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1/18

1

H phng trnh

I. H phng trnh dng hon v vng quanh. Bi 1. ( thi HSG quc gia nm 1994 )

Gii h phng trnh :( )( )

( )

3 2

3 2

3 2

3 3 ln 1

3 3 ln 1

3 3 ln 1

x x x x y

y y y y z

z z z z x

+ + + =

+ + + =

+ + + =

Gii :

Xt hm s : ( ) ( )3 2f 3 3 ln 1t t t t t = + + +

Ta c : ( )2

2

2

2 1f' 3 1 0, R

1

tt t x

t t

= + + >

+

Vy hm s ( )f t ng bin trn R. Ta vit li h phng trnh nh sau :

( )( )( )

ff

f

x yy z

z x

==

=

Khng mt tnh tng qut, gi s : { }min , ,x x y z= . Lc :

( ) ( ) ( ) ( )f f f f x y x y y z y z z x . Hay : x y z x x y z = =

Vi : x y z= = , xt phng trnh : ( )3 22 3 ln 1 0x x x x + + + = Do hm s : ( ) ( )3 22 3 ln 1x x x x x = + + + ng bin trn R nn pt c nghim duy nht : 1x= .

Vy h phng trnh c nghim duy nht : 1x y z= = = .

Bi ton tng qut 1 .Xt h phng trnh c dng:( ) ( )( ) ( )

( ) ( )( ) ( )

1 2

2 3

1

1

f g

f g

....

f g

f gn n

n

x x

x x

x x

x x

= = =

=

Nu hai hm s f v g cng tng trn tp A v ( )1 2, ..., nx x x l nghim ca h ph ng trnh , tron

, 1,2,...,ix A i n = th 1 2 ... nx x x = = = .

Chng minh :

Khng mt tnh tng qut gi s : { }1 1 2min , ..., nx x x x = .

Lc ta c : ( ) ( ) ( ) ( )1 2 1 2 2 3 2 3 1f f g g ... nx x x x x x x x x x .

Vy : 1 2 1.... nx x x x

T suy ra : 1 2 ... nx x x = = = .

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2/18

2

Bi 2.

Gii h phng trnh :

3 2

3 2

3 2

2

2

2

1

4

1

4

1

4

x x

y y

z z

y

z

x

+

+

+

=

=

=

Gii:V v tri ca cc phng trnh trong h u dng nn h ch c nghim : , , 0x y z > .

Xt hm s : ( )

3 221

f4

t t

t

+

=

, ta c : ( ) ( ) ( )3 22

2 1f' 2 ln 4 3 . 0, 04

t t

t t t t

+

= + < >

.

Vy hm s ( )f t nghch bin trn khong ( )0; + .

Khng mt tnh tng qut, gi s : { }min , ,x x y z= . Lc :

( ) ( ) ( ) ( )f f f f zx y x y y z y z x ( ) ( )f f zx z x y x = = = .

Vy h phng trnh c nghim duy nht :1

2x y z= = = .

Bi ton tng qut 2 .Xt h phng trnh c dng(vi n l):( ) ( )( ) ( )

( ) ( )

( ) ( )

1 2

2 3

1

1

f g

f g

....

f g

f g

n n

n

x x

x x

x x

x x

= = =

=

Nu hm s f gim trn tp A , g tng trn A v ( )1 2, ..., nx x x l nghim ca h ph ng trnh , tro

, 1,2,...,ix A i n = th 1 2 ... nx x x = = = vi n l .Chng minh :


Lc ta c :

( ) ( ) ( ) ( ) ( ) ( )1 2 1 2 2 3 2 3 1 1 1 2f f g g ... f f n nx x x x x x x x x x x x x x .

1 2x x=

T suy ra :1 2

...n

x x x = = = .

Bi 3.

Gii h phng trnh :

( )

( )

( )

( )

2

2

2

2

1 2

1 2

1 2

1 2

x y

y z

z t

t x

= =

=

=

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3/18

3

Gii :V v tri ca cc phng trnh trong h khng m nn phng ch c nghim : , , , 0x y z t .

Xt hm s : ( ) ( )2

f 1s s= , ta c : ( ) ( )f' 2 1s s= . Do hm s tng trn khong ( )1; + v gi

trn [ ]0; 1 ( Do f(s) lin tc trn R ).

Khng mt tnh tng qut, gi s : { }min , , ,x x y z t = .

+ Nu ( ) ( )1; , , , 1;x x y z t + + , do theo bi ton tng qut 1, h c nghimduy nht : 2 3x y z t = = = = + .

+ Nu [ ]0; 1x ( )0 f 1 0 2 1x y , hay [ ]0;1y , tng t [ ], 0; 1z t .

Vy [ ], , , 0; 1x y z t . Do ta c :

( ) ( ) ( ) ( )f f f f zx y x y y z y z x x z = .

Vi x z= ( ) ( )f f zx y t = = .

Lc h phng trnh tr thnh :( )

( )

( )2

2

2

1 21 2

1 2

x yx y

x y

y x x y

= = =

= =

2 3x y = =

Vy h phng trnh cho c 2 nghim : 2 3x y z t = = = = + v 2 3x y= = .

Bi ton tng qut 3 .Xt h phng trnh c dng(vi n chn ):( ) ( )( ) ( )

( ) ( )( ) ( )

1 2

2 3

1

1

f g

f g

....

f g

f gn n

n

x x

x x

x x

x x

= = =

=

Nu hm s f gim trn tp A , g tng trn A v ( )1 2, ..., nx x x l nghim ca h phng trnh , tro

, 1,2,...,ix A i n = th1 3 1

2 4

...

...n

n

x x x

x x x = = =

= = =vi n chn .

Chng minh :


Lc ta c :.

( ) ( ) ( ) ( )1 3 1 3 2 4

2 4

f f g gx x x x x x

x x

( ) ( ) ( ) ( )2 4 3 5

3 5

f f g g

.........

x x x x

x x

( ) ( ) ( ) ( )2 1 1

1 1

f f g g

.........

n n n

n

x x x x

x x

( ) ( ) ( ) ( )1 1 2 2f f g gn n nx x x x x x

Vy : 1 3 1 1 1 3 1.... ...n nx x x x x x x = = = ; 2 4 2 2 4.... ...n nx x x x x x x = = =



4/18

4

Phn bi tp ng dng phng php

1. Gii h phng trnh :3 2

3 2

3 2

2 7 8 2

2 7 8 2

2 7 8 2

x x x y

y y y z

z z z x

+ =

+ = + =

2. Chng minh vi mi a R , h phng trnh :2 32 3

2 3

x y y a

y z z a

z x x a

= + +

= + + = + +

c mt nghim duy nht .

3. Cho h phng trnh :2

2

2

x y a

y z a

z x a

= +

= + = +

Tm a h phng trnh ch c nghim vi dng x y z= = . 4. Gii h phng trnh :

31 1 2

32 2 3

399 99 100

3100 100 1

3 2 2

3 2 2

.........

3 2 2

3 2 2

x x x

x x x

x x x

x x x

+ =

+ = + = + =

5. Cho n l s nguyn ln hn 1. Tm a h phng trnh :2 3

1 2 2 2

2 32 3 3 3

2 31

2 31 1 1

4

4

.........4

4n n n n

n

x x x ax

x x x ax

x x x ax

x x x ax

= +

= +

= + = +


6. Cho n l s nguyn ln hn 1 v 0a . Chng minh h phng trnh :2 3

1 2 2 2

2 32 3 3 3

2 31

2 31 1 1

4

4

.........

4

4n n n n

n

x x x ax

x x x ax

x x x ax

x x x ax

= +

= + = + = +

c nghim duy nht .

7. Chng minh vi mi a R , h phng trnh :2 3 2

2 3 2

2 3 2

x y y y a

y z z z a

z x x x a

= + + +

= + + + = + + +




5/18

5

Ii. H phng trnh gii c bng phng php lng gic ho. 1. Gii h phng trnh :

( ) ( )

2 21 1 1 (1)

1 1 2 (2)

x y y x

x y

+ =

+ =

Gii. K :2

2

11 0

11 0

xx

yy

t cos ; y=cosx = vi [ ], 0; , khi h phng trnh :

( )( )

cos .sin cos .sin =12

1 cos 1 cos 2sin cos sin .cos 1 0

+ + = + = =

t21

sin cos , t 2 sin .cos2

tt

= =

Khi ta c :2

21 1 0 2 3 12

tt t t t

= + =

Vi 1t= , ta c :0

2sin 1 04 2 1

x

y

= = = = =

Nu : ( )0x a a > , ta t cosx a = , vi [ ]0;

2. Gii h phng trnh : ( ) ( ) ( )( )2 2

2 1 4 3 1

1 2

x y xy

x y

+ =

+ =

Gii . Do [ ]2 2 1 , 1; 1x y x y+ = . t sin , y cosx = = vi [ ]0; 2 .

Khi (1) ( )( )2 sin cos 1 2sin2 3 + =

12. 2sin .2. sin2 3

4 2

+ =

4sin sin2 sin 34 6

+ =

8sin sin cos 34 12 12

=

4cos cos cos 2 312 3 6

+ =

2cos 4cos cos 2 312 12 6

=

2cos 2 cos 3 cos 312 4 12 + =

2cos 3 3

4 =

( )0 0

0 0

35 1203cos 3

4 2 65 120

kk R

k

= + = = +

T suy ra h c 6 nghim ( ) ( ) ( ) ( )0 0 0 0 0 0, { sin65 , cos65 , sin35 , cos35 , sin85 , cos85x y = ,

( ) ( ) ( )0 0 0 0 0 0sin5 , cos5 , -sin25 , cos25 , sin305 , cos305 }



6/18

6

Nu : ( )2 2 0x y a a+ = > , ta t sin , cosx a y a = = , vi [ ]0; 2

3. Gii h phng trnh :2

2

2

2

2

2

x x y y

y y z z

z z x x

+ =

+ = + =

Gii : T cc phng trnh ca h , suy ra : , , 1x y z . Do ta c :

2

2

2

2 (1)1

2(2)

1

2(3)

1

xyx

yz

y

zx

z

=

=

=

t t tgx = vi ;2 2

(4) v sao cho tg , tg2 , tg4 1 (5).

Tng t bi 2. H phng trnh c 7 nghim2 4

, , , 0, 1,...,7 7 7

k k kx tg y tg z tg k

= = = =

Vi mi s thc xc mt s vi ;2 2

sao cho tgx =


2 3

2 3

3 3 0

3 3 0

3 3 0

x z x z z

y x y x x

z y z y y

+ =

+ = + =

Gii . Vit li h phng trnh di dng :( )

( )

( )

2 3

2 3

2 3

1 3 3

1 3 3

1 3 3

x z z z

y x x x

z y y y

=

=

=

(I)

T , d thy nu ( ), ,x y z l nghim ca h cho th phi c x, y, z1

3 . Bi th :

(I)

3

2

3

2

3

2

3(1)

1 3

3 (2)1 3

3(3)

1 3

z zx

z

x xyx

y yz

y

=

=

=

(II)

t tgx = vi ;2 2

(4) v sao cho

1tg , tg3 , tg9

3 (5).

Khi t (2), (3), (1) s c : tg3 , tg9y z = = v tg27x =



7/18

7

T y d dng suy ra ( ), ,x y z l nghim ca (II) khi v ch khi tg3 , tg9y z = = , tgx = , v c xc nh bi (4), (5) v tg tg27 = (6).

Li c : ( ) ( )6 26 k k Z =

V th tho mn ng thi (4) v (6) khi v ch khi26

k = vi knguyn tho mn :

12 12k . D dng kim tra c rng, tt c cc gi tr c xc nh nhva nu u thomn (5).

Vy tm li h phng trnh cho c tt c 25 nghim, l :3 9

, , , 0, 1,... 1226 26 26

k k kx tg y tg z tg k

= = = =

5. Gii h phng trnh :1 1 1

3 4 5

1

x y zx y z

xy yz zx

+ = + = +

+ + =

Gii. Nhn xt : 0; , ,xyz x y z cng du . Nu ( ), ,x y z l mt nghim ca h th

( ), ,x y z cng l nghim ca h, nn chng ta s tm nghim , ,x y z dng .

t ( )0tg ; tg ; tg 0 , , 90x y z = = = < < .

H( )

( )

1 1 13 tg 4 tg 5 tg 1

tg tg tg

tg tg tg tg tg tg 1 2

+ = + = +

+ + =

(1)2 2 21 tg 1 tg 1 tg

3 4 5tg tg tg

+ + + = =

3 4 5

sin2 sin2 sin2 = =

T (2) suy ra : ( )tg tg tg 1 tg tg + = ( )

( )tg tg

tg tg1 tg tgco

+ = = +

( )tg tg2 2

= + + + =

.

Do

= = < < + + =

3 4 5

sin2 sin2 sin2

0 , , ;2 2

nn 2 , 2 , 2 l cc gc ca mt tam gic c s o 3 cnh 3,4

Do tam gic c 3 cnh 3,4,5 l tam gic vung nn 0 02 90 45 z tg 1 = = = =

2 2

2tg 3 2x 3 1tg2 x

1 tg 4 1 x 4 3

= = = =

2 2

2tg 4 2y 4 1tg2 y

1 tg 3 1 y 3 2

= = = =



8/18

8

Tuyn tp cc bi ton hay

II . H phng trnh 2 n. 1. Gii h phng trnh : 4 2

2 2

698(1)

81

3 4 4 0 (2)

x y

x y xy x y

+ =

+ + + =

Gii : Gi s h phng trnh c nghim . Ta thy (2) tng ng vi :

( ) ( )22 3 2 0x y x y+ + =

phng trnh ny c nghim i vi x ta phi c :

( ) ( )2 2 7

3 4 2 0 13

y y y = (3)

Mt khc phng trnh (2) cng tng ng vi : ( )2 24 3 4 0y x y x x + + + = phng trnh ny c nghim i vi y ta phi c :

( ) ( )2 2 44 4 3 4 0 0

3

x x x x = + (4)

T (3) v (4) ta c : 4 2256 49 697 698

81 9 81 81x y+ + = < , khng tho mn (1).

Vy h phng trnh cho v nghim .

2. ( thi HSG Quc Gia nm 1995-1996.Bng A )

Gii h phng trnh :

13 1 2

17 1 4 2

xx y

yx y

+ = +

= +

3. ( thi HSG Quc Gia nm 1995-1996.Bng A )Hy bin lun s nghim thc ca h phng trnh vi nx, y :

3 4 2

2 2 3 22

x y y a

x y xy y b

=

+ + =

Gii . iu kin c ngha ca h :x, y R .Vit li h di dng :

( ) ( )

( ) ( )

3 3 2

2 2

1

2

y x y a

y x y b

=

+ =

Xt cc tr

ng hp sau :Trng hp 1 : 0b = . Khi :

( )0

2y

y x

=

= v do vy : H cho

( )( )

( )( )

3 3 2

3 3 2

0yI

y x y a

y xII

y x y a

=

=

=

=



9/18

9

C (II)4 22

y x

x a

=

=

T : + Nu 0a th (I) v (II) cng v nghim, dn n h v nghim .+ Nu 0a = th (I) c v s nghim dng ( ), 0x R y = , cn (II) c duy nht nghim

( )0, 0x y= = . V th h cho c v s nghim .

Trng hp 2 : 0b . Khi , t (1) v (2) d thy , nu ( ),x y l nghim ca h cho th

phi c x, y >0 . V th ( ) ( )2 3b

x yy

= .

Th (3) vo (1) ta c :

3

3 2by y y ay

=

t 0y t= > . T (4) ta c phng trnh sau :

( ) ( )3

32 2 6 2 9 3 2 0 5b

t t t a t b t a t t

= + =

Xt hm s : ( ) ( )39 3 2f t t b t a t = + xc nh trn [ )0;+ c :

( ) ( ) [ )28 3 2 2f' 9 9 0, 0;t t b t t a t = + + + .

Suy ra hm s ( )f t ng bin trn [ )0; + , v v th phng trnh (5) c ti a 1 nghim tron

[ )0; + . M ( )3

f 0 0b= < v ( ) 3 23f b 0b b a= + > , nn phng trnh (5) c duy nh

nghim, k hiu l 0t trong ( )0; + . Suy ra h c duy nht nghim2 2

0 0

0

,b

x t y t t

= =

.

Vy tm li : + Nu 0a b= = th h cho c v s nghim .` + Nu a tu , 0b th h cho c duy nht nghim .+ Nu 0, 0a b = th h cho v nghim .

4. Tm tt c cc gi tr ca m h phng trnh : 2 22 2

2 1x xy y

x xy y m

+ =

+ + =(1) c nghim .

Gii . + Vi 0y = h tr thnh2

2

2 1x

x m

=

=. H c nghim khi

1

2m =

+ Vi 0y , tx

t

y

= , h tr thnh

2

2

2

2

12 1

1

t ty

mt t

y

+ = + + =

( )

2

2

2 2

12 1

(2)

1 2 1

t ty

t t m t t

+ = + + = +

Vy h PT (1) c nghim ( ),x y khi v ch khi h PT (2) c nghim ( ),t y .



10/18

10

Xt h (2), t 22

12 1t t

y+ = suy ra 2

1

2 1 0 1

2

t

t tt

< + > >

. Do h (2) c nghim ( ),t y

2

2

1

2 1

t tm

t t

+ + =

+ c nghim ( )

1, 1 ,

2t

+

. Xt hm s ( )2

2

1f

2 1

t tt

t t

+ +=

+ trn khong

( ) 1, 1 ,2

+

. Ta c : ( )( )

2

22

6 2f'2 1t ttt t

+ += +

, ( ) 3 7f' 03 7

ttt

= = = +

Lp bng bin thin :

t 3 7 3 7

f(t) - 0 + + 0 -

f(t)

1

2 +

14 5 7

28 11 7

+

+

+

1

2

Nhn vo bng bin thin ta thy h c nghim :14 5 7

28 11 7m

+

+.

5. Gii h phng trnh :( ) ( )

( ) ( )

3

3

2 3 1 1

2 3 2

x y

x y

+ =

=

Gii . R rng nu 3 2y = h v nghim.

Vi 3 2y , t (2) suy ra3

3

2x

y=

, thay vo (1) ta c :

( )

( )33

27 2 31

2

y

y

+=

(3) . Xt hm s : ( )

( )

( )33

27 2 3f 1

2

yy

y

+=

, ta c : ( )

( )

( )

3 2

33

81 8 6 2f'

2

y yy

y

+ +=

Suy ra : ( )f' 0 1y y= = Ta c bng bin thin :

y -1 +

f(y) + 0 - -

f (y)

0

+

-1 1

2

3 2



11/18

11

Nhn vo bng bin thin suy ra pt(3) khng c nghim trn cc khong ( ); 1 v ( )31; 2 .Phng trnh c 1 nghim 1y = v 1 nghim trong khong ( )3 2, +

D thy 2y = l 1 nghim thuc khong ( )3 2, + .

Vy h phng trnh cho c 2 nghim : ( )1; 1 v1

; 2

2

.

6. ( thi HSG Quc Gia nm 2004 Bng B )Gii h phng trnh sau :

3 2

2 2

3 49

8 8 17

x xy

x xy y y x

+ =

+ =

7. ( thi HSG Quc Gia nm 1998-1999 Bng A )Gii h phng trnh :

( )

( )

2 1 2 2 1

3 2

1 4 .5 1 2

4 1 ln 2 0

x y x y x y

y x y x

+ + + = +

+ + + + =

Gii . K: 2 2 0y x+ > t 2t x y= th phng trnh th nht ca h tr thnh :

( )1

1 1 1 4 1 21 4 .5 1 25 5

t tt t t

t

+ + + ++ = + = (1)

V tri l hm nghch bin, v phi l hm ng bin trn nn t=1 l nghimduy nht ca (1).

Vy1

2 12

yx y x

+ = = th vo phng trnh th hai ca h ta c :

( ) ( )3 22 3 ln 1 0 2y y y y+ + + + + = V tri l hm ng bin do y =-1 l nghim duy nht ca (2).p s : 0, 1x y= = .

8. ( thi HSG Quc Gia nm 2000-2001 Bng B )Gii h phng trnh :

7 2 5

2 2

x y x y

x y x y

+ + + =

+ + =

Gii : K c ngha ca h phng trnh : { }min 7 ,2x x y

t : 7x y a+ = v 2x y b+ = . T h phng trnh cho ta c h :

( )( )

5 1

2 2

a b

b x y

+ =

+ =

Nhn thy : 2 2 5a b x = . Kt hp vi (1) suy ra :( )5

2

xb

= , th vo (2) ta c :

( )5

2 2 1 32

xx y x y

+ = =

Th (3) vo (2) ta c :11 77

5 2 1 22

y y y

+ = =

Th vo (3) suy ra nghim ca h l: 10 77,x= 11 77

2y

= .



12/18

12

9. Cho h phng trnh 2 n x, y :( )

( ) ( )

2 4 23 3

8 2 2 4 43 3 3 3

1

1 1 2

k x x x yx

k x x x k x y x

+ + + = + + + + =

1. Xc nh k h phng trnh c nghim .2. Gii h phng trnh vi k= 16.

10. ( thi HSG Quc Gia nm 1995-1996 Bng A )

Gii h phng trnh :

13 . 1 2

17 . 1 4 2

xx y

yx y

+ = +

= +

Gii . K c ngha ca h : 0, 0x y v 2 2 0x y+ .

D thy , nu ( ),x y l nghim ca h cho th phi c x >0, y>0 . Do :

H cho

1 213

1 4 21

7

x y x

x y y

+ = +

= +

( )

( )

1 1 2 2 13 7

1 2 21 2

3 7

x y x y

x y

= + = +

Nhn (1) vi (2) theo v ta c :

( )( ) ( ) ( )1 1 8

21 7 3 6 7 4 0 63 7

xy x y y x y x y x y x x y x y

= = + + = =+

( vx >0, y>0)

Thay vo (2) v gii ra ta c :11 4 7 22 8 7

,21 7

x y+ +

= = .Th li ta thy tho mn yu cu bt.

Iii. H phng trnh 3 n. 1. ( thi HSG Tnh Qung Ngi 1995-1996)

Gii h phng trnh :

3 2

3 2

3 2

6 12 8 0

6 12 8 0

6 12 8 0

y x x

z y y

x z z

+ =

+ = + =


2 3

2 3

12 48 64

12 48 64

12 48 64

x x y

y y z

z z x

+ =

+ = + =


19 5 2001

19 5 2001

1890

1890

1890

x y z z

y z x x

z x y y

+ = +

+ = + + = +

Gii . Chng ta s chng minh h phng trnh trn c nghim duy nht 0x y z= = = .



13/18

13

Gi s ( ), ,x y z l mt nghim ca h phng trnh khi ( ), ,x y z cng l mt nghim ch phng trnh , nn khng mt tnh tng qut ta c th gi thit : c t nht hai trong ba s , ,x y z

khng m. V d 0, 0x y . T phng trnh th nht ta suy ra 0z .

Mt khc nu 0 1u< th 2000 18 41890 2u u u+ > +

Nu 1u > th 2000 2000 2000 1000 18 41890 1 2. 2.u u u u u u+ > + > = > +

Do

2001 19 5

1890u u u u+ > + vi mi u>0.Bi vy nu cng tng v ca HPT ta suy ra 0x y z= = = .pcm

6. Tm iu kin cn v ca m h phng trnh sau c nghim duy nht :( )( )( )

2 3 2

2 3

2 3

2 3

2 3

2 3

x m y y my

y m z z mz

z m x x mx

= + +

= + + = + +

7. ( thi HSG Quc Gia nm 2004 Bng A )

Gii h phng trnh sau :

( )

( )( )

23

23

23

2

3016

x x y z

y y z x z z x y

+ =

+ =+ =

8. Gii h phng trnh :( ) ( )

( ) ( )

( ) ( )

2 3

2 3

2 3

1 2 1

1 2 1

1 2 1

x x y x

y y z y

z z x z

+ = +

+ = +

+ = +

Gii . Vit li h cho di dng :

( ) ( )

( ) ( )( ) ( )

3 2 3

3 2 3

3 2 3

2 2 1 f

2 2 1 f 2 2 1 f

x x x y x g y

y y y z hay y g zz z z x z g x

+ + = + =

+ + = + = + + = + =

Trong ( ) 3 2f 2t t t t = + + v ( ) 3g 2 1t t= + . Nhn xt rng g(t), f(t) l hm ng bi

trn R v : ( ) 2f' 3 2 2 0,t t t= + + > ( ) 2g 6 0,t t t= R.

Suy ra h cho tng ng vi h :( )

( )4h 0

x y z

x

= = =

Trong ( ) 3 2h 2 1t t t t = + . Nhn xt rng ( )h t lin tc trn R v : ( ) ( )h 2 0, h 0 0, < >

( ) ( )h 1 0, h 2 0< > nn phng trnh ( )h 0t = c c 3 nghim phn bit u nm trong ( )2; 2

t ( )2cos , 0;x u u = . Khi sin 0u v (4) c dng :( )

3 2

2cos , 0;

8cos 4cos 4cos 1 0

x y z u u

u u u

= = =

+ =hay

( )

( )3 22cos , 0;

sin 8cos 4cos 4cos 1 0

x y z u u

u u u u

= = =

+ =

Hay( )2cos , 0;

sin4 sin3

x y z u u

u u

= = =

=(5).



14/18

14

Gii h phng trnh (5) ta thu c3 5

; ;7 7 7

u

v

( )2cos , 0;

3 5; ;

7 7 7

x y z u u

u

= = =

9. Tm tt c cc b ba s dng ( ), ,x y z tho mn h phng trnh :2004 6 6

2004 6 6

2004 6 6

22

2

x y zy z x

z x y

= += +

= +

Gii :

Gi s ( ), ,x y z l mt b ba s dng tho mn h PT cho . Khng mt tnh tng qugi s 0 x y z< . Nhvy :

2004 6 6 6 6

2004 6 6 6 6

2

2

x y z x x

z x y z z

= + +

= + +

2004 6

2004 6

11

1

xx xx y z

zz z

= = =

o li, d thy 1x y z= = = l mt b ba s dng tho mn yu cu bi ton .

10. Tm iu kin ca m h phng trnh sau c nghim :2 2 2

2 2

2 2

1

2

x y z xy yz zx

y z yz

x z xz m

+ + =

+ + = + + =


5 4 2

5 4 2

2 2

2 2

2 2

x x x y

y y y z

z z z x

+ =

+ = + =

12. Gii h phng trnh : ( )( )( )

3 2 2

3 2 2

3 2 2

3 3 33 3 3

3 3 3

x y y y

y z z z

z x x x

+ + =

+ + =

+ + =

13. Tm tt c cc s thc a sao cho h phng trnh sau c nghim thcx, y, z :1 1 1 1

1 1 1 1

x y z a

x y z a

+ + =

+ + + + + = +

Gii. K: 1, 1, 1x y z H phng trnh tng ng vi h phng trnh :

( ) ( ) ( )( ) ( ) ( )

1 1 1 1 1 1 2

1 1 1 1 1 1 2

x x y y z z a

x x y y z z

+ + + + + + + + =

+ + + + + =

t u = 1 1x x + + ; 1 1v y y= + + ; 1 1s z z= + +

Do 1, 1, 1x y z nn 2, 2, 2u v s . Ngc li nu 2, 2, 2u v s , ta c :

2 21 1

1 1x x

ux x+ = =

+ + 2

2

1 2 1 41 1

2 4x u x u

u u

+ = + = +

Tng t i vi y, z .



15/18

15

Do bi ton ca ta a v bi ton tng ng : Tm tt c cc s thc a sao cho h

phng trnh sau c nghim 2, 2, 2u v s :

( )2

11 1 11

u v s a

u v s

+ + =

+ + =

+ iu kin cn : Gi s h phng trnh (1) c nghim . Theo bt ng thc Bunhia ta c

( ) 1 1 1 92 92

a u v s au v s

= + + + +

+iu kin : Gi s9

2a . Chng ta s chng minh h phng trnh (1) c nghim

Ly 3s = ( tho mn 2s ) . Khi (1) tng ng vi : ( )

2 3

3 2 3.

2

u v a

au v

+ =

=

,u v l hai nghim ca tam thc bc hai : ( )( )2 3 2 32 2 3

2

at a t

+

( )( )2 3 2 3 2 9,

2

a a au v

=

Ch : t ( ) ( ) ( )2 2

2 9 0 6 2 2 3 6h a h h h h= + > + > + . Tc l :

( ) ( )( )2 3 2 2 2 3 2 9a a a > 2, 2u v > > .

Nh vy h phng trnh (1) c nghim 2, 2, 2u v s .

Tm li cc s thc a cn tm l tt c cc s thc9

2a .


20 11 2007

1

x y zx y z

xy yz zx

+ = + = +

+ + =

15. ( thi HSG Quc Gia nm 2005-2006 Bng A )

Gii h phng trnh :

( )

( )

( )

23

23

23

2 6.log 6

2 6.log 6

2 6. log 6

x x y x

y y z y

z z x z

+ =

+ =

+ =

Gii . K xc nh , , 6x y z < . H cho tng ng vi :

( ) ( )

( ) ( )

( ) ( )

3 2

3 2

3 2

log 6 12 6

log 6 22 6

log 6 32 6

xy

x x

yz

y y

zx

z z

=

+

= +

= +



16/18

16

Nhn thy ( )f x =2 2 6

x

x x +l hm tng, cn ( ) ( )3g log 6x x= l hm gim vi x


17/18

17

18 . Gii h phng trnh :( )2 2

2

2 2

2

3 8 8 8 2 4 2

x y y x z

x x y yz

x y xy yz x z

+ = +

+ + = + + + = + +

Gii . H cho tng ng vi :

( ) ( )( ) ( )

( ) ( ) ( ) ( )2 2 2 2

01 2 1 0

4 4 1 2 1

x x y y y zx x y z

x y y z x z

+ + + =+ + + =

+ + + = + + +

Xt : ( ) ( ) ( ); , ; , 1; 2 1a x y b x y y z c x z= = + + = + +

2 2

. 0, . 0, 4a b a c b c = = =

+ Nu 0a =

th1

0,2

x y z= = = .

+ Nu 0a

th b

v c

cng tuyn nn : 2c b=

, t ta c :1

0,2

x y z= = = .

Tm li h c hai nghim :

1 1 1

0; 0; , 0; ;2 2 2

.

iV. H phng trnh n n. ( n >3, nN ) 1. Gii h phng trnh :

19961 2 3

19962 3 4

19961995 1996 1

19961996 1 2

.........

x x x

x x x

x x x

x x x

+ =

+ =

+ = + =

Gii : Gi X l gi tr ln nht ca cc nghim , 1,...1996ix i = v Y l gi tr b nht ca chngTh th t phng trnh u ta c :

2X 19961 2 3x x x + =

T i vi cc phng trnh ca h ta c : 2X 1996 , 1,2,....,1996kx k =

Hay l ta c : 2X 1996X suy ra : 19952 X ( v X >0 ) (1)Lp lun mt cch tng t ta cng i n : 19952 Y (2)

T (1) v (2) suy ra1995 1995

X Y 2= = Ngha l ta c : 19951 2 1996.... 2x x x = = = =

2. Gii h phng trnh : 1 1 2 21 21 2

...

....

n n

n

n

x ax a x a

b b b

x x x c

= = =

+ + + =

vi 1 21

, , ..., 0, 0n

n ii

b b b b=



18/18

18

Gii . t : 1 1 2 2

1 2

... n n

n

x ax a x at

b b b

= = = =

Ta c :1 1 1

n n n

i i i i i ii i i

x tb a x a t b= = =

= + = + 11 1

1

n

in ni

i i ni i

ii

c a

c a t b t

b

=

= =

=

= + =

1

1

n

ii

i i i n

ii

c a

x a b

b

=

=

= +


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