 
I, THANH 1



I. PHNG PHÁP DÙNG TÍNH CHIA HT
1. 
ng dùng :
–  m và a ±

b  m thì b  m.
–  b, b  c thì a  c.
–   c.
–  m, b n thì ab mn.
–  b, a  bc.
–  
 Tìm x, y  
G :
 3, 159  3, suy ra 17y   3.
 y = 3k (k   :
3x + 17.3k = 159  x + 17k = 53  x = 53  17k.

x 53 17k
y 3k





(k Z).
 Tênh : x

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PHNG TRÌNH NGHIM NGUYÊN
2 I, THANH
2. 
 Tìm x, y   : xy  x  y = 2 (3)
 :
Ta có (3)  xy  x  y + 1 = 3  x(y  1)  (y  1) = 3  (x  1)(y  1) = 3
Suy ra x  1  3).   {  1 ;  3} 
x  1
1
1
3
3
y  1
3
3
1
1
x
2
0
4
2
y
4
2
2
0
 ; 2), (2 ; 4), (0 ; 2), (2 ; 0).


    

      


– ng  :
x 1 y 5 x y 6 x 4
x 1 y 1 x y 2 y 2
     
  

  
     
  

– ng 4 :
x 1 y 5 x y 4 x 2
x 1 y 1 x y 0 y 2
        
  

  
     
  

 {2 ; 4}.
3. Tách ra các giá t
  
 :

2
0
4
2
 ; 2), (2 ; 4), (0 ; 2), (2 ; 0).

1. Tìm các  :
a) 2x  3y  156 ; b) 3xy  x  y  1 ; c) 2x
2
 3xy  2y
2
 7 ;
d) x
3
 y
3
 91 ; e) x
2
 xy  6x  5y  8 ; f) x
2
 2y
2
 5.
2.  


II. PHNG PHÁP 
1. 
 Tìm  2  y
2

a) x
2
 y
2
 2006 (7)
b) x
2
 y
2
 2007 (8)
Gii :
a) Cách 1. nh (7 : (x  y)(x  y) = 2006 (7.1)
Vì (x  y)  (x  y)   y) và (x  
) suy ra (x  y) và (x   y)(x  
2006 khôuy ra (7.1

Cách 2. 
2
, y
2
chia cho 4

2
 y
2
chia cho 4 có





 
4. ô êãn :
x
3
 y
3
 z
3
 x  y  z  2008
y 2007  2008)
5. n : n
3
+ 2006n  2008
2007
+ 1
y 2006  2007)
6. 
49cs0 50cs0
A 100 0500 01
 
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 
I, THANH 5
III. PHNG PHÁP DÙNG BT NG THC
1. 
 Tìm ba s nguyên dng sao cho tng ca chúúng.

Suy ra
2
3
1
x

x
2
 1 
Thay x = 1 vào (9.1) : 1  y  z  yz  (y  1)(z  1)  2.
Do
0 y 1 z 1   
, nên ta t t 
y  1  1 và z  1  2 hay y  2 và z  3.
 ; 2 ; 3.
2. 
  :
1 1 1
x y 3

(10)
Gii :
Cách 1. Do x, y có 
1 x y
10) ta suy ra
12
y6
3y
  


y  3
9
9
3
x
4
2
6
y
12
6
6
3. 
 
               

  :
x x x
2 3 5
(11)
Gii :
(11)  :
xx
xx
xx
2 3 2 3
11
5 5 5 5
   
    

4. c
M ý :
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 
I, THANH 7
- si :
ab
ab
2


(a, b   a = b).
–  : (a
2
+ b
2
)(x
2
+ y
2
)  (ax + by)
2
. D

–  :
|x|  x x  0 ; - x x  0.
-|x|  x  |x|
|x| + |y|  |x + y|,  xy  0.
 

Cách 2. (12)  x
4
+ x
2
y
2
+ x
2
+ y
2
 4x
2
y = 0  (x
2
 y)
2
+ (xy  x)
2
= 0.
 :
2 2 2
x y 0 y x y x
x y 1
xy x 0 x(y 1) 0 y 1 0
  
   
    
  
     
  




y 2007  2008)
10.  :
a)
xx
2 3 35
; b)
x x x
3 4 5
; c)
x x x
5 12 17 .

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PHNG TRÌNH NGHIM NGUYÊN
8 I, THANH
11.  .
 .
12.  : x
3
+ x
2
 x  1  y
3
.
13.  : x! + y! = (x + y)!
14.  :


) 

2. 
 

 
2
< x
2
< (a + 1)
2
.
 
2
< x
2
< (a + 2)
2
thì x
2
= (a + 1)
2
.
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 
I, THANH 9
              


+2x
2
+ x + 3 là  

Gii :

4
+ 2x
3
+2x
2
+ x + 3 = y
2
(18
Ta có : y
2
= (x
4
+ 2x
3
+ x
2
) + (x
2
+ x + 3) = (x
2
+ x)
2
+ (x
2

2
 y
2
= (x
2
+ x + 2)
2
 [(x
2
+ x)
2
+ (x
2
+ x + 3)]
= [(x
2
+ x)
2
+ 4(x
2
+ x) + 4]  [(x
2
+ x)
2
+ (x
2
+ x + 3)]
= 3x
2
+ 3x + 1 = 3(x +

+ x + 1)
2

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

 x
2
+ x  2 = 0  2.

2

 {-2 ; 1} 
15. g trình :
22
3x 4y 6x 13.  

2
 x  1)  4y(y  1).
21.  : x
4
 x
3
 x
2
 x  y
2
 y.
22.  : x
4
 2y
2
 1. V. PHNG PHÁP 
1. 
  : x
3
+ 2y
3
= 4z
3
(15)
Gii :
5) suy ra x   2x
1

; y
1
; z
1
5)
 2x
1
, y  2y
1
, z  2z
1
.
       
2
; y
2
; z
2
     5  
x
1
 2x
2
, y
1
 2y
2
, z
1
 2z

9) nên :
3 3 3
0 0 0
x 2y 4 z
. Suy ra x
0
 2.

0
 2x
1
 :
3 3 3 3 3 3
1 0 0 1 0 0
8x 2y 4z 4x y 2z    
. Suy ra y
0
 2.

0
 2y
1
 :
3 3 3 3 3 3
1 1 0 1 1 0
4x 8y 2z 2x 4 y z    
. Suy ra z
0
 2.


x y z
x y z x y z
2 2 2
       
.

0
; y
0
; z
0
).
 y  z  (19) 
nào khác.

23. T :
a) x
3
 3y
3
 9z
3
; b) x
2
 y
2
 3z
2
;
c) x