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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