
1.1 Tìm x nguyên để P nguyên
1. 


+
+
x
x
2. 


+
+−
x
x



−
+
=
x
x
P
Bài 4. Cho biê
̉
u thư
́
c:
a 3 3 a


'&

!+#"#
$
%

&!

'&

+#,-

.-/

&!

'&

+#
"

.
Bài 5 Cho biểu thức
1 1
A 1
a 1 a 1
= − −
− +
!01 2345%"6.78'9:(&;.<%=


=+
axa
a
a
x

−
=⇒
H1
/
≥
x
#
/

≥
−
a
a
HI&!
/
≥
/
≥−⇔
a
a
⇔

≤


\4

OJ0
Lời giải :J0S-"S-7I&J+\40J4

B4+B+

0OJ]0
H14

B4+B+

^/7I& :&4Z+#_J]0^+\4^/
Đinh Quang Thành GV trường THCS Ninh Khánh TP Ninh Bình – ĐT 0303503650
1
F`.5%ZO4OA478+\4N4

B4+B+

"T+#US-#!%D(?
`.E)>!V
+\4O784

B4+B+

NJ0
+\4784

B4+B+

⇔
4J+B0BJ+B0/
⇔
J+B0J4B0/
Bài 6. 4

c4+d4c+ce
⇔
4

c4c4+B+B4\\
⇔
4J4\0c+J4c0B
J4\0\
⇔
J4\0J4c+B0\
Bai 7. 4B\4+Be+


⇔
e+

ce4+B4+\4
⇔
e+J+\40B
Bai 8. (y
2
+ 4)(x
2
+ y


=

Do đó có các nghiệm: (0; 0); (2; 2); (2; -2)
Bài 9. +

4B4B+B4

B+

B4+
⇔
4

B4+cJ4B+0c+

J4\0
⇔
4J4B+0c
J4B+0c+

J4\0
⇔
J4B+0J4\0c+

J4\0
⇔
J4\0J4B+c+

0

⇔
4J4B+0BJ4B+0B4B\
⇔
J4B+0J4B0BJ4B0\
⇔
J4B0J4B+B0\
P8&23
 4B+4+
 4+c4B+
 4B4+B+O
4. 4

c4+d4c+ce
5. 4B\4+Be+

6. J+

B*0J4

B+

0e4+

7. +

4B4B+B4

B+

B4+

x+BxfB^x+Bxfjx+^+j
^+^xf/J7w,W0
.F%+^xf^f
H2+.&[ +#US-%X!3.S-'1.J0,8%5%.576%X!JNN0
P8&231 .&[ +#%5%3.S-'1.V
4B+Bf4+f *J4+B+fB4f0*4+f
Phương pháp 3 : Sử dụng tính chất chia hết
.S-3.538+>iUyW.%.o%.&!.Q";%.< &.3.S-'1.7w.&[ 
.F%1 .&[ %X!3.S-'1.
Bài 15. :1 .&[ +#%X!3.S-'1.V
4

\+

J*0
Lời giải :_3.S-'1.J*0!^43.E&,8>?,z.!+4`BJ`.@%G078
J*0Z!"Sa%V
*`

B*`B\+


S-"S-J`

B`\0+


^+

,8>?%.{^+,8>?%.{


\+78f

\f%p%.&!.Q%._"D!%DV4

B+

Bf

\4\+\f%.&!
.Q%.
H1///`.w%.&!.Q%.#4

B+

Bf

\4\+\ft///7I& :&>?+#4Z+Z
f<%,83.S-'1.J0`.w%D.&[ +#
Bài 18. x + x + x = 4y + 4y ² ³ ²
⇔
(x + 1)(x +1) = (1 + 2y) (1)² ²
§Æt (x + 1; x + 1) = d (d ²
∈
N
*
)
Ta cã x + 1
M
d

ULV4J+B0\J+B0S-"S-J4\0J+B0
P8&23J.S-3.530V1 4Z+ G
20. /*OAOAOO
21. 
22. OO
23.Z J4Z+ GB0
24. J4Z+ GB0
25.Z J4Z+ GB0
Phương pháp 4 : Sử dụng bất đẳng thức
sm(o"h.<%";"5.&5 @Y8"D78_>•"5.&58+^%5%&5'6
+#%X!Y8+
Bài 24 :1 .&[ +#%X!3.S-'1.V
4

\4+B+

JA0
Lời giải :
JA0S-"S-7I&J4\+x0

\+

x*
H1J4\+x0

‚/^\*+

x*‚/
^\j+j
Kƒ,Sa.!++\NN\NN/783.S-'1.";W.4!%D%5%.&[ 


*
*
yyx
xyx
„+'!4

+

xyyxxyyxyxyxyx
++>++≥+++=+≥

0J
`.w.k! l3.S-'1.|.<
k

≤
x
78

≤
y
Q4

±
.F%+

±
.13.S-'1.`.w.k! l
HI&4/Z4Z4\!.o+3.S-'1.%D.&[ +#

B+

\4\+eJe0
Lời giải :Je0…^*4

B*+

\*4\*+
…^J*4

\*4B0BJ*+

\*+B0*
…^†4\†

B†+\†

B
P}3.S-3.53.i%.:!.o+*%.‡%DU+.o @UL3.MW%..8.R
%X!.!&>?%.W.3.S-

78


s"D3.S-'1..k! l%.‡'.!&`.E)V
&E&%5%.['#^3.S-'1.Je0%D(?.&[ +#,8J4N+0ˆuN0NJN0N
J\N\0NJ\N\0v
Bài 341 .&[ +#%X!3.S-'1.

Bài 35. Ninh Bình 09- 10. Tìm x, y nguyên thoả mãn:


B

B

CM+,83.S-'1.%.<!Y4+%D7!&'g.S.!Z<%,8Q%D.&[ 4!
78+(.1%p%D.&[ 4(78+!
V
4 + 
4  
+  

− =

− =


− =

⇒
4 
+ /
=


=

.F%
4 /
+ 


=

V
4 + 
4  
+  

− =

− =


− =

⇒
4 
+ 
=


=

.F%
4 /
+ 
=


= −


x
y
=


=

N
/

x
y
=


= −

N 

/
x
y
= −


=


Bài 361 .[ +#%X!3.S-'1.>!


/

Be






=−
=
/
*


yx
x

⇔
.F%





=−
=




c*4B*B+

Bd+BOO
⇔
J4\+0

BJ4\0

BJ+\0

O

B

B

Phương pháp 6 : Lùi vô hạn
Bài 37 :1 .&[ +#%X!3.S-'1.4

\+

/JO0
Lời giải :
&E>iJ4
/
N+
/
0,8.&[ %X!JO0.1V4
/

+

NJ+

ˆG0
_"D!%DV4


\+


/…^4


\+


/
H2+QJ4
/
N+
/
0,8.&[ +#%X!JO0.1J4
/
xN+
/
x0%p,8.&[ +#%X!
JO0
Đinh Quang Thành GV trường THCS Ninh Khánh TP Ninh Bình – ĐT 0303503650
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