
4 4 1 1x x= + +

64.

( )
2
2cos 2 3 sin cos 1 3 sin 3 cosx x x x x+ + = +

65.

( ) ( )
3 3
2 2
1 1 1 1 2 1x x x x
 
+ − − − + = + −
 
 

66.
1 3
2

1 5
x
x
x x
+
= −
+ − −

67.

( )
2
2 3x x − =

68.

3
3 2x x x− = +

69.

3 2
3
16 24 12 3x x x x− + − =

70.

( )
7 3
2log log 2t t= +


71.

( )
cos6 cos4 4 1 cos3x x x− = +
72.

2 2
3 3 2
3 3
1 1 81
sin cos cos 4
sin cos 4
x x x
x x
   
+ + + =
   
   

73.

( )
sin 4 cos4 1 4 sin cosx x x x− = + −
74.

( ) ( )
3
3 2 2
1 2 1x x x x+ − = −


75.

2 2 3 2
3
4 10 2 3 2 9 4 4x x x x x x x− + + = − + − +
76.

2 2
2 3 2
8
16
2
8 3 3 4 2
xy
x y
x y
x x x x y
y y

+ + =

+



+ = + −




77.

4
4
2
1
2
4
2
1
1
4
x y
x
x y
x y
y
x y

 
+
+ =

 
 
+

 

 

+

− =
 

 
+
 


78.

3 2 3
3
4 12 9 6 5
xy x y
x x x y y
− + =


+ + = − + +


79.

2 6 2
2 3 2
x
y x y
y

x x y x y

+ = − −



+ − = + −


80.

2 2 3 4
4 1 1 5 4 2x x x x x x+ + = + + − −

81.

2
3
6 1 1x x x+ + − = −


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