  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. 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        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   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     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    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 :   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 . 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 .   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 ).        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.