Phuong trinh nghiem nguyen


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

M
2 a 6 2 a 6
+ −
= −
− +
vơ
́
i
a 0;a 9.≥ ≠







!"#
$
%

&!

'&

()

*


&!

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

(01 %5%>?+#?!";&5'6(&;.<%=,8 @>?+#.
1.2. Tìm x để P nguyên.
Bài 

A
+
+
x
x
B


+
x
C;%D&5'6+#.1


+
x
3.E&,8>?+#
!"F


+
x
!
∈
G
⇒


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

HI&!
/
<
/
≤−⇔
a
a
⇔

≥
JKL&0
HM+
/
≤<
a
!
∈
G#!N!HI&!.14O7I&!
.14
*

D ,L&%D&5'6+#.14O784
*

1.3 Tìm căp số x, y nguyên
Phương pháp 1 : Đưa về dạng tích
P&Q"R&3.S-'1.7TULV7Q'5&,8W%.%X!%5%"!.<%%.<!YZ7Q3.E&,8W%.
%X!%5%>?+#
Bài 1 :1 .&[ +#%X!3.S-'1.V
+


\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+


ONJ0
+\4784

B4+B+

ANJ0
+\4A784

B4+B+

NJH0
CQ"M+Z(8&5%&.S"Sa%&E&b+Q
P8&1 .&[ +#%X!3.S-'1.
...
Bài 34B+4+
⇔
4+c4c+B
⇔
4J+c0\J+\0
⇔
J+\0J4\0/
Bài 44+c4B+c
⇔
4J+\0cJ+\0
⇔
J+\0J4\0
Bài 54B4+B+O
⇔
4B4+B+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

2
) = 8xy
2
⇔ (xy – 2y)
2
+ (y
2
– 2x)
2
= 0
2
2
y 0
xy 2y 0
x 2
y 2x 0
y 2x

=

− =



=
⇔ ⇔
 

− =



=

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

Bài 10. *4
*
Be4

+B+

c*+c/
⇔
*J4
*
B4

+B+

0cJ+

B*+B*0

⇔
*J4

B+0

cJ+B0


⇔
Bài 11*4

B4+B*4B+B/

⇔
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+

8. *4
*
Be4

+B+

c*+c/
Đinh Quang Thành GV trường THCS Ninh Khánh TP Ninh Bình – ĐT 0303503650
2
Phương pháp 2 : Sắp thứ tự các ẩn
Q%5%Y4Z+ZfZ%D7!&'g(1."hZ!%D.;&E>i4j+jfj.F%Sa%
,L&";1 %5%.&[ .k! l"&T`&[8+_"DZUm3.n3.576";^%5%
.&[ %X!3.S-'1."l%.
ZQ%5%Y%D%o'9%&?.!.S,p+._!%m(2%Z%5%>?+#,&#&Q3.1
!>q`.iY";"S!7TULbr.@%.F%WY.-
Bài 11. :1 .&[ +#US-%X!3.S-'1.V
4B+Bf4+fJ0
Lời giải :
s7!&'g(1."h%X!4Z+Zf'3.S-'1.Z'SI%.Q!4n4j+jf
H14Z+Zf+#US-#4+ft/ZU4j+jf^4+f4B+Bfjf^4+j
^4+.@%uNNv
Q4+^4+Z.!+78J0!%DVBffZ7w,W
Q4+ZU4j+#478+Z.!+78J0Z^f
Q4+ZU4j+#478+Z.!+78J0Z^f
H2+.&[ +#US-%X!3.S-'1.J0,8%5%.576%X!JNN0
Bài 12. :1 .&[ +#US-%X!3.S-'1.V
x4Bx+BxfJ0
Lời giải :s7!&'g(1."h%X!4Z+ZfZ'SI%.Q!4n4j+jf!%DV
x4Bx+Bxfjx4^4jx^4
.!+478J0!%DV

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>?%.{

CF+J.@%G0Z!%DV
J`

B`\0*


S-"S-`J`B0

BJ]]0
.24nV`J`B0,8>?%.{Z

B,8>?,z^3.S-'1.J]]07w.&[ 
H2+3.S-'1.J*0`.w%D.&[ +#
Bài 17. :|.< &.'}`.w~L&%5%>?+#4Z+Zf.k! lV
Đinh Quang Thành GV trường THCS Ninh Khánh TP Ninh Bình – ĐT 0303503650
3
4

B+

Bf

4B+BfB///J0
Lời giải :!%D4

\4J4\04J4B0,8W%.%X!>?+#,&#&Q3J7I&4,8>?
+#0s"DV4

\4%.&!.Q%.
S-•+


\+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

⇒
x + x ²
M
d
⇒
(x + x) – (x + 1) ² ²
M
d
⇒
x – 1
M
d
⇒
(x + 1) – (x – 1)
M
d
⇒
2
M
d (2)
Tõ (1) ta cã x + 1 vµ x +1 ®Òu lµ sè lÎ (3)²
Tõ (2) vµ (3) ta cã d = 1 (4)
Bài 19. 1 .&[ +#%X!3.S-'1.V
4+B4\+Jd0
Lời giải :!%DJd0S-"S-+J4\0\4BH14`.w.k! l3.S-
'1.#Jd0S-"S-7I&V
+J\4B0xJ4\0S-"S-+\BxJ4\0
!.o+V+,8>?+#S-"S-7I&4\,8SI%%X!.!+4\.F%4\
\S-"S-7I&4.F%4_"D!%D.&[ J4N+0,8JN\078JN/0
|.9€V|D.;Um3.S-3.53";&E&(8&58+Z.•"S!3.S-'1.Jd07T

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%.&[ 

+#%X!3.S-'1.,8V
J4N+0.@%uJ\N\0NJN0NJ\N\0NJN0NJ\N0NJN\0v
Bài4

B4+B+

4

+


Đinh Quang Thành GV trường THCS Ninh Khánh TP Ninh Bình – ĐT 0303503650
4
HI&

≥
x
78

≥
y
!%D





≥
≥



*
*
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.&[ +#

J4N+0
( ) ( ) ( ){ }
NNNN/N/
−−∈
Bài tập tổng hợp
&E&%5%3.S-'1..&[ +#V
25.4

\*4+
d4\+B/
27.O4

Be+

AO
28.4

B/4+Be+

Od
1 4Z++#US-.k! lV
O *4+\J4B+0O
/ J4+B+fBf40*4+f
31. 4+xfB+fx4Bf4x+
 x4Bx+BxfxOO
Phương pháp 5 : Đưa về dạng tổng, lập phươngV
P&Q"R&3.S-'1.7TULV7Q'5&,8R%X!%5%(1.3.S-Z7Q3.E&,8R
%X!%5%>?%.W.3.S-
Bài 33 :1 .&[ +#%X!3.S-'1.4


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:

4B+B4+B4

B+

⇔4

B+

c4c+c4+*
⇔J4

c4+B+

0BJ4

c4B0BJ+

c+B0d
⇔J4c+0

BJ4c0

BJ+c0

d
Đinh Quang Thành GV trường THCS Ninh Khánh TP Ninh Bình – ĐT 0303503650
5
s4Z+
∈
G#%.‡%D.;3.MW%.d


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 
+ /
= −


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


‰Q,2V.S-'1.%Dd%F3.&[ ,8V

x o
y
=


=

N
x
y o
=


=




x
y
=


=

N


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

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

4
d
c4

+B+

d*
⇔
4
d
cJ4
d

\4

+B+
0
d*
⇔
J4

0

BJ4

\+0

/

B*

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

e
/
yx
x
⇔

H2+%D*%F3.&[ JNe0NJN\e0NJ/Ne0NJ/N\e0
Bài 124

c4+B+

4c+c
⇔
J4

c4+B+

\4B+B0/
⇔
4

c4+B+

\*4Bd+B*/
⇔
4

c4+B+

B4


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
/


\+
/

/^4
/
%.&!.Q%.Z"F4
/
4

N
J4

ˆG0Z!%DV4


\+
/

/…^4


\+
/

/
^+
/
%.&!.Q%.Z"F+
/

+

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
6