Chuyên đề bồi dưỡng học sinh giỏi toán 8 Phương trình nghiệm nguyên

 
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

2
 2y
2
= 5 (2)
 :
 (2)   x = 2k + 1 (k  Z) 
4k
2
+ 4k + 1  2y
2
= 5  2(k
2
+ k  1) = y
2

Suy ra y
2
 
 Z), ta c :
2(k
2
+ k - 1) = 4t
2
 k(k + 1) = 2t
2
+ 1 (2.1)
  2t
2
nên nh (2.1) vô 
nh (2) không c ên.

www.VIETMATHS.com
www.VIETMATHS.com
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).

 Tìm x  
2
 2x  4 là m
 :

2
 2x  4 = y
2
(y  Z)  (x  1)
2
 y
2

= 5  (x  1  y)(x  1 + y) = 5 (4).
1.5 = (1).(5), nên  :
–  :
x 1 y 1 x y 2 x 4
x 1 y 5 x y 6 y 2
     
  

  
     
  

– ng  :
x 1 y 1 x y 0
x y 2
x 1 y 5 x y 4
     


    

      


– 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
  
 :

 : x(y  1)  y  2 (5)
  
0x    :
y 2 3
x1
y 1 y 1

  


www.VIETMATHS.com
www.VIETMATHS.com
 
I, THANH 3
Vì x  Z nên
3
y1
 Z, suy ra y  

y  1
1
1
3
3
x
4
2
2
0
y

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

 y (6)
Gii :
 : 9x  2  y(y  1) (6.1)
 
 

 3k  1 (k  Z) thì y  1  3k   :
9x  2  (3k  1)(3k  2)  9x  9k(k  1)  x  k(k  1).
 1) và y = 3k  
 1) và y = 3k  1 (k  Z)
2.  
 
Ch ý  b và a

b (a, b


www.VIETMATHS.com
www.VIETMATHS.com
PHNG TRÌNH NGHIM NGUYÊN
4 I, THANH
 b)

(a + b)

2a là m b và a


 
  :

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

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

b) x
2
, y
2
x
2
+ y
2


(8

3.  :
a) 3x
2
 4y
2
 13 ; b) 19x
2
 28y
2
 2009 ;
c) x
2
 2y
2
 8y + 3 ; d) x
2
 4y
2

 
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
 



www.VIETMATHS.com
www.VIETMATHS.com
 
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.

Gii :
Cách 1. 
x  y  z  xyz (9)

1  x  y  z


9) ta có xyz  x  y  z  3z  xy  3 (do z > 0).
 t 
–  1, ta có x  1 và y  1. Thay vào (9 z  
–  2, ta có x  1, y  2. Thay vào (9 3.
–  3, ta có x  1, y  3. Thay vào (9 
y z.

 ; 2 ; 3.
Cách 2. 9) cho xyz > 0  :
1 1 1
1
xy yz zx
  
(9.1)

1 x y z  
9.1) suy ra :
2 2 2 2
1 1 1 1 1 1 3
1
xy yz zx x x x x
      
.

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
  

www.VIETMATHS.com
www.VIETMATHS.com
PHNG TRÌNH NGHIM NGUYÊN
6 I, THANH
 ta suy ra :
11
y3
y3
  

 y  {4 ; 5 ; 6}. Xét ba   :
  4
  
2
15
().
   6.
106 ; 6).
Cách 2. 
1 1 1
xy 3(x y) 0 (x 3)(y 3) 9
x y 3
         
.
 trên 
suy ra x  3 và y  3   {  1 ;  3} g sau :
x  3
1
1
3

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
   
    

   
   

–  : 1  1  1, l
–  1,  :
23
1
55


– 
xx
2 2 3 3
,
5 5 5 5
   

   
   
, suy ra
xx
2 3 2 3
1
5 5 5 5
   
   
   
   

11) là x = 1.

4. c
M ý :
www.VIETMATHS.com
www.VIETMATHS.com
 
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.
 

(x
2
+ 1)(x
2
+ y
2
) = 4x
2
y (12)
Gii :
Cách 1. Áp dsi ta có :
x
2
+ 1   x = 1.
x
2
+ y
2
  x = y.
 :
(x
2
+ 1)(x
2
+ y
2
)  4x
2

(12) 

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
  
   
    
  
     
  




7.  :
a) x
2
+ xy + y
2
= 2x + y ; b) x
2
+ xy + y
2
= x + y ;
c) x
2
 3xy + 3y
2
= 3y ; d) x
2
 2xy + 5y
2
= y + 1 ;
8.  :
1 1 1
x y 4


9. 
x y 3
xy 2




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

www.VIETMATHS.com
www.VIETMATHS.com
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.  :

x
17
 y
17
 19
17


IV. PHNG PHÁP 
1. 
ng dùng :
– 
– 
2
.
– 
– 
– 
–  
 
Gii :
 :
36x + 20 = 4y
2
+ 4y  3(12x + 7) = (2y + 1)
2
(13)
3 

2

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

Gii :

 : x
2
+ x + 1 = (k + 1)
2
.
Do x > 0 nên x
2
< x
2
+ x + 1 < x
2
+ 2x + 1 hay x
2
< (k + 1)
2
< (x + 1)
2
(14)

2
và (x + 1)
2
4

 
4
+ 2x
3

+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

+ x + 3).

2
< y
2
< (a + 2)
2

2
+ x.
 : y
2
 a
2
= x
2
+ x + 3 = (x +
1
2
)
2
+
11
4
> 0, suy ra y
2
> a
2
.
(a + 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 +

1
2
)
2
+
1
4
> 0, suy ra y
2
< (a + 2)
2
.
Do a
2
< y
2
< (a + 2)
2
nên y
2
= (a + 1)
2
, hay (x
2
+ x)
2
+ (x
2
+ x + 3) = (x
2

+ 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.  

16. 
2
+ y và y
2


17.  :
2 2 2 2
(1 2 3 x)(1 2 3 x ).       

www.VIETMATHS.com
www.VIETMATHS.com
PHNG TRÌNH NGHIM NGUYÊN
10 I, THANH
18.  
pp
23
trong

19.  :
x
4
 x
3
+ x
2
 x  1.

20.  : x(x

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


1
nguyên. Thay vào (15

3 3 3
1
4x y 2z
(15.1)
5.1) suy ra y   2y
1

1
nguyên. Thay vào (15
cho 
3 3 3
11
2x 4y z
(15.2)
5.2) suy ra z   2z
1

1
nguyên. Thay vào (15

3 3 3
1 1 1
x 2y z
(15.3)
5) thì (x
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

2
.

k

 y  z  0.
9).
2. 
 9
x  y  z  0.
www.VIETMATHS.com
www.VIETMATHS.com
 
I, THANH 11
Gii :
x  y  z  0, (19 ; y  ; 0 ; 0).

0
; y
0
; z
0
   

Vì (x
0
; y
0
; z
0

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.


0
 2z
1

3 3 3 3 3 3
0 0 0 0 0 0
2x 4y 8z x 2y 4z    
.

1
; y
1
; z
1
).

1
; y
1
; z
1
 ; 0 ; 0), (x
0
; y
0
; z
0
 ; 0 ; 0)
0 0 0
1 1 1 0 0 0

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

2
 y
2
 6(z
2
 t
2
) ; d) x
2
 y
2
 z
2
 2xyz ;
24.  : x
2
+ y
2
 7z
2
.


www.VIETMATHS.com
www.VIETMATHS.com