5.pthpt_2

1

TUYN CHN CC BI TON PHNG TRNH, H PHNG TRNH, BT PHNG TRNH

TRONG THI HC SINH GII CC TNH, THNH PH NM HC 2010 - 2011

(L Phc L - tng hp v gii thiu)

Bi 1.

1/ Gii phng trnh 2 1 3 4 1 1x x x x .

2/ Gii phng trnh vi n s thc 1 6 5 2x x x

( thi HSG tnh Vnh Long) Bi 2. Gii phng trnh 5 4 3 211 25 14 0x x x x x

( thi HSG tnh ng Nai)

Bi 3. Gii h phng trnh 2 2 4

2 5 2 5 6

x y

x y

( HSG B Ra Vng Tu)

Bi 4. Gii h phng trnh sau

1 3 3

12 8

x x yy

x y y

( thi HSG Hi Phng, bng A)


2 2

4 4 14 2 4 2

x y xyx y xy

( thi HSG tnh Lm ng)

Bi 6. Gii h phng trnh trn tp s thc 4

2 2

5 65 6

x yx y x

( thi chn i tuyn ng Nai)

AOTRANGTB.COM

Download ti liu hc tp ti : http://aotrangtb.com

2

Bi 7. Gii h phng trnh 2 2

2 2

3 2 112 4

yx y x

xx y y

( thi HSG H Tnh)

Bi 8. Gii phng trnh 23 6 7 1x x x

( thi chn i tuyn Lm ng)


1 1

2 0

x x y

y x y x y x

( thi HSG tnh Qung Bnh)

Bi 10.

1/ Gii bt phng trnh 2 2( 4 ) 2 3 2 0x x x x .

2/ Gii h phng trnh sau

2

2

7

12

xy y x yx x y

( thi HSG in Bin)

Bi 11. Gii h bt phng trnh 6 8 10

2007 2009 2011

11

x y zx y z

.

( thi chn i tuyn Bnh nh)

Bi 12.

1/ Gii phng trnh 1 121 3

x xx x

2/ Gii h phng trnh 2

2

2

2

x x y

y y x

( thi HSG tnh Bn Tre)


AOTRANGTB.COM

3

Bi 13.

1/ Gii phng trnh 2 4 3 5x x x .

2/ Gii phng trnh 3 2 3 1 2 2x x x x trn [ 2, 2]

( thi HSG tnh Long An)

Bi 14. Gii h phng trnh sau 2 2

1 2 2

1 1 3 3( )

y xx yx

y x x

( chn i tuyn trng Chuyn L Qu n, Bnh nh).


2

2 3 4 97 6 2 9

x y xy x yy x x

( thi chn i tuyn Nha Trang, Khnh Ha)

Bi 16.

1/ Gii phng trnh 22 7 2 1 8 7 1x x x x x


2 2 3 2

6 1 4

x y x y

x y

( thi HSG tnh Vnh Phc)

Bi 17. Gii phng trnh sau 2

4 3 2 3 12 2 2 1 ( ) xx x x x x xx

( thi HSG tnh H Tnh)

Bi 18. Gii phng trnh 2 2 3 2 2 5 0sin sin cosx x x .

( thi chn i tuyn trng THPT chuyn L Khit, Qung Ngi)

Bi 19.

1/ Gii phng trnh 2 24 2 4x x x x .


AOTRANGTB.COM

4

2/ Gii h phng trnh 2 2

2 2

2 ( ) 3( ) 10y x y x

x x y y

( thi chn i tuyn THPT Chuyn Lam Sn, Thanh Ha)

Bi 20. Gii phng trnh 23 6 7 1x x x .


Bi 21. Gii h phng trnh

5( ) 6( ) 46 5

6( ) 4( ) 54 6

4( ) 5( ) 65 4

x y x zx y xy x z xz

z y x yz y zy x y xy

x z y zx z xz y z yz

( chn i tuyn trng PTNK, TPHCM)

Bi 22.

1/ Gii phng trnh 12 1 3 2 ( 11)2

x y z x y z


2 2

1212 27 9

3 4 4 0

x

x x

x y xy x y

( thi HSG tnh Qung Nam)

Bi 23.

1/ Tm tt c cc gi tr ca , a b phng trnh 2

2

2 2 1

x ax b mbx ax

c hai nghim phn bit vi

mi tham s m.


3 3 3

61 19y xy x

x y x

( thi HSG vng tnh Bnh Phc)

Bi 24.


AOTRANGTB.COM

5

1/ Gii h phng trnh 2 2 2 2

3 3 3 3

20102010

x y zx y z

2/ Gii phng trnh 3 32 2 2 33 3 3 2 0x x x x x x

( thi chn i tuyn Ninh Bnh)

Bi 25.

1/ Gii bt phng trnh sau 2

2

2 1 2( 1) 2(2 )

4 1 17 0

x y x x x y

y x x

2/ Vi n l s nguyn dng, gii phng trnh 1 1 1 1... 0sin 2 sin 4 sin8 sin 2nx x x x

.

( thi HSG tnh Khnh Ha)

Bi 26.

1/ Gii phng trnh sau 3 sin 2 cos 2 5sin (2 3) cos 3 3 12cos 3

x x x xx

.

2/ Gii phng trnh 23 22 1log 3 8 5

( 1)x x x

x

( thi HSG tnh Thi Bnh)

Bi 27.

1/ Gii h phng trnh

2 2

2

1

2 1

x y xy yyx yx

2/ Gii phng trnh lng gic 2 2 2 2sin 2tan cot 2

xx x

( thi HSG tnh Ph Th)

Bi 28. Gii phng trnh 2 1 124 60 36 05 7 1

x x x x

( thi HSG tnh Qung Ninh)

AOTRANGTB.COM

6

Bi 29. Gii phng trnh 3 2 3 2 23 2 2 3 2 1 2 2 2x x x x x x x

( thi chn i tuyn trng THPT Chuyn HSP H Ni)


2 2

2 3 4 2 3 4 187 6 14 0

( )( )x x y yx y xy x y


Bi 31. Gii h phng trnh 32 2 1 2 1 2 3 2

4 2 2 4 6

( ) ( )x x y y

x y

( thi chn i tuyn trng THPT chuyn Lng Th Vinh, ng Nai)

Bi 32. Gii h phng trnh 4 3 3 2 2

3 3

9 97( )

x x y y y x x y xx y x

( thi chn HSG tnh Hng Yn)

Bi 33. Gii h phng trnh 3

2

2 2 1 3 1

2 1 2 1

y x x x y

y x xy x

( thi chn i tuyn chuyn Nguyn Du, k Lk)


2 2

352 3 4 9x y

x y x y

( thi HSG tnh Yn Bi)

Bi 35. Gii phng trnh 3 232 2 1 27 27 13 2x x x x

( thi HSG Hi Phng, bng A1)


2 2

2 2

1 1 2( )2

1 12

x yx y

y xx y

( thi chn i tuyn Qung Ninh)

7


3

3

3

3 12 5012 3 227 27

x x yy y zz x z

( thi chn i tuyn trng THPT Phan Chu Trinh, Nng)

Bi 38. Gii phng trnh 9 2

39 1 2 13

x x x

( thi chn i tuyn Ph Yn)

Bi 39.

1/ Gii phng trnh sau 21 1 2 2x x x x

2/ Gii h phng trnh sau 3 3 2

2

3 4 2

1 2 1

y y x x x

x y y

( thi HSG tnh Ngh An)

Bi 40.

1/ Gii h phng trnh 3 3 2

4 4

8 4 12 8 2 0x y xy

x y x y

2/ Chng minh phng trnh sau c ng mt nghim 2011 3 3 2( 1) 2( 1) 3 3 2x x x x x .

( d b thi HSG tnh Ngh An)


3

3

3

3 124 6

9 2 32

x y xy z yz x z

( thi chn i tuyn KHTN, vng 1)


2

2

2 2

11

3 2 6 2 2 1log ( ) log ( )

y x xey

x y x y

8

( thi chn i tuyn trng THPT Cao Lnh, ng Thp)

Bi 43. Gii phng trnh sau 2 2

2

2 2

2 11 2 1 4

x x x x xx x x x

( thi HSG tnh Bnh Phc)

Bi 44.

1/ Gii phng trnh 3 23 3 4 3 2x x x x

2/ Tm s nghim ca phng trnh

2011 2009 4 2011 2009 2 2(4022 4018 2 ) 2(4022 4018 2 ) cos 2 0x x x x x x x

( thi chn i tuyn Chuyn Nguyn Du)

Bi 45. Gii h phng trnh sau 2 2 2 2 2

(2 )(1 2 )(2 )(1 2 ) 4 10 12 2 1 0

x x y y zx y z xz yz x y

( thi chn i tuyn H Tnh)

Bi 46.

1/ Gii phng trnh sau 22010 ( 1 ) 1x x x .


3 3

4 2 52 2

xy x

x y

y xx y

( thi chn i tuyn trng THPT So Nam, tnh Qung Nam)

Bi 47. Gii h phng trnh 11 10 22 12

4 4 2 237 13 8 2 (3 3 1)

x xy y y

y x y x x y

( thi chn i tuyn TP.HCM)


2

2

2

2009 2010 ( )2010 2011 ( )2011 2009 ( )

x y x yy z y zz x z x

( thi chn i tuyn chuyn Quang Trung, Bnh Phc)

9


2 2

2

15574 3 (3 1)25

x y

x x y x

( thi chn i tuyn Ngh An)

Bi 50. Cho cc tham s dng , ,a b c . Tm nghim dng ca h phng trnh sau :

2 2 24x y z a b c

xyz a x b y c z abc

( kim tra i tuyn Ninh Bnh)

Bi 51. Gii h phng trnh sau trn tp hp s thc 2 2

2 2

3 3

3 0

x yxx yx yyx y

( thi chn i tuyn Chuyn Vnh Phc, tnh Vnh Phc)


2 2 3

23( )

x x y yx y

( kim tra i d tuyn trng THPT Chuyn HSP H Ni)

Bi 53. Gii phng trnh 2 3 532 .sin .cos 2 1 1x x x x x x x x

( thi chn i tuyn H Ni)


2 2

2

2 2

( 2) ( 3) ( 3)( 2)5 9 7 15 3

8 18 18 18 84 72 24 176

x y y x zx x z y yzx y xy yz x y z

( thi chn i tuyn HSP H Ni, ngy 2)

Bi 55.

Tm , ,x y z tha mn h

2 2

2 2

2 2

2 ( ) 11 2 2 2

(3 1) 2 ( 1)

z x y x yy z xy zx yzy x x x

( thi chn i tuyn trng H KHTN H Ni, vng 3)

10

LI GII CHI TIT V NHN XT

Bi 1.

1/ Gii phng trnh 2 1 3 4 1 1x x x x .

2/ Gii phng trnh vi n s thc 1 6 5 2x x x

( thi HSG tnh Vnh Long)

Li gii.

1/iu kin 1x . Phng trnh cho tng ng vi

2 2( 1 1) ( 1 2) 1 1 1 1 2 1x x x x (*)

-Nu 1 1x th (*) ( 1 1) ( 1 2) 1 3 2 1 1 1 1x x x x , loi.

-Nu 1 1 2 2 5x x th (*) ( 1 1) ( 1 2) 1 1 1x x , lun ng.

-Nu 1 2x th (*) ( 1 1) ( 1 2) 1 2 1 3 1 1 2x x x x , loi.

Vy phng trnh cho c nghim l mi x thuc 2;5 .

2/ iu kin 52

x . Phng trnh cho tng ng vi

2

2

1 5 2 6

(1 ) ( 5 2 ) 2 (1 )( 5 2 ) 6

(1 )( 5 2 ) 5 (1 )( 5 2 ) 10 25

7 30 0 3 10

x x x

x x x x x

x x x x x x x

x x x x

Th li, ta thy ch c 3x l tha mn.

Vy phng trnh cho c nghim duy nht l 3x .

Nhn xt. Cc dng ton phng trnh v t ny kh c bn v quen thuc, chng hon ton c th gii bng cch bnh phng kh cn m khng cn lo ngi v tnh gii c ca phng trnh hay khng. n gin trong vic xt iu kin, ta c th gii xong ri th li cng c.

11

Bi 2. Gii phng trnh 5 4 3 211 25 14 0x x x x x

( thi HSG tnh ng Nai)

Li gii.

Phng trnh cho tng ng vi

5 4 4 3 3 2 2

4 3 2

4 3 2

( 2 ) ( 2 ) ( 2 ) ( 9 18 ) (7 14) 0( 2)( 9 7) 0

29 7 0

x x x x x x x x xx x x x xxx x x x

Phng trnh th hai trn c th vit li l 4 3 2 4 3 3 2 2

2 2

( 9 6) 1 0 ( 2 2 3 3 6 6) 1 0( 1) ( 3 6) 1 0

x x x x x x x x x x xx x x

Do 2 2( 1) ( 3 6) 1 0,x x x x nn phng trnh ny v nghim.


Nhn xt. y l mt phng trnh a thc thng thng, c nghim l 2x nn vic phn tch thnh nhn t kh n gin; ci kh l bit nh gi phng trnh cn li v c nn tip tc tm cch gii n hay khng hay tm cch chng minh n v nghim. Trng hp bi cho phn tch thnh cc a thc khng c nghim n gin, bi ton tr nn kh khn hn rt nhiu; thm ch l ngay c vi nhng a thc bc bn. Chng hn nh khi gii phng trnh

4 3 22 3 10 16 3 0x x x x , nu tnh ton trn giy th khng phi d dng m c c phn tch 2 2(2 5 1)( 3) 0x x x x gii tng phng trnh tch.


2 5 2 5 6

x y

x y

( HSG B Ra Vng Tu)

Li gii. iu kin: , 0x y . Cng tng v hai phng trnh ca h, ta c:

( 2 5 2 ) ( 2 5 2 ) 10x x y y

Tr phng trnh th hai cho phng trnh th nht, v theo v, ta c:

12

5 2( 2 5 2 ) ( 2 5 2 ) 2 22 5 2 2 5 2

x x y yx x y y

t 2 5 2 0, 2 5 2 0a x x b y y . Ta c h sau:

2

10 10 10 55 5 5 5 52 2 50 20 2

10

a b b a b a aba a

a b a a

Xt phng trnh 22 5 2 5 2 5 (5 2 ) 2 5 25 2 10 2 2 2 2x x x x x x x x x .

Tng t, ta cng c 2y .

Vy h phng trnh cho c nghim l ( , ) (2, 2)x y .

Nhn xt. Ngoi cch gii tn dng tnh cht ca cc cn thc, ta cng c th t n ph ri bin i; trong phng trnh th hai, cc s hng t do c th khc nhau m li gii vn c tin hnh tng t. Chng hn, gii h phng trnh sau

2 2 6

2 5 2 9 8

x y

x y


1 3 3

12 8

x x yy

x yy

( thi HSG Hi Phng, bng A)

Li gii.

iu kin 10, 0, 3y x x yy

.

t 1 , 3, , 0a x b x y a by

. H cho vit li l 2 23 2, 1

1, 25a b a b

a ba b

-Vi 2, 1a b , ta c

13

2

1 41 12, 3 1 4, 4 44

1 3, 14 8 15 0, 44

5, 144

xx x y x x y x

y y y x

x yx x x xx

x yy xy x

-Vi 1, 2a b , ta c

2

1 11 11, 3 2 1, 7 77

4 10, 3 108 6 0, 77 4 10, 3 10

xx x y x x y x

y y y x

x yx x xy x x y

Th li, ta thy tt c u tha.

Vy h phng trnh cho c 4 nghim l

( , ) (3,1), (5, 1), (4 10,3 10), (4 10,3 10)x y .

Nhn xt. Dng h phng trnh gii bng cch t n ph ny thng gp nhiu k thi, t H-C n thi HSG cp tnh v khu vc. Chng ta s cn thy n xut hin nhiu cc thi ca cc tnh c nu di y.


2 2

4 4 14 2 4 2

x y xyx y xy


Li gii.

Ly phng trnh th nht tr phng trnh th hai, v theo v, ta c:

4 2 3 2 2 2 2 2

2

2 4 4 1 0 ( 1) 4 ( 1) 0 ( 1)( 1 4 ) 01 1 1 4 0

y y xy xy y xy y y y xyy y y xy

-Nu 1y , thay vo phng trnh u tin, ta c: 24 1 4 1 ( 1) 0 0 1x x x x x x .

Th li, ta thy c hai nghim u tha mn.

14

-Nu 1y , thay vo phng trnh u tin, ta c:

24 1 4 1 ( 1) 0 0 1x x x x x x .

Th li, ta thy c hai nghim u tha mn.

-Nu 2

2 11 4 04

yy xy xy

(d thy trong trng hp ny 0y ), thay vo phng trnh

u tin, ta c: 22 2

4 3 2 2 4 2 2 21 14 4 1 (1 ) 4 4(1 ) 4 ( 1)(5 7) 04 4

y yy y y y y y yy y

.

Suy ra 1, 0y x v hai nghim ny nu trn.

Vy h phng trnh cho c 4 nghim phn bit l ( , ) (1,1), (0,1), ( 1, 1), (0, 1)x y .

Nhn xt. y l mt dng h phng trnh a thc kh kh, r rng nu phng trnh th hai, ngi ta chia hai v cho 2 th kh c th t nhn bit gi tr ny m nhn vo ri tr tng v nh trn. Vic pht hin ra gi tr 2 nhn vo c th dng cch t tham s ph ri la chn.

Bi 6. Gii h phng trnh trn tp s thc 4

2 2

5 65 6

x yx y x

( thi chn i tuyn ng Nai)

Li gii.

Tr tng v hai phng trnh ca h, ta c 4 2 2 2 25( ) 0 ( ) ( ) 5 0 ( ) 5x x y y x x y x x y x y x x y

-Nu x y , t phng trnh th nht ta c 4 25 6 0 ( 3)( 2)( 1) 0 2 1x x x x x x x x , tng ng vi 2 1y y .

Th li thy tha, ta c hai nghim ( , ) ( 2, 2), (1,1)x y .

-Nu 2 25( ) 5x x y y xx

, thay vo phng trnh th nht ca h, ta c

4 6 3 22

55 6 5 6 25 0x x x x xx

15

ng thi, t h cho ta cng c 2 2 65 6 65

x x y x .

Do 3 2

3 2 6 3 26 6 216 96 3125 4 5. 4. 25 5 6 25 05 5 25 25

x x x x x

.

Suy ra trong trng hp ny, h v nghim.

Vy h cho c hai nghim l ( , ) ( 2, 2), (1,1)x y .


2 2

3 2 112 4

yx y x

xx yy

( thi HSG H Tnh)

Li gii.

iu kin: 2 20, 1xy x y . t 2 2 1, , 0xa x y b aby

.

H cho tr thnh 3 2 3 2 1, 11 1 2 3 0

2 33, 92 32 3 2 3

b ab ba b b b

b aa ba b a b

-Vi 1, 1a b , ta c 2 2 2,x y x y , ta tm c hai nghim l ( , ) (1, 1), ( 1,1)x y .

-Vi 9, 3a b , ta c 2 2 10, 3x y x y , ta tm c hai nghim l ( , ) (3,1), ( 3, 1)x y .

Th li, ta u thy tha mn.

Vy h cho c 4 nghim phn bit l ( , ) (1, 1), ( 1,1), (3,1), ( 3, 1)x y .

Bi 8. Gii phng trnh 23 6 7 1x x x

( thi chn i tuyn Lm ng)

Li gii.

iu kin 1x .

16

Ta c

23

2 33

2 33

2 33

( 6 2) ( 4) ( 1 1) 02 2( 2)( 2) 0

1 1( 6) 2 6 4

1 1( 2) 2 01 1( 6) 2 6 4

21 12 0

1 1( 6) 2 6 4

x x xx xx x

xx x

x xxx x

x

xxx x

D thy phng trnh th hai v nghim v v tri lun dng nn phng trnh cho c nghim duy nht l 2x .

Nhn xt. Cch n gin hn dnh cho bi ny l chng minh hm ng bin, tuy nhin, cn ch xt 1x trc khi o hm.


1 1

2 0

x x y

y x y x y x

( thi HSG tnh Qung Bnh)

Li gii.

iu kin , 1 0x x y .

Phng trnh th nht ca h tng ng vi

2 2

1 1 1 2 1 1 2 1

4( 1) ( 2) 4 2 2

x x y x x y x y y x y

y x y y x y x

Phng trnh th hai ca h tng ng vi

2 2 2 22 0 ( )y x y x y x y x xy y x y x

Ta c h mi l 2

11

2 2 2 22 2 42 ( 2) ( 2) 2 0 2

4

y

xy x y xy xy y y y y yy x y x y

x

17

So snh vi iu kin ban u, ta thy c hai nghim trn u tha mn.

Vy h phng trnh cho c hai nghim l 1( , ) ( , 1), (2,4)4

x y .

Bi 10.

1/ Gii bt phng trnh 2 2( 4 ) 2 3 2 0x x x x .

2/ Gii h phng trnh sau

2

2

7

12

xy y x yx xy

( thi HSG in Bin)

Li gii.

1/ iu kin 2 12 3 2 0 22

x x x x . Ta c

22 2

2

4 04 0( 4 ) 2 3 2 0 1 22 3 2 0

2

x xx xx x x x

x xx x

Kt hp cc iu kin trn, ta c 12 42

x x x .

Vy bt phng trnh trn c nghim l 1( , ] {2} [4, )2

x .

2/ iu kin 0y . H cho tng ng vi 7

( ) 12

xx yyxx yy

t , xu x y vy

, ta c h 7 3, 4

12 4, 3u v u vuv u v

-Vi 3, 4u v , ta c 4, 3 3, 1xx y x yy

, tha iu kin.

18

-Vi 4, 3u v , ta c 12 33, 4 ,5 5

xx y x yy

, tha iu kin.

Vy h cho c hai nghim l 12 3( , ) (3,1), ( , )5 5

x y .

Bi 11. Gii h bt phng trnh 6 8 10

2007 2009 2011

11

x y zx y z

.

( thi chn i tuyn Bnh nh)

Li gii.

T bt phng trnh th nht ca h, ta c 1 , , 1x y z .

T hai bt phng trnh ca h, ta c 2007 2009 2011 6 8 10 6 2001 8 2001 10 2001(1 ) (1 ) (1 ) 0x y z x y z x x y y z z

T iu kin 1 , , 1x y z , ta d dng thy rng 6 2001 8 2001 10 2001(1 ), (1 ), (1 ) 0x x y y z z .

Do , phi c ng thc xy ra, tc l 6 2001 8 2001 10 2001(1 ) (1 ) (1 ) 0 , , 1 , , 0x x y y z z x y z x y z .

Kt hp vi iu kin 6 8 10 1x y z , ta thy h bt phng trnh cho c cc nghim l

( , , ) (1,0,0), (0,1,0), (0,0,1)x y z .

Bi 12.

1/ Gii phng trnh 1 121 3

x xx x


2

2

2

x x y

y y x

( thi HSG tnh Bn Tre)

Li gii.

1/ iu kin 1,3 0, 1 3 1 3, 1x x x x x x .

19


2

2 1 1 31 ( 1) (3 ) ( 1 3 )( 1 3 )1 3 1 3

1 3 0

( 1 3 ) 1

x x xx x x x x xx x x x

x x

x x

D thy phng trnh th nht v nghim nn ta ch xt 2

2

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

2 79 4( 1)(3 ) 4 8 3 02

x x x x x x x x

x x x x x

Vy phng trnh cho c hai nghim l 2 72

x .

2/ iu kin , 0x y . D thy nu 0x th 0y v ngc li nn h c nghim ( , ) (0,0)x y .

Ta xt , 0x y . Xt hm s 2

( ) , 02

t tf t t , ta thy 1( ) 0, 04

f t t tt

nn y l

hm ng bin.

H cho c vit li l ( )( )

x f yy f x

. Suy ra x y , thay vo h cho, ta c

2

12 1 2 ( 1)( 1) 0 3 5

2

xx x x x x x x x x

x

Tng ng vi hai gi tr ny, ta cng c 1

3 52

y

y

Vy h cho c ba nghim l 3 5 3 5( , ) (0,0), (1,1), ( , )2 2

x y .

Nhn xt. Bi phng trnh th nht nu khng c bin i ph hp m t n ph th li gii s kh di dng v rc ri, chng ta cn ch tn dng nhng tnh cht ca cn thc, lng lin hp khai thc c im ring ca bi ton.

20

Bi 13.

1/ Gii phng trnh 2 4 3 5x x x .

2/ Gii phng trnh 3 2 3 1 2 2x x x x trn [ 2, 2]

( thi HSG tnh Long An)

Li gii.

1/ iu kin 5x .


2 2 3 23 2

4( 4 3) 5 ( 4)( 4 6 1) 0

4 6 1 0x

x x x x x x xx x x

Ta xt phng trnh 3 24 6 1 0x x x (*)

Hm s 3 2( ) 4 6 1f x x x x c 2( ) 3 8 6 0f x x x nn l ng bin; hn na, (0). (1) ( 1).2 0f f nn phng trnh ( ) 0f x c ng mt nghim thuc (0,1) .

Ta s gii phng trnh (*) bng phng php Cardano.

t 43

x y , ta c 3 2 61(*) 03 27

y y . t y u v , ta c

3 3 61 2( ) (3 )( ) 027 3

u v uv u v .

Chn u v v sao cho

3 3 6127

29

u v

uv

.

Gii h phng trnh ny, ta chn nghim 3 1 2( 61 3 417),54 9

u vu

.

T , ta tm c nghim ca phng trnh (*) l

30

3

1 2 4( 61 3 417) 0.18946454 319 ( 61 3 417)

54

x x

21

Vy phng trnh cho c hai nghim l 04,x x x .

2/ iu kin 2x .


3 2 2 5 4 3 2

5 4 3 2

( 3 1) 4( 2) ( 1)( 6 2 9 7) 01

6 2 9 7 0

x x x x x x x x x xxx x x x x

Phng trnh 5 4 3 26 2 9 7 0x x x x x c ng mt nghim thuc [ 2, 2] v n c gi tr gn ng l 0 1.916086228x x .

Vy phng trnh cho c hai nghim phn bit l 01,x x x .

Nhn xt. R rng phng trnh bc ba trn phi gii trc tip bng cng thc tng qut, iu ny t khi xut hin cc k thi HSG. i vi phng trnh th hai, vic xt [ 2, 2]x nu trong bi c th gi dng lng gic; tuy nhin, cch t 2cosx cha c kt qu, mong cc bn tm hiu thm. Mt bi tng t xut hin trong k thi HSG BSCL nh sau

Gii phng trnh 5 4 3 232 32 16 16 2 1 0x x x x x .

Phng trnh ny c gii bng cch t n ph 2y x ri bnh phng ln, nhn vo hai v

cho 2y a v phng trnh quen thuc 3 3 2y y y .

Bi ton nh th ny kh nh v phc tp!


1 2 2

1 1 3 3( )

y xx yx

y x x

( chn i tuyn trng Chuyn L Qu n, Bnh nh).

Li gii.

iu kin xc nh: 0 0,x y .


2 21 2 2 2 2 2 2 0( )y x y x y x x xy y y x x x xx yx

22

Xem y l phng trnh bc hai theo bin y, ta c

2 2 22 8 4 4 2 0( ) ( )x x x x x x x x x x x .

Do , phng trnh ny c hai nghim l

1 2

2 2 2 2 22 2

( ) ( ) ( ) ( ), ,x x x x x x x xy x y x .

Xt hai trng hp

-Nu y x , thay vo phng trnh th hai ca h, ta c:

2 21 1 3 3( )x x x .

D thy: 2 21 1 0 3 3( )x x x nn phng trnh ny v nghim.

-Nu 2y x , thay vo phng trnh th hai ca h, ta c:

2 2 2 2 22 1 1 3 3 1 2 3 2 12 3

( ) .( ) xx x x x x x xx

(*)

(d thy 32

x khng tha mn ng thc nn ch xt 32

x v php bin i trn l ph

hp). Xt hai hm s: 2 1 0( ) ,f x x x v 2 02 3

( ) ,xg x xx

.

Ta c: 2

01

( ) xf xx

nn l hm ng bin, 2

2 3 02 3

( )( )

g xx

nn l hm nghch bin.

Suy ra phng trnh (*) c khng qu mt nghim.

Nhm thy 3x tha mn (*) nn y cng chnh l nghim duy nht ca (*).

Vy h cho c nghim duy nht l 3 2 3( , ) ( , )x y .

Nhn xt. Quan h ca x v y c che giu ngay trong phng trnh u tin, nu nhn thy iu th cc bc tip theo s rt d nhn bit. Bi ny tnh ton tuy rm r nhng hng gii rt r rng nn khng qu kh.

23


2

2 3 4 97 6 2 9

x y xy x yy x x

( thi chn i tuyn Nha Trang, Khnh Ha)

Li gii.

T phng trnh th nht, ta c 2

2

42 3 9

xyx x

, t phng trnh th hai, ta c 22 9 6

7x xy .

Suy ra

2 22 2 2

2

2

4 2 9 6 28 (2 9 6)(2 3 9)2 3 9 7

1 9 3 33( 2)(2 1)(2 9 27) 0 22 4

x x x x x x x xx x

x x x x x x x

-Nu 2x , ta c 22 9 6 16

7 7x xy ; nu 1

2x , ta c

22 9 6 17 7

x xy .

-Nu 9 3 334

x vi 22 9 27x x th 22 9 6 3

7x xy .

Vy h phng trnh cho c bn nghim l 16 1 1 9 3 33( , ) ( 2, ), ( , ), ( ,3)7 2 7 4

x y .

Nhn xt. Bi ny c th cn nhiu bin i n gin hn nhng r rng cch rt y ra ri thay vo mt phng trnh nh trn l t nhin hn c.

Bi 16.

1/ Gii phng trnh 22 7 2 1 8 7 1x x x x x


2 2 3 2

6 1 4

x y x y

x y

( thi HSG tnh Vnh Phc)

Li gii.

1/ iu kin 1 7x . t 27 , 1, , 0 8 7a x b x a b ab x x .

24

Phng trnh cho tr thnh 2 2 2 ( )( 2) 0 2b a b ab a b b a b b .

-Nu a b th 7 1 7 1 3x x x x x , tha iu kin bi.

-Nu 2b th 1 2 3x x .

Vy phng trnh cho c nghim duy nht l 3x . 2/ iu kin 2 0, 1x y y . Phng trnh th nht ca h tng ng vi

(2 ) 2 2 3 0 2 1 2 3 2 1 1 2x y x y x y x y x y y x .

Thay vo phng trnh th hai ca h, ta c 3 6 2 4x x .

D thy v tri tng theo bin x nn phng trnh trn c khng qu mt nghim. Ta thy 2x tha mn, suy ra 2, 3x y .

Vy h cho c nghim duy nht l ( , ) (2, 3)x y .

Bi 17. Gii phng trnh sau 2

4 3 2 3 12 2 2 1 ( ) xx x x x x xx

( thi HSG tnh H Tnh)

Li gii.

iu kin ( , 1] (0,1]x .

Nu 1x th 4 3 2 2 2 2 3 22 2 2 1 ( ) ( 1) 0, ( 1) 0x x x x x x x x x x x nn phng trnh trn khng c nghim tha 1x .

ng thi 1x khng l nghim ca phng trnh nn ta ch xt (0,1)x .


222 2 2 2

22

2 (1 )1( 1) 2 (1 ) ( 1) (1 ) 11(1 )

x xxx x x x x xxx x

t 2

2

1 0(1 )xt

x x

, phng trnh trn tr thnh 22 1 2 0 2t t t tt

(do 0t ).

Khi

25

22 2 2 4 2 3

2

2 2 2

1 2 ( 1) 4 (1 ) 2 1 4 4 0(1 )

( 2 1) 0 2 1 0 1 2

x x x x x x x xx x

x x x x x

So snh vi iu kin nu, ta thy phng trnh trn c nghim duy nht l 1 2x .

Bi 18. Gii phng trnh 2 2 3 2 2 5 0sin sin cosx x x .

( thi chn i tuyn trng THPT chuyn L Khit, Qung Ngi)

Li gii.

t 1 1sin , cos ,a x b x a b . T phng trnh cho, ta c h sau:

2 2

4 3 2 2 5 01

ab a ba b

Ta c:

2

2

4 3 2 2 5 0 4 3 2 2 5 0

4 2 2 2 2 3 2 2 2 0

2 2 2 1 2 2 0

2 2 1 2 2 0

( ) ( )

( ) ( ) ( )

( ) ( )

ab a b ab a b

ab a b a b

a b a b a b

a b a b

Mt khc: 2 2 1a b nn 2 22 2 2 0( )a b a b a b .

ng thc xy ra khi v ch khi 22

a b .

Do , t (*), suy ra:2 2 2 1 02 2 1 0

22 2 02

( )

( )

a ba b

a b a b

D thy h ny v nghim.

Vy phng trnh cho v nghim.

26

Nhn xt. y l dng phng trnh lng gic gii bng cch nh gi quen thuc. Ngoi cch t n ph a v i s hon ton nh trn, ta c th bin i trc tip trn phng trnh ban u, tuy nhin iu d lm chng ta lc sang cc hng thun ty lng gic hn v vic gii bi ton ny gp nhiu kh khn hn.

Bi ny chnh l thi Olympic 30-4 nm 2000, lp 10 do trng L Hng Phong TP.HCM ngh. Li gii chnh thc cng ging nh trn nhng nguyn sin , cosa x b x .

Bi 19.

1/ Gii phng trnh 2 24 2 4x x x x .


2 2

2 ( ) 3( ) 10y x y x

x x y y

( thi chn i tuyn THPT Chuyn Lam Sn, Thanh Ha)

Li gii.

1/ iu kin 2 2x . Phng trnh cho tng ng vi

2 2 2 2 2( 2) ( 1) 4 ( 2) ( 1) (4 ) ( 2)( 2) 0

0 2 2

x x x x x x x x x

x x x

Th li ta thy tha.

Vy phng trnh cho c 4 nghim l 0, 2, 2x x x .

2/ Ta thy nu 0x th 0y v ngc li nn h phng trnh cho c nghim ( , ) (0,0)x y .

Xt trng hp 0xy .

Chia tng v phng trnh th nht cho phng trnh th hai, ta c

2 22 2 2 2 2 2 4 2 2 4

2 2

2 2 2 2 2 2 2 2

2 ( ) 3 20 ( ) 3 ( ) 3 17 20 0( ) 10

5( 4 )(3 5 ) 0 4 ,3

y x y x y x y x x y x x y yx x y y

x y x y x y x y

-Nu 2 24x y , h cho tr thnh 2 33

2 4

2 .3 3 2 2212.5 10 2 2

y y x y x xy xyxyx y y y

.

27

-Nu 2 253

x y , h cho tr thnh

233 4

4 42

1522 . 3 4 94 9 2 13538 4 15 16 135 135. 103 2

xy y x y xy xxy yx y y y

.

Vy h cho c 5 nghim l 4 4

4 4

15 135 15 135( , ) (0,0),(2,1), ( 2, 1), ( , ), ( , )2 22 135 2 135

x y .

Bi 20. Gii phng trnh: 23 6 7 1x x x .


Li gii.

iu kin 1x .

D thy 1x khng l nghim ca phng trnh nn ta ch xt 1x . Ta c 2 23 36 7 1 6 1 7x x x x x x (*)

Xt hm s 2322

1 1( ) 6 1, 1 ( ) 2 0, 12 13 ( 6)

f t t t t t f t t ttt

. Do

hm ny ng bin. T suy ra phng trnh (*) trn c khng qu mt nghim; mt khc (2) 7f nn phng trnh cho c nghim duy nht l 2x .


5( ) 6( ) 46 5

6( ) 4( ) 54 6

4( ) 5( ) 65 4

x y x zx y xy x z xz

z y x yz y zy x y xy

x z y zx z xz y z yz

( chn i tuyn trng PTNK, TPHCM)

Li gii.

t , ,6 4 5

x y y z z xa b cx y xy y z yz z x zx

. H cho tr thnh

28

14 585 6 4 6 36 4 5 4 6 54

4 5 6 4 5 95 4 66 16

a aca cb a a b bc b ab c

Do

1 1 1 6 1 33 146 8 7 14 337( ) 6

3 1 1 1 45 1412 124 4 14 45

7( ) 45 149 1 1 45 1 1231245 16 7 14

x yxx y xy x y xx y xy

y z y z yz yy z yz y z y

z x zxz x z

z x zx z x z

Vy h cho c nghim l 14 14 14( , , ) ( , , )33 45 123

x y z .

Nhn xt. Bi ny c hnh thc kh phc tp v cc h s xem ra rt khc nhau; tuy nhin, nu quan st k, chng ta s d dng tm ra cc n ph cn thit lm n gin ha bi ton.

Bi 22.

1/ Gii phng trnh 12 1 3 2 ( 11)2

x y z x y z


2 2

1212 279

3 4 4 0

x

x x

x y xy x y

( thi HSG tnh Qung Nam)

Li gii.

1/ iu kin 0, 1, 2x y z . Phng trnh cho tng ng vi

2 2

2 4 1 6 2 11

( 1) ( 1 2) ( 2 3) 0

1 1 2 2 3 0 1, 5, 11

x y z x y z

x y z

x y z x y z

Vy phng trnh cho c nghim l ( , , ) (1,5,11)x y z .

29

2/ Phng trnh th hai ca h tng ng vi 2 2( 4) 3 4 0y x y x x .

y l phng trnh bc hai theo bin y nn cn c iu kin 2 2 2 4( 4) 4( 3 4) 3 4 0 0

3x x x x x x .

Do 2

2 2 32 4 4 16 24 1212 27 ( ) 2. 27 93 3 9 9 9

x

x x

T bt ng thc trn v phng trnh th nht ca h, ta suy ra 43

x .

Do 2 2 2 24 4 4 8 16 4 4( 4) ( ) 3.( ) 4 0 0 ( ) 03 3 3 3 9 3 3

y y y y y y

Vy h cho c nghim l 43

x y .

Bi 23.

1/ Tm tt c cc gi tr ca ,a b phng trnh 2

2

22 1

x ax b mbx ax

c hai nghim phn bit vi

mi tham s m.


3 3 3

61 19y xy x

x y x

( thi HSG vng tnh Bnh Phc)

Li gii.

1/Trc ht, ta s tm nghim chung, nu c, ca hai phng trnh bc hai sau:

2 2 0x ax b v 2 2 1 0bx ax . Gi s 0x l nghim chung , ta c:

20 02 0x ax b v

20 02 1 0bx ax . Tr tng v hai phng trnh ny, ta c:

20 0(1 )( 1) 0 1 1b x b x .

-Nu 1b th phng trnh cho tr thnh 2

22

2 1 1 , 2 1 02 1

x ax m m x axx ax

. D thy

nu 1m th phng trnh ny v nghim, nu 1m th phng trnh ny c v s nghim, khng tha mn bi.

30

-Nu 1b th 0 1x , tng ng vi 1 2 0a b hoc 1 2 0a b .

Do , khi 1 2 0a b hoc 1 2 0a b th tng ng hai phng trnh cho c nghim chung l 0 1x v 0 1x .

Phng trnh ban u tng ng vi 2 2 2

2 2

2 2 ( 2 1) 02 1 2 1

x ax b x ax b m bx axmbx ax bx ax

hay 2(1 ) 2( ) 0bm x a am x b m (*) v 2 2 1 0bx ax .

Ta thy rng phng trnh (*) c khng qu hai nghim nn mun phng trnh cho c hai nghim phn bit vi mi m th hai phng trnh 2 2 0x ax b v 2 2 1 0bx ax khng c nghim chung, ng thi phng trnh (*) phi c ng hai nghim phn bit, tc l

2

1 2 0,1 2 01 0,( ) (1 )( ) 0,

a b a bbm m

a am bm b m m

T iu kin th hai, ta thy 0b , khi , h iu kin trn tr thnh

22 2 2 2 2 2 4

2

1 11 2 0,1 2 02 2

( ) 0, (2 1) 0, (2 1) 4 0, 0

1 , 0 1 12

2 24 1 0

a a a aa am m m a m a m a m a a a

a aa a

a

Vy cc gi tr ,a b tha mn bi l 1 12 2

a a v 0b .


3 3 3

61 19y xy x

x y x

Ta thy 0y khng l nghim ca h phng trnh nn ta ch xt 0y , ta c bin i sau

2 2

3 3 3 33

1 16( ) 6( )

1 1 3 119( ) ( ) ( ) 19( )

x xx xy y y y

x x xx x xy y y y y y

31

Thay 21 6( )xxy y vo phng trnh th hai, ta c

6 3 3 6 3 1216( ) 18( ) 19( ) 216( ) ( ) 06

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

.

-Nu 0 0x xy , thay vo h cho, ta thy khng tha mn.

-Nu 6y x , thay vo phng trnh th nht ca h, ta c

3 2 3 2 1 16 36 6 6 0 (3 1)(2 1) 03 2

x x x x x x x x x x x .

Vi 13

x , ta c 6 2y x ; vi 12

x , ta c 6 3y x .

Th li u thy tha.

Vy h cho c hai nghim phn bit l 1 1( , ) ( , 2), ( ,3)3 2

x y .

Nhn xt. bi 1, bc tm nghim chung ca hai phng trnh lm n gin ha vic xt iu kin ca nghim xem c tha mn phng trnh hay khng, v r rng

( ) ( ) 0( ) ( ) ( ) 0( ) 0( ) ( )

f x mg xf x f x mg xmg xg x g x

nn nu x l nghim ca phng trnh

cho m li khng tha mn iu kin xc nh ca mu th n l nghim chung ca ( ), ( )f x g x ( y l xt vi mi m nn c c nhng gi tr m khc 0). Bi 2 xut hin kh nhiu trong cc ti liu luyn thi H v vic tm ra cch chia nh th cng kh l m mn, chng ta c th rt y t phng trnh di thay ln ri nh gi phng trnh mt n x thu c.

Bi 24.

1/ Gii h phng trnh: 2 2 2 2

3 3 3 3

20102010

x y zx y z

2/ Gii phng trnh: 3 32 2 2 33 3 3 2 0x x x x x x

( thi chn i tuyn Ninh Bnh)

Li gii.

1/ T phng trnh th nht ca h, ta c , , 2010x y z .

32

Suy ra 3 3 3 3 3 3 2 2 2 32010( ) 2010x y z x y z x y z .

T phng trnh th hai suy ra ng thc phi xy ra, tc l

2

2

2

(2010 ) 0 0 2010(2010 ) 0 0 2010

0 2010(2010 ) 0

x x x xy y y y

z zz z

Kt hp vi phng trnh th nht, ta thy h cho c ba nghim phn bit l

( , , ) (2010,0,0), (0, 2010,0), (0,0,2010)x y z .

2/ Phng trnh cho tng ng vi

3 32 2 3 2 33 2 2 3 2x x x xx x x x

Xt hm s ( ) 3 ,tf t t t , ta c ( ) 3 .ln 3 1 0,tf t t nn y l hm ng bin.

Phng trnh trn chnh l 3 3 3 3

3 2

(2 2) ( 2 ) 2 2 23 2 0 ( 2)( 1) 0 2 1

f x x f x x x x x xx x x x x x

Vy phng trnh cho c hai nghim l 2, 1x x .

Bi 25.

1/ Gii bt phng trnh sau 2

2

2 1 2( 1) 2(2 )

4 1 17 0

x y x x x y

y x x

2/ Vi n l s nguyn dng, gii phng trnh 1 1 1 1... 0sin 2 sin 4 sin8 sin 2nx x x x

.

( thi HSG tnh Khnh Ha)

Li gii.

1/ iu kin 1x . Bnh phng hai v ca bt phng trnh th nht ca h, ta c

2 2

2 2

(2 ) 1 2(2 ) 1 2( 1) 2(2 )

(2 ) 1 2(2 ) 1 0 (2 1) 0

2 1 0 2 1

x y x x y x x x y

x y x x y x x y x

x y x y x x

33

Thay vo phng trnh th hai ca h, ta c

2

2

(2 1) 4 1 17 0 4 ( 1) 4 1 4 1 17 094 18 0 2

4

x x x x x x x x x x

x x x x

Ta thy ch c 2x l tha mn, khi , tng ng ta c 3y .

Vy h bt phng trnh cho c nghim duy nht l ( , ) (2,3)x y .

2/ iu kin 2 2 , 1, 2,3,..., ; , 0,1, 2,..., 12

iix k i n k x k i n

.

Trc tin, ta s rt gn v tri ca phng trnh cho.

Ta c bin i sau 2cos cos 2 2cos cos 2 1cot cot 2

sin sin 2 sin 2 sin 2a a a aa aa a a a

.

Do

11 1

1 1 1 1... 0sin 2 sin 4 sin8 sin 2

1 0 (cot 2 cot 2 ) 0 cot 2 cotsin 2

2 ,2 1

n

n ni i n

ii i

nn

x x x x

x x x xx

kx x k x k

D thy nghim ny tha mn iu kin ban u.

Vy phng trnh cho c nghim l ,2 1n

kx k

.

Bi 26.

1/ Gii phng trnh sau 3 sin 2 cos 2 5sin (2 3) cos 3 3 12cos 3

x x x xx

.

2/ Gii phng trnh 23 22 1log 3 8 5

( 1)x x x

x

( thi HSG tnh Thi Bnh)

34

Li gii.

1/ iu kin 3 5cos 22 6

x x k .


2

2

3 sin 2 cos 2 5sin (2 3) cos 3 3 2cos 3

3 sin 2 cos 2 5sin 3 cos 3 0

2 3 sin .cos 1 2sin 5sin 3 cos 3 0

2sin sin (2 3 cos 5) 3 cos 2 0

x x x x x

x x x x

x x x x x

x x x x

t sin , 1t x t . Ta c 22 (2 3 cos 5) 3 cos 2 0t t x x (*)

y l phng trnh bc hai bin t c 2 2 2(2 3 cos 5) 8( 3 cos 2) 12cos 12 3 cos 9 (2 3 cos 3)x x x x x

Do , phng trnh (*) c hai nghim l

(2 3 cos 5) (2 3 cos 3) 1 (2 3 cos 5) (2 3 cos 3) 3 cos 24 2 4

x x x xt t x

-Nu 1 1 7sin 2 2 ,2 2 6 6

t x x k x k k (tha mn).

-Nu 3 cos 2 sin 3 cos 2 sin( ) 1 2 ,3 6

t x x x x x k k (tha mn).

Vy phng trnh cho c ba h nghim l 72 , 2 , 2 ,6 6 6

x k x k x k k .

2/ iu kin 12 1 0, 1 0 , 12

x x x x .

Phng trnh cho tng ng vi 2 2 2

3 3 32

2 23 3

2 1log 3 8 4 log (2 1) log 3( 1) 3( 1) (2 1)3( 1)

log (2 1) (2 1) log 3( 1) 3( 1)

x x x x x x xx

x x x x

35

Xt hm s 3( ) log , 0f t t t t , ta c 1( ) 1 0, 0ln 3

f t tt

nn y l hm ng bin.

Phng trnh trn chnh l 2 2 2 2(2 1) (3( 1) ) 2 1 3( 1) 3 8 4 0 2

3f x f x x x x x x x .

Ta thy hai nghim ny u tha mn nn phng trnh cho c hai nghim l 2 , 23

x x .

Nhn xt. bi phng trnh lng gic, n lc rt gn c thnh mt phng trnh ch cha sin ,cosx x ; ta thng dng cch t n ph nh trn i s ha vic gii bi ton, khng phi d dng c th tm ra cch phn tch nhn t nh trn, nht l nhng bi ton di dng hn. Nu t sint x khng thnh cng, ta hon ton c th chuyn sang cost x th v chng hn, nh bi ton trn, nu t cost x th li gii s khng cn d dng na.

Trong thi H khi B nm 2010 cng c mt bi tng t, gii phng trnh sau

sin 2 cos 2 3sin cos 1 0x x x x . Bng vic p dng cng thc nhn i a phng trnh v dng (sin ,cos ) 0f x x , ta tin hnh t n ph sint x phn tch thnh nhn t, li gii kh r rng v t nhin.

Cc bn th gii thm bi ton sau

3 24sin sin .cos (7sin 3cos ) (sin 2 cos 2 ) 5(sin cos ) 2cos 0x x x x x x x x x x .

Bi 27.

1/ Gii h phng trnh

2 2

2

1

21

x y xy yyx yx

.

2/ Gii phng trnh lng gic 2 2 2 2sin 2tan cot 2

xx x

( thi HSG tnh Ph Th)

Li gii.

1/ Ta thy h phng trnh ny khng c nghim tha 0y nn ta ch xt 0y , khi ,

phng trnh th nht ca h tng ng vi 2 1 1x x yy

.

36

t 2 1,xu v x yy

. Ta c h

2 2

4 4 4 4 31 1 12 2 2 1 ( 1) 0

a b b a b a b a bab a a a a

a a

Ta c

22 2

11 21 1 3 2 0

3 3 23

5

xx yx x x x

yy x y x x

x yy

Vy h phng trnh cho c hai nghim l ( , ) (1, 2), ( 2,5)x y .

2/ iu kin

2 2

sin cos 2cos 0,sin 2 0, tan cot 2 0 sin .cos 0, 0cos sin 2

2sin 1 2sinsin .cos 0, 0 sin cos 0sin 2

( 2 , 2 ) ( 2 , 2 ),2 2

x xx x x x x xx x

x xx x x xx

x k k k k k

Ta bin i phng trnh cho tng ng vi

2 1 1 2 sin 2 ( 2 1) sin 2 1 2 sin 21

sin 2

x x x

x

t sin 2 ,0 1t x t . Phng trnh trn chnh l

2 1( 2 1) 1 2 ( 1)( 2 1) 0 12

t t t t t t (tha iu kin).

-Nu 1 sin 2 1 ,4

t x x k k .

-Nu 1 1 5sin 2 ,2 12 122

t x x k k k .

So snh vi iu kin ban u, ta thy phng trnh cho c ba h nghim l

37

52 , 2 , 2 ,4 12 12

x k x k x k k .

Bi 28. Gii phng trnh: 2 1 124 60 36 05 7 1

x xx x

( thi HSG tnh Qung Ninh)

Li gii.

iu kin: 75

x . Xt hm s 2 1( ) , 11

f t t tt

. Ta c:

1( ) 2 0, 12( 1) 1

f t t tt t

nn hm ny ng bin.

Do 7 15

x v 75 6 5. 6 15

x nn phng trnh cho tng ng vi

2 21 1 3(5 6) (5 6) ( ) 5 625 7 1

x x f x f x x x xx x

.

Th li ta thy tha iu kin bi.

Vy phng trnh cho c nghim duy nht l 32

x .

Bi 29. Gii phng trnh 3 2 3 2 23 2 2 3 2 1 2 2 2x x x x x x x


Li gii.

iu kin xc nh 3 2

3 2

3 2 2 03 2 1 0x x

x x x

Theo bt ng thc AM GM th

3 2 3 23 2 3 2 1 3 2 2 3 2 33 2 2 1 3 2 2

2 2( ). x x x xx x x x

ng thc xy ra khi v ch khi 3 23 2 2 1 1x x x .

38

3 2 3 23 2 3 2 1 3 2 1 3 23 2 1 1 3 2 1

2 2( ). x x x x x xx x x x x x

ng thc xy ra khi v ch khi 3 23 2 1 1 1x x x x .

3 2 3 2 23 2 3 2 3 2 3 3 2 3 2 33 2 2 3 2 1

2 2 2x x x x x x xx x x x x

2 2 223 2 3 3 2 3 1 2 2 2

2 2( ) ( )x x x x x x x

ng thc xy ra khi v ch khi 21 0 1( )x x .

Do , ta lun c 3 2 3 2 23 2 2 3 2 1 2 2 2x x x x x x x .

ng thc phi xy ra, tc l 1x . Th li thy tha.


Nhn xt. Bi ny khng qu kh v ch p dng cc nh gi rt quen thuc ca BT. Tuy nhin, xc nh c hng i ny cng khng phi n gin; thng thng sau khi nhm ra c nghim l 1x v ng trc mt phng trnh v t c cha cn th ny, ta hay dng cch nhn lng lin hp; th nhng, cch ri cng s i vo b tc cng nhng tnh ton phc tp.


2 2

2 3 4 2 3 4 187 6 14 0

( )( )x x y yx y xy x y


Li gii.

Xt ng thc 2 2 7 6 14 0x y xy x y (*)

Ta xem (*) l phng trnh bc hai theo bin x, vit li l 2 27 6 14 0( )x x y y y .

39

Phng trnh ny c nghim khi

2 2 2 77 4 6 14 0 3 10 7 0 13

( ) ( )y y y y y y y .

Hon ton tng t, xem (*) l phng trnh bc hai theo bin y, vit li l 2 26 7 14 0( ) ( )y y x x x .

Phng trnh ny c nghim khi

2 2 2 106 4 7 14 0 3 16 20 0 23

( ) ( )x x x x x x x .

Ta xt hm s 2 32 3 4 4 3 0 14

( ) , ( )f t t t t f t t t .

Suy ra, trn 1[ , ) , hm s ny ng bin. Ta c

2 6 1 3 3 6 18( ) ( ) , ( ) ( ) ( ). ( ) .f x f f y f f x f y .

T phng trnh th nht ca h th ta thy ng thc phi xy ra, tc l 2 1,x y .

Thay hai gi tr ny vo (*), ta thy khng tha.

Vy h phng trnh cho v nghim.

Nhn xt. tng gii ca bi ny khng kh v cng kh quen thuc khi ch cn tm min xc nh ca bin thng qua vic tnh Delta ca mt phng trnh bc hai; tuy trong li gii trn c kho st hm s nhng thc ra cc kt qu c th chng minh bng bt ng thc i s thun ty nn cng c gii chnh ca bi ny l i s. V do vic hai biu thc ca x v y phng trnh u ca h ging nhau c th dn n nh gi sai hng m dng gii tch, xt hm s khai thc phng trnh u tin trong khi iu khng em li kt qu g. Cc h s c chn ra s rt p chnh l u im ni bt ca bi ton ny.

Bi 31. Gii h phng trnh 32 2 1 2 1 2 3 2

4 2 2 4 6

( ) ( )x x y y

x y

( thi chn i tuyn trng THPT chuyn Lng Th Vinh, ng Nai)

Li gii.

40

iu kin xc nh: 1 22

,x y .

Xt hm s: 32 0( ) , ( ; )f t t t t .

Suy ra: 26 1 0( )f t t nn y l hm ng bin.

T phng trnh th nht ca h, ta c: 2 1 2 2 1 2( ) ( )f x f y x y .

Thay vo phng trnh th hai, ta c:

4 4 8 2 4 6y y (*)

Ta thy hm s: 4 4 8 2 4 6 2( ) , ( , )g y y y y c o hm l:

34

1 1 0 22 44 8

( ) , ( , )( )

g y yyy

nn ng bin.

Hn na: 46 4 6 8 2 6 4 6 0( ) . .g nn (*) c ng mt nghim l 6y .

Vi 6y , ta c 12

x .

Vy h cho c nghim duy nht l 1 62

( , ) ( , )x y .

Nhn xt. Dng ton ng dng trc tip tnh n iu vo bi ton n gin ha biu thc rt thng gp. Hng gii ny c th d dng pht hin ra t phng trnh th nht ca h, x v y nm v mi v ca phng trnh v quan st k hn s thy s tng ng ca cc biu thc v dn n xt mt hm s nh nu trn. tng bi ny hon ton ging vi bi 5 thi H mn ton khi A nm 2010.

Gii h phng trnh 2

2 2

(4 1) ( 3) 5 2 0

4 2 3 4 7

x x y y

x y x

.

Bi 32. Gii h phng trnh 4 3 3 2 2

3 3

9 97( )

x x y y y x x y xx y x

41

( thi chn HSG tnh Hng Yn)

Li gii.

Ta c

4 3 3 2 2 4 3 3 2 2

2 2 2 2

9 9 9 09 0 9 0( ) ( ) ( )

( ) ( ) ( ) ( )

x x y y y x x y x x xy x y x y x yx y x x xy y x y x y x x y

T phng trnh th hai ca h, ta thy x y nn t bin i trn, suy ra:

2 29 0 9( ) ( )x x y x x y (*)

Ta c: 3 3 3 3 337 77( )x y x y x y xx x

.

Thay vo (*), ta c: 3 23 7 9( )x x xx

.

Ta s chng minh rng v tri l mt hm ng bin theo bin x. Tht vy:

23 2 2 3 33 3 3

2233 2 3 3 3 6 2 433 3

7 7 72

7 72 2 7 7

( )

.

x x x x x x x xx x x

x x x x x x x x x x xx x

T (*) suy ra 0x v trong biu thc trn, cc s m ca bin x u dng nn y l hm ng bin; suy ra n c khng qu mt nghim.

Thay trc tip 1x vo biu thc, ta thy tha.

Vy h cho c ng mt nghim l: 1 2( , ) ( , )x y .

Nhn xt. im c bit ca bi ny l x l c h phng trnh mi sau khi bin i, nu nh ta dng cch i s trc tip, phn tch ra c mt nghim x = 1 th phng trnh bc cao cn li kh m gii c. Cch lp lun theo tnh n iu ca hm s th ny va trnh c iu va lm cho li gii nh nhng hn.

Bi 33. Gii h phng trnh 3

2

2 2 1 3 1

2 1 2 1

y x x x y

y x xy x

42

( thi chn i tuyn chuyn Nguyn Du, k Lk)

Li gii.

iu kin 1 1x . t 21 0 1a x x a .


3 2 3 32 2(1 ) 3 2 2y a a a y y y a a

D thy hm s 3( ) 2 ,f t t t t c 2( ) 6 1 0,f t t t nn ng thc trn c vit li l ( ) ( ) 1f y f a y a y x .


2 21 2 1 2 1x x x x

t cos , 0,x t t , phng trnh trn tr thnh

2 2

2

1 cos 2cos 1 2cos 1 cos

2sin cos 2 sin 2 2 sin 2 sin(2 )2 2 4

42 .26 34 2sin sin(2 ) , ,

3 42 4 2 .24 2 10 5

t t t t

t tt t t

t t kt kt t k ktt k t k

Do 0,t nn t hai h phng trnh trn, ta ch nhn gi tr 310

t .

Khi 3 3 3cos , 1 cos 2 sin10 10 20

x y .

Vy h cho c nghim duy nht l 3 3( , ) (cos , 2 sin )10 20

x y .

Bi 34. Gii h phng trnh: 3 3

2 2

352 3 4 9x y

x y x y

( thi HSG tnh Yn Bi)

43

Li gii.

Phng trnh th hai ca h tng ng vi 2 2(6 12 8) (9 12 27) 35x x y y

Thay vo phng trnh th nht ca h, ta c:

3 3 2 2 3 3(6 12 8) (9 12 27) ( 2) ( 3) 5x y x x y y x y x y

Li thay vo phng trnh th hai ca h, ta c:

2 2 22( 5) 3 4( 5) 9 5 25 30 0 ( 2)( 3) 0 2 3y y y y y y y y y y .

Vi 2y , ta c 3x , vi 3y , ta c 2x .

Th li ta thy tha.

Vy h phng trnh cho c hai nghim l ( , ) ( 2,3), ( 3, 2)x y .

Nhn xt. Dng ton da trn hng ng thc ny xut hin kh nhiu, chng hn trong thi VMO 2010 va qua; nu chng ta thy cc biu thc ca x v y trong h phng trnh cha y cc bc th kh nng gii theo cch dng hng ng thc l rt cao.

Mt bi ton tng t, gii h phng trnh sau 3 3

2 2

92 4

x yx y x y

.

Bi 35. Gii phng trnh 3 232 2 1 27 27 13 2x x x x

( thi HSG Hi Phng, bng A1)

Li gii.

Phng trnh cho tng ng vi 3 3(3 1) 2(3 1) (2 1) 2 2 1x x x x .

Xt hm s 3( ) 2 ,f t t t t . Ta thy 2( ) 6 1 0,f t t t nn y l hm ng bin.

Phng trnh trn chnh l

33 3

3 2 2

(3 1) ( 2 1) 3 1 2 1 (3 1) 2 127 27 7 0 (27 27 7) 0 0

f x f x x x x xx x x x x x x


44


2 2

2 2

1 1 2( )2

1 12

x yx y

y xx y

( thi chn i tuyn Qung Ninh)

Li gii.

iu kin , 0x y . Cng tng v ca hai phng trnh, ta c 2 2 3 22 3 2 3x y x xyx

Ly phng trnh th nht tr phng trnh th hai, v theo v, ta c 2 2 2 31 3 1 3x y x y y

y

Do 3

3 2 3 2 3 2 3 3

3 2 3 2 3 2 3 3

3 12 3 3 3 3 3 ( ) 3 2

11 3 1 3 3 1 ( ) 3 12

xx xy x xy y x y x y x yx yy x y x xy y x y x y

y

Th li, ta thy nghim ny tha.

Vy h phng trnh cho c nghim l 3 33 1 3 1,

2 2x y .

Nhn xt. Dng ton ny cng xut pht t vic khai trin cc hng ng thc, nhng y l da trn tnh i xng tm ra s bt i xng nhm sng to ton th v. Cch gii bi ny theo hng trn l quen thuc v tt hn c, mt bi tng t trong thi HSG ca TPHCM l

Gii h phng trnh

4 4

2 2 2 2

1 1 2( )2

1 1 (3 )( 3 )2

y xx y

x y x yx y

Bi ny to ra t khai trin nh thc Newton bc nm, nu ta xt khai trin bc by th bi ton thu c s rt n tng.


45

3

3

3

3 12 5012 3 227 27

x x yy y zz x z

( thi chn i tuyn trng THPT Phan Chu Trinh, Nng)

Li gii.

Ta c

3 3 23 12 50 48 12 3 2 12 4 2 1( ) ( )( )x x y y x x y x x (1)

3 3 212 3 2 3 18 12 16 3 6 4 2( ) ( )( )y y z z y y z y y (2)

3 3 227 27 27 54 27 54 27 2 6 3( ) ( )( )z x z x z z x z z (3)

-Nu 1x th 22 1 0( )( )x x , t (1) suy ra 4y hay 24 2 0( )( )y y , t (2) suy ra

6z hay 26 3 0( )( )z z , t (3) suy ra 2x , mu thun.

Do , 1x khng tha mn h, ta ch xt 21 1 0( )x x .

Chng minh hon ton tng t, ta cng c: 2 22 0 3 0( ) ,( )y z .

T (2) suy ra 4 6, y z cng du.

T (3) suy ra 2 6, x z cng du.

T , ta c: 2 4,x y cng du.

Hn na, t (1), ta thy 2 4, ( )x y cng du, tc l: 0 2 4 0( )( )x y .

Do : 2x hoc 4y .

T cc phng trnh (1), (2), (3), d thy c hai trng hp trn u cho ta kt qu l:

2 4 6, ,x y z .

Vy h cho c nghim duy nht l 2 4 6( , , ) ( , , )x y z .

Nhn xt. Mu cht ca bi ton l phi c c cc phn tch (1), (2), (3) trn. iu ny ch c th thc hin c khi on c nghim ca bi ton l 2 4 6, ,x y z v c th tc

46

gi bi ton cng xut pht t cc ng thc bin i c nh trn. Dng ny cng tng xut hin trong thi HSG ca TPHCM nm 2006 2007 vi cch gii tng t.

Gii h phng trnh

3

3

3

3 42 6 63 9 8

x y xy z yz x z

Bi 38. Gii phng trnh 9 2

39 1 2 13

x x x

( thi chn i tuyn Ph Yn)

Li gii.

Phng trnh cho tng ng vi 9 2 3 9 2 3 2

9 3 3 2 9 3 3

9 1 3(2 1) 9 1 24 36 18 33 (27 27 9 1) 9 3 3 (3 1) 3(3 1)

x x x x x x x xx x x x x x x x x x

Xt hm s 3( ) 3 ,f t t t t , ta c 2( ) 3 3 0,f t t t nn y l hm ng bin.

Phng trnh trn c vit li l 3 3( ) (3 1) 3 1f x f x x x . (*)

Trc ht, ta xt cc nghim tha mn 2 2x ca (*). t 2cos , [0, ]x , khi

3 2(*) 8cos 6cos 1 2cos3 1 cos3 cos3 9 3

x k .

M [0, ] nn ta ch chn 3 nghim ca h trn l 5 7, ,9 9 9

, tng ng, ta c

cc nghim ca (*) l 5 72cos , 2cos , 2cos9 9 9

x x x . R rng ba nghim ny l phn

bit v (*) l phng trnh bc ba, c khng qu ba nghim nn y cng chnh l tt c cc nghim ca n.

Vy phng trnh cho c cc nghim l 5 72cos , 2cos , 2cos9 9 9

x x x .

Bi 39.

47

1/ Gii phng trnh sau 21 1 2 2x x x x

2/ Gii h phng trnh sau 3 3 2

2

3 4 2

1 2 1

y y x x x

x y y

( thi HSG tnh Ngh An)

Li gii.

1/ iu kin 1 2x . t 1 2 0t x x , ta c 2

2 2 2 233 2 2 ( ) 22

tt x x x x .


22 2

4 2

4 3 3 2 2

3 2

3 2

3( 1 2 ) ( 2) 3 2 ( ) 3 22

6 4 (3 4 2) 0

(1 2) (1 2) (3 2 2) (2 2 3) (1 2) (5 2) (3 4 2) 0

( 2 1)[ (1 2) (2 2 3) (5 2)] 0

2 1

(1 2) (2 2 3) (5 2) 0

tx x x x t

t t t

t t t t t t t

t t t t

t

t t t

D thy phng trnh th hai khng c nghim dng nn ta ch xt 2 1t . Khi

2 21 2 2 1 3 2 2 3 2 2 0 0 1x x x x x x x x , tha.

Vy phng trnh cho c hai nghim l 0, 1x x .

2/ iu kin 1 1,0 2x y .

Phng trnh th nht ca h tng ng vi 3 3( 1) ( 1)y y x x .

Xt hm s 3( ) ,f t t t t , ta c 2( ) 3 1 0,f t t t nn y l hm ng bin.

Phng trnh trn c vit li l ( ) ( 1) 1f y f x y x . Thay vo phng trnh th hai

ca h, ta c 2 21 1 1 1 1 1 1 1x x x x x x (*)

48

t 2

2 2 2 21 1 0 2 2 1 12

tt x x t x x .

Do 2

22(*) 1 2 0 22

t t t t t .

Khi 2 21 1 2 2 2 1 2 1 1x x x x x , tha iu kin. Tng ng vi mi gi tr x, ta c 0, 2y y .

Vy h phng trnh cho c hai nghim l ( , ) ( 1,0), (1, 2)x y .

Nhn xt. bi 1, cch t n ph v phn tch nh th ch mang tnh cht tham kho v n kh thiu t nhin. Ta hon ton c th kho st hm s 2( ) 1 1 2 2f x x x x x trn [ 1,2] , tnh o hm cp 2 chng minh phng trnh ( ) 0f x c khng qu hai nghim phn bit ri nhm nghim hoc ta cng c th dng phng php nhn lng lin hp gii quyt cng kh thun tin.

Bi 40.


4 4

8 4 12 8 2 0x y xy

x y x y


( d b thi HSG tnh Ngh An)

Li gii.

1/ t 2y t , h cho tr thnh 3 3 2 3 2 2

4 4 3 3

1 1 ( )4 4 0 4 ( 1) ( 1) 0x t xt t x t x

x t x t x x t t

Thay 3 2 21 ( )t x t x t phng trnh th nht vo phng trnh th hai ca h, ta c

3 2 23 3 2

04 ( 1) ( ) 0

4( 1) 0x

x x xt t xx t tx

D thy 0x khng l nghim ca h nn ta ch xt 3 3 2 3 3 24( 1) 0 4 4x t tx x t tx .

Phng trnh th nht ca h c vit li l 3 3 24 4 4 4x t xt . Do

49

3 3 2 3 3 2 3 24 4 4 4 0x t xt x t tx t tx t t x .

D thy 0t khng tha mn h cho nn ch xt t x .

-Nu t x , ta c h 3 3 2 3

4 4 4

. 1 11

4 4 0 5 5 0x x x x x

xx x x x x x

.

Khi 112

t x y , h cho c nghim 1( , ) (1, )2

x y .

-Nu t x , ta c h 3 3 2 3

4 4 4

( ) 1 14 ( ) 4 ( ) 0 5 3 0x x x x x

x x x x x x

, h ny v nghim.

Vy h phng trnh cho c nghim duy nht l 1( , ) (1, )2

x y .


iu kin 1x . Phng trnh cho tng ng vi 2011 3 3( 1) 2( 1) ( 1) 1x x x .

t 1 0t x . Ta cn chng minh phng trnh 2011 3 62 1t t t c ng mt nghim dng.

Xt hm s 2011 6 3( ) 2 1, 0f t t t t t . Ta c (0) 1, lim ( )x

f f t

v ( )f t lin tc trn

(0, ) nn phng trnh ( ) 0f t c t nht mt nghim dng.

Ta c 2011 3 6 2011 3 22 1 ( 1) 0 0t t t t t t , m vi 0t , ta c 2011 3 2( 1) 1 1t t t .

Khi 2010 5 2 2010 2010 5 2010 2( ) 2011 6 6 1999 6( ) 6( ) 0f t t t t t t t t t nn y l hm ng bin, tc l n c khng qu mt nghim.

Kt hp cc iu ny li, ta thy rng phng trnh 2011 3 62 1t t t c ng mt nghim dng, tc l phng trnh cho c ng mt nghim. Ta c pcm.

Nhn xt. Bi 2 tuy hnh thc kh phc tp nhng qua php t n ph a v mt phng trnh a thc thng thng. tng gii nh trn tng xut hin trong B tuyn sinh ca B GD T, l bi ton sau

Chng minh phng trnh 5 2 2 1x x x c ng mt nghim.

50


3

3

3

3 124 6

9 2 32

x y xy z yz x z

( thi chn i tuyn H KHTN H Ni, vng 1)

Li gii.

H cho tng ng vi

3 2

3 2

3 2

3 6 6 3( 2) ( 2)( 2 3)4 16 10 4( 4) ( 2)( 2 5)2 4 9 28 2( 2) ( 4)( 4 7)

y x x y x x xz y y z y y yx z z x z z z

Nhn tng v cc phng trnh ca h, ta c

2 2 2

2 2 2

24( 2)( 2)( 4) ( 2)( 2)( 4) ( 2 3)( 2 5)( 4 7)

( 2)( 2)( 4) 0( 2 3)( 2 5)( 4 7) 24

x y z x y z x x y y z z

x y zx x y y z z

-Nu ( 2)( 2)( 4) 0 2 2 4x y z x y z .

Ta thy rng nu 2x th theo phng trnh th nht, 2y ; theo phng trnh th hai, 4z v h cho c nghim ( , , ) (2, 2, 4)x y z .

Tng t nu 2y hoc 4z .

-Nu 2 2 2 2 2 2( 2 3)( 2 5)( 4 7) 24 ( 1) 2 ( 1) 4 ( 2) 3 24x x y y z z x y z

Ta thy rng 2 2 2( 1) 2 ( 1) 4 ( 2) 3 2.3.4 24x y z nn ng thc phi xy ra, tc

l 1, 1, 2x y z .

Th li, ta thy b ny khng tha mn h cho.

Vy h phng trnh cho c nghim duy nht l ( , , ) (2, 2, 4)x y z .

Nhn xt. Bi ny cng c dng tng t nh mt bi cp trn v ni chung th dng ny kh quen thuc; tuy nhin, im mi ca bi ny l khng phi nhn c nghim ( , , ) (2, 2, 4)x y z thng qua cc bt ng thc so snh vi nghim na m l qua vic chng minh b nghim qua vic chng minh n l duy nht vi vic dng cc tam thc bc hai.

51


2

2

2 2

11

3 2 6 2 2 1log ( ) log ( )

y x xey

x y x y

( thi chn i tuyn trng THPT Cao Lnh, ng Thp)

Li gii.

iu kin xc nh 2 6 0

2 0x yx y

Xt hm s: 1 0( ) ( ), [ , )tf t e t t . Ta c 1 2 0( ) ( ) ( )t t tf t e t e e t nn y l hm ng bin. Do

2 2 2 22

2 2 2 2 2 22

1 1 11

( ) ( ) ( ) ( )y x x yxe e x e y f x f y x y x yy

.

Phng trnh th hai ca h tng ng vi

2 23 2 6 2 2 1log ( ) log ( )x y x y

3 2 3 22 22 6 2 2 2 6 2 2log ( ) log ( ) ( ) ( )x y x y x y x y (*)

Xt hai trng hp

-Nu x y th thay vo (*), ta c 3 23 6 2 2 2( ) ( )x x .

Theo iu kin ban u th 2 2 0 2 4 2 2 0x x x .

Hn na: 3 2 2 3 23 6 2 2 4 2 27 46 0 3 6 2 2 4( ) ( ) ( ) ( ) ( ) ( )x x x x x x .

Do : 3 2 23 6 2 2 4 2 2 2( ) ( ) ( )x x x nn phng trnh ny v nghim.

-Nu x y , thay vo (*), ta c 3 2 36 2 2 6 8 6 2 4( ) ( ) ( )x x x x .

Suy ra: 4y x . Th li thy tha.

Vy h cho c nghim duy nht l 4 4( , ) ( , )x y .

Nhn xt. H dng ny rt quen thuc vi tng chnh l dng tnh cht ca hm n iu: ( ) ( )f a f b a b . bi trn cng ch cc nh gi trong trng hp x = y, bi v khi

phng trnh bc ba thu c phi gii theo cng thc tng qut, iu thng b trnh cc k

52

thi HSG; do , vic tm mt nh gi thch hp chng minh nghim ca n khng tha bi l mt iu kh t nhin.

Bi 43. Gii phng trnh sau: 2 2

2

2 2

2 11 2 1 4

x x x x xx x x x

( thi HSG tnh Bnh Phc)

Li gii.

iu kin xc nh 2 2 1 170 2 4,0 4 12

x x x x x .


2 22

2 2

2 11 4 ( 2) 1 4 ( )

x x x x xx x x x

Xt hm s ( ) , 0,41 4

tf t tt

, ta c

21 4 1( ) ( ) 0, 0, 4

2 2 4 (1 4 )t tf t t

t t t

nn y l hm ng bin.

Phng trnh trn chnh l 2 2 2( 2) ( ) ( 1) 0f x x f x x x (*).

Ta xt hai trng hp

-Nu 2 2 2 21 171 2 ( 2) ( )2

x x x x x f x x f x x , ng thi 2 1 0x ,

khi 2 2 2( 2) ( ) ( 1) 0f x x f x x x .

-Nu 2 2 2 21 1 2 ( 2) ( )x x x x x f x x f x x , ng thi 2 1 0x , khi 2 2 2( 2) ( ) ( 1) 0f x x f x x x .

Th trc tip thy 1x tha mn (*)


53

Nhn xt. Vic pht hin ra hm s ( ) , 0,41 4

tf t tt

nh trn khng kh, c th thy

ngay t vic quan st biu thc v t iu kin xc nh; tuy nhin, vic ny cng d khin ta lm tng n vic xt hm s no m khng ngh ra cch nh gi kiu nh trn.

Mt bi ton c cng cch nh gi nh trn l 33 2( ) ln( 1)x xe x x x e .

Cc bn th gii thm bi ton sau 2 2

2 2

2 11 2 1 4

x x x x xx x x x

.

Bi 44.

1/ Gii phng trnh 3 23 3 4 3 2x x x x

2/ Tm s nghim ca phng trnh

2011 2009 4 2011 2009 2 2(4022 4018 2 ) 2(4022 4018 2 ) cos 2 0x x x x x x x

( thi chn i tuyn Chuyn Nguyn Du)

Li gii.

1/ Phng trnh cho tng ng vi 33 3 4 2 3 ( 1)x x x

t 31 3 4y x . Ta c h phng trnh 3

3

( 1) 2 4( 1) 3 4x x yy x

Tr hai phng trnh ca h, v theo v, ta c 2 2

2 2

( ) ( 1) ( 1)( 1) ( 1)

0( 1) ( 1)( 1) ( 1) 1

x y x x y y y x

x yx y

x x y y

Suy ra 3 3 2 231 3 4 ( 1) 3 4 3 4 ( 1)( 2) 0 1 2x x x x x x x x x x .

Th li ta thy tha.

Vy phng trnh cho c hai nghim phn bit l 1, 2x x .

2/ t 2011 20094022 4018 2t x x x . Ta c

54

2 2 24 2 2 2 2 2

2 2 2

1 sin 2 (sin cos )2 cos 2 0 ( 1) sin 2

1 sin 2 (sin cos )t x t x x

t t x t xt x t x x

Ta c bn phng trnh sau sin cos , (sin cos ), sin cos , sin cost x x t x x t x x t x x .

Ta thy hm s 2011 2009( ) 4022 4018 2t x x x x l l nn ch cn xt cc phng trnh

( ) sin cos , ( ) cos sint x x x t x x x .

Ta c

2011 2009( ) sin cos 4022 4018 2 sin cost x x x x x x x x .

Xt hm s 2011 2009( ) 4022 4018 2 (sin cos )g x x x x x x c 2010 2008( ) 4022.2011 4018.2009. 2 (cos sin ) 0g x x x x x nn l hm ng bin.

Hn na (0) 1, (1) 0 (0). (1) 0g g g g , ng thi ( )g x lin tc trn (0,1) nn phng trnh ( ) 0g x c ng mt nghim thuc (0,1) , tc l phng trnh ( ) sin cost x x x c ng mt nghim thc.

Tng t, phng trnh ( ) cos sint x x x cng c ng mt nghim thc thuc (0,1) .

Do , mi phng trnh ( ) cos sint x x x v ( ) cos sint x x x cng c mt nghim thc.

Vy phng trnh cho c ng 4 nghim thc.

Bi 45. Gii h phng trnh sau 2 2 2 2 2

(2 )(1 2 )(2 )(1 2 ) 4 10 12 2 1 0

x x y y zx y z xz yz x y

( thi chn i tuyn H Tnh)

Li gii.

Ta c 2 2 2 2 2 2 22 2 1 0 ( ) ( 1) 0x y z xz yz x y x y z xy hay

1 10, 1 , ( ) ( )x y z xy y z x y xx x

.

Thay vo phng trnh th nht ca h, ta c

55

2 22

2 2

1 2 1(2 )(1 2 )(2 )(1 ) 4 1 10( )

1 2 2 1(2 )(1 2 )( )( ) 4 1 10( )

(4 )(1 4 ) 1 1 14 1 10( ) 4( ) 17 4 1 10( )

x x xx x x

x xx x xx x x

x x x x xx x x x

t 1 2t x tx

. Ta c 2 2 212t xx

, thay vo phng trnh trn, ta c

2 2 2 2

2

4( 2) 17 4 1 10 4 25 4 1 10 (4 25) 16(1 10 ) 07(4 20 29)(2 3)(2 7) 02

t t t t t t

t t t t t

Vi gi tr t ny, ta c 21 7 7 332 7 2 02 4

x x x xx

.

-Vi 7 334

x , ta tnh c 7 33 7,4 2

y z .

-Vi 7 334

x , ta tnh c 7 33 7,4 2

y z .

Th li ta thy tha.

Vy h cho c hai nghim phn bit l 7 33 7 33 7 7 33 7 33 7( , , ) ( , , ), ( , , )

4 4 2 4 4 2x y z .

Nhn xt. Vic pht hin ra hng ng thc trn l khng kh nhng vic thay cc gi tr vo v tm ra cch t n ph thch hp qu l khng n gin, cn c cch bin i chnh xc. Bi ton c hnh thc v tng cng kh th v.

Bi 46.

1/ Gii phng trnh sau 22010 ( 1 ) 1x x x .


3 3

4 2 52 2

xy x

x y

y xx y

56

( thi chn i tuyn trng THPT So Nam, tnh Qung Nam)

Li gii.

1/ Phng trnh cho tng ng vi 22010 1x x x .

Ta s chng minh phng trnh ny c nghim duy nht l 0x . Tht vy

Xt hm s 2( ) 2010 ( 1 )xf x x x , ta c 2

( ) 2010 .ln 2010 ( 1)1

x xf xx

-Nu 0x th

2

1( ) ln 2010 ( 1) 011

f x

x

nn y l hm ng bin, m (0) 0f nn

phng trnh ny c ng mt nghim 0x vi 0x .

-Nu 1x , ta c 2 3

2 3 2 5

1 3 1( ) 2010 .(ln 2010) , ( ) 2010 .(ln 2010) . 02( 1) ( 1)

x xf x f xx x

.

Suy ra ( )f x l hm ng bin nn 2(ln 2010) 1( ) ( 1) 0

2010 2 2f x f nn ( )f x l hm

nghch bin, suy ra 2( ) lim ( ) lim [2010 ( 1 )] 0xx x

f x f x x x

nn phng trnh (0) 0f

khng c nghim vi 1x .

-Nu 112

x th 2 21 1 5 11 ( ) 12 2 2

x x , 1 5 1201022010

x nn trong

trng hp ny phng trnh v nghim.

-Nu 1 02

x th 2

( ) 2010 .ln 2010 ( 1) 01

x xf xx

nn y l hm ng bin, suy ra

( ) (0) 0f x f .

Tm li, phng trnh cho c nghim duy nht l 0x .


3 3

4 2 52 2

xy x

x y

y xx y

T phng trnh th hai ca h v tnh ng bin ca hm s 3( ) 2tf t t , ta c x y .

57

Thay vo phng trnh th nht ca h, ta c 2 24 2 4 4 ( 1) 3 4

2 2

4 2 5 5 4 2 8 4 3 0( 1) ( 2 3) 0 1

x x xx x x x x xx x x x

Th li, ta thy tha; tng ng vi gi tr x ny, ta c 1y .

Vy h phng trnh cho c nghim duy nht l ( , ) (1,1)x y .

Bi 47. Gii h phng trnh 11 10 22 12

4 4 2 237 13 8 2 (3 3 1)

x xy y y

y x y x x y

( thi chn i tuyn TP.HCM)

Li gii.

Ta thy h ny khng c nghim tha 0y nn ta ch xt 0y , khi ta c

11 10 22 12 11 11( )x xx xy y y y yy y

.

Xt hm s 11 10( ) , ( ) 11 1 0,f t t t t f t t t nn y l hm ng bin.

ng thc trn chnh l 2( ) ( )x xf f y y x yy y

.


2 2 23 32 3 2

7 13 8 3 17 13 8 2 (3 3 1) 2 3x x x x x xx x x x x

.

t 1 0tx

. Ta c

3 32 3 2 3 2 27 13 8 2 3 3 (2 1) 2(2 1) (3 3 ) 2 3 3t t t t t t x t t t t

Xt hm s 3 2( ) 2 , 0 ( ) 3 2 0f a a a a f a a nn hm ny ng bin. Phng trnh trn chnh l

3 32 2 3 2 2(2 1) ( 3 3 ) 2 1 3 3 (2 1) 3 3 ( 1)(8 5 2) 0f t f t t t t t t t t t t t Do 0t nn khng c gi tr no tha mn.

Vy h phng trnh cho v nghim.

58

Nhn xt. Do phng trnh thu c sau khi tm ra quan h gia x v y khng c nghim dng nn ta c th dng bt ng thc nh gi thay v dng hm s nh trn.

Bi 48. Gii h phng trnh:

2

2

2

2009 2010 ( )2010 2011 ( )2011 2009 ( )

x y x yy z y zz x z x

( thi chn i tuyn chuyn Quang Trung, Bnh Phc)

Li gii.

t 2009 0a , ta xt h tng qut hn l

2

2

2

( 1) ( )( 1) ( 2) ( )( 2) ( )

ax a y x ya y a z y za z ax z x

(*)

Ta tnh c 2 2 2( ) ( ) ( ) ( )( )

2x y z x y zax x y x z

Tng t ( 1) ( )( ), ( 2) ( )( )a y y z y z a z z x z y

T y suy ra 2.( 1) .( 2) ( )( )( ) 0ax a y a z x y y z z x

Mt khc, t (*) ta thy rng tng ca tng cp trong ba gi tr , ( 1) , ( 2)ax a y a z u khng m, ta s chng minh rng c ba gi tr ny u khng m.

Tht vy, gi s 0 0ax x , t phng trnh th nht v phng trnh th ba ca (*), suy ra ( 1) 0, ( 2) 0 , 0a y a z y z hay , 0 ( )( ) 0x y x z ax x y x z , mu thun.

Do 0ax . Tng t, ta cng c ( 1) , ( 2) 0a y a z .

Nhng tch ca ba s ny li khng m nn ta phi c ( 1) ( 2) 0ax a y a z x y z .

Th li thy tha. Vy h cho c ng mt nghim l 0x y z .

Nhn xt. R rng cc h s ban u l chn da theo thi quen chn h s trng vi nm cho nn ta hon ton c th xt bi ton tng qut vic bin i thun tin hn. Bi ton thc s th v sau khi c ( )( ), ( 1) ( )( ), ( 2) ( )( )ax x y x z a y y z y z a z z x z y . Nu khng dng bt ng thc nh gi m c gng dng cc php th th kh c th thnh cng.

59


2 2

2

15574 3 (3 1)25

x y

x x y x

( thi chn i tuyn Ngh An)

Li gii.

H cho tng ng vi

2 22 2

22 2

102( )5( ) 125

57 474 3 3 2 2 3 325 25

x yx y

x x xy y x y x xy y

Ta thy 2 2 47 472 2 3 3 (2 )( 2 ) (2 ) ( 2 )25 25

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

t 2 , 2x y a x y b , ta c

2 2 2 2

2

75

121 ( ) 2 1 2 ( ) 1

2547 94 144 172 2( ) ( 1)25 25 25 25

13225

a b

aba b a b ab ab a b

ab a b ab a b a b a b

ab

Ta thy h phng trnh th hai v nghim, h th nht c hai nghim l 3 4 4 3( , ) ( , ), ( , ) ( , )5 5 5 5

a b a b , tng ng l 2 1 11 2( , ) ( , ), ( , )5 5 25 25

x y .

Vy h phng trnh cho c hai nghim phn bit l 2 1 11 2( , ) ( , ), ( , )5 5 25 25

x y .

Nhn xt. Cch phn tch phng trnh th hai qu tht rt kh thy. Bi ton ny thc cht xut pht t mt h i xng thng thng, nhng qua cc php th v tch biu thc, n tr nn phc tp v vic bin i ngc li thng phi m mn. Ta cng c th nhn phng trnh th nht vi 25 v phng trnh th hai vi 200 ri cng li, ta c 225(3 1) 144x y , cc gi tr 25 v 50 ny chn bng phng php h s bt nh vi mong mun tm ra mt quan h p gia x v y, nh bnh phng trn chng hn.

60

Bi 50. Cho cc tham s dng , ,a b c . Tm nghim dng ca h phng trnh sau :

2 2 24x y z a b c

xyz a x b y c z abc

( kim tra i tuyn Ninh Bnh)

Li gii.

Phng trnh th hai ca h tng ng vi 2 2 2

4a b c abcyz zx xy xyz

.

t 1 1 1, ,a b cx y zyz zx xy

, suy ra 2 2 21 1 1 1 1 1 4x y z x y z (*).

D thy 1 1 10 , , 2x y z nn tn ti cc gi tr u, v tha 0 , 2u v v 1 12sin , 2sinx u y v .

Thay vo (*), ta c 2 2 21 14 .sin .sin 4sin 4sin 4 0z z u v u v .

y l phng trnh bc hai theo bin 1z , ta c 2 2 2 2 2 2 2(2sin .sin ) (4sin 4sin 4) 4(1 sin )(1 sin ) 4cos .cos 0u v u v u v u v .

Suy ra phng trnh ny c hai nghim l 11

2sin sin 2cos cos 02sin sin 2cos cos 0

z u v u vz u v u v

Do 2 .sin , 2 .sin , 2 (cos cos sin sin )a yz u b zx v c xy u v u v .

Thay vo phng trnh th nht ca h, ta c

2 2

2 .sin 2 .sin 2 (cos cos sin sin )

( cos cos ) ( sin sin ) 0

cos cos sin sin 0

x y z yz u zx v xy u v u v

x v y u x v y u z

x v y u x v y u z

Ta tnh c sin sin22 2 2

a yb x a b a bz x v y u zzx yz z

Tng t, ta cng c ,2 2

c a b cy x .

61

Vy h phng trnh cho c nghim duy nht l ( , , ) ( , , )2 2 2

b c c a a bx y z .

Nhn xt. y l bi ton trong IMO Shortlist, bi v li gii thc s rt hay, l mt kt hp p gia i s v lng gic. Ta cng c th gii bng bin i i s nh cch t n ph

, ,2 2 2

b c c a a bx u y v z w v nh gi bng bt ng thc.

Bi 51. Gii h phng trnh sau trn tp hp s thc 2 2

2 2

3 3

3 0

x yxx yx yyx y

( thi chn i tuyn Chuyn Vnh Phc, tnh Vnh Phc)

Li gii.

Ta s gii h phng trnh ny bng s phc.

Nhn phng trnh th hai ca h vi i (n v o 2 1i ) ri cng vi phng trnh th nht,

ta c 2 2 2 2 2 23 3 3( ) ( )3 ( ) 0x y xi yi x yi i x yix yi x yi

x y x y x y

t 2 21 x yiz x yiz x y

. ng thc trn c vit li l

23 3 (1 2 )0 3 3 0 2 12

i iz z z i z z i z iz

.

-Nu 2z i , suy ra 2 2, 1x yi i x y .

-Nu 1z i , suy ra 1 1, 1x yi i x y .

Th li ta thy tha.

Vy h cho c hai nghim l ( , ) (2,1), (1, 1)x y .

Nhn xt. Dng ton ny cng kh ph bin v u chung tng l gii quyt bng s phc.

Cc bi ton tng t 2 2

2 2

3 10 1

10 3 2

x yxx y

x yyx y

, 2 2

2 2

2 2

2 0

x yxx y

x yyx y

( chn i tuyn H Ni 2007)

62

Trn thc t, ta cng c th gii bng cch dng bin i i s, nhn x v y thch hp vo tng v ca cc phng trnh ri tr li thu c quan h n gin hn gia cc bin ny.

Bi 52. Gii h phng trnh:

4 4

2 2 3

23( )

x x y yx y

( kim tra i d tuyn trng THPT Chuyn HSP H Ni)

Li gii.

t 33, ,x y a x y b c .

T phng trnh th hai ca h, ta c 3 3ab c ab c .

Ta c 2 2

,a b a bx y . Suy ra

2 24 4 2 2 2 2

2 2 2( )( )( ) ( )a b a b abx y x y x y x y ab a b

, hn na:

3322 2 2

( )( ) a b a b a c bx y a b

Do , phng trnh th nht ca h cho tng ng vi

32 2 2 2 3

2 2( ) ( )ab a c ba b c a b a c b

Ta c h mi l

2 2 3 2 42 4 3 3 4 3 3

2 1 0( ) ( ) ( )( )c a b a c b c cc a a ca c a ac ca a c

aaab c

1a a cc

.

Suy ra h ny c hai nghim l 211( , ) ( , );( , )a b c cc

.

63

Xt hai trng hp

- Nu 1,a c b th 3 31 3 1 3 1

2 2 2,cx y .

- Nu 21 ,a b cc

th 3 3

2 2

3 3

1 1 1 2 1 1 1 12 2 2 23 3

,c cx c y cc c c c

Vy h cho c hai nghim l: 3 3

3 3

3 1 3 1 2 12 2 3 3

( , ) , , ,x y

.

Bnh lun. y l mt h phng trnh rt p, hnh thc ca n d lm chng ta bi ri khi khng th nhm c nghim no cng nh tm c mt hm s no kho st nh tng thng thng. Li gii thun ty i s v cch t n ph nh bi cn phi ch , n tng xut hin trong VMO 2005

3 2

2 2

3 498 6 17

x xyx xy y y x

Mt bi ton tng t nh trn cng c li gii rt th v 4 4

2 2 5

3 14 2

( ) 5 0

x yy x

x y

Bi 53. Gii phng trnh 2 3 532 .sin .cos 2 1 1x x x x x x x x

( thi chn i tuyn H Ni)

Li gii.

Ta thy phng trnh khng c nghim 12

x nn ta ch xt 12

x .

Xt hm s 2 5 33 1( ) 2 .sin .cos 2 1 1,2

f x x x x x x x x x x .

Ta c 2 4 223

2( ) 3 .sin (2 1) cos 5 3 13 (2 1)

f x x x x x x xx

Ta s chng minh nh gi mnh hn l 2 4 23 .sin (2 1) cos 5 3 1 0,x x x x x x x (*)

64

Ta thy biu thc ny khng thay i khi thay x bi x nn ta ch cn xt 0x .

Ta cn chng minh bt ng thc sau 3 2

sin ,cos 1 , 06 2x xx x x x .

Xt hm s 2

( ) cos 12xg x x , 0x , ta c

( ) sin , ( ) cos 1 0 ( ) sin (0) 0g x x x g x x g x x x g . Do , ( )g x l hm

ng bin trn [0, ) , suy ra 2 2

( ) (0) 0 cos 1 0 cos 12 2x xg x g x x .

Tng t, ta cng c 3

sin6xx x .

T hai nh gi ny, ta c

3 22 4 2 2 4 23 .sin (2 1) cos 5 3 1 3 ( ) (2 1)(1 ) 5 3 1

6 2x xx x x x x x x x x x x .

Hn na, ta cng c

3 2 4 4 22 4 2 2 2 4 4 23 7 33 ( ) (2 1)(1 ) 5 3 1 3 1 5 3 1 0

6 2 2 2 2x x x x xx x x x x x x x x x

nn 2 4 23 .sin (2 1) cos 5 3 1 0,x x x x x x x .

Do (*) ng hay ( ) 0,f x x . Suy ra ( )f x l hm ng bin nn phng trnh cho c khng qu mt nghim. Mt khc (0) 0f nn 0 l nghim ca phng trnh cho.


Nhn xt. im quan trng nht ca bi ton l chng minh ( ) 0f x , nhng l mt biu thc va c cha c sin ,cosx x v cn thc, ng thi s hng t do ca hm s li m nn tht s rt kh d on c phi lm g trong trng hp ny. Vic b i biu thc cha cn trn rt quan trng v n gip ta c c mt hm s chn v ch cn xt biu thc trn min [0, ) ;

trn min , ta cn c thm hai nh gi 3 2

sin ,cos 16 2x xx x x nn bi ton a v chng

minh bt ng thc thng thng. Nu khng a cc yu t lng gic v a thc th phi tip tc o hm v cha chc iu ny kh thi. Bt ng thc (*) c th lm mnh thm na l

2 4 23 93 .sin (2 1) cos 1 0,2 2

x x x x x x x .

65


2 2

2

2 2

( 2) ( 3) ( 3)( 2)5 9 7 15 3

8 18 18 18 84 72 24 176

x y y x zx x z y yzx y xy yz x y z

( thi chn i tuyn HSP H Ni, ngy 2)

Li gii.

t 2, 3a x b y .

Thay vo tng phng trnh ca h cho, ta c

2 2 2 2 2 2( 2) ( 3) ( 3)( 2) ( 4) 4 0x y y x z a b b a z a ab b bz b ,

2 25 9 7 15 3 7 3 0x x z y yz a a b bz

2 2

2 2

2 2

8 18 18 18 84 72 24 1768 2 18 72 18 18 30 94 08 2 18( 4 ) 30 94 0

x y xy yz x y za a b b ab bz za a b ab bz b z

Suy ra

2 2

2

2 2

4 07 3 0

8 2 18( 4 ) 30 94 0

a ab b bz ba a b bza a b ab bz b z

(*)

T phng trnh th nht v phng trnh th ba, ta c 2

2 2 2 5 478 2 18 30 94 0 10 2 30 94 015

a aa a a z a a z z .

Thay vo phng trnh th hai, ta c 2 2 2

2 22

5 47 5 12 5( )7 05 5 5 12

a a a a a aa a b b b a a ba a

.

Nhn phng trnh th nht ca h (*) vi 3 ri tr cho phng trnh th hai, ta c

2 22 3 3 5 0a a ab b b

Thay 25 47

15a az v

2

2

5( )5 12

a aba a

vo phng trnh ny, ta c

66

22 2 22

2 2 2

2 2 2 2 2 2 2 2

6 5 4 3 2

2 2

15 ( ) 5( ) 25( )2 3 05 12 5 12 5 12

(2 )(5 12) 15 ( ) 25( ) (5 12) 75( ) 0

50 70 208 94 482 156 0( 2)(5 14 13)(5 11 3) 0

0

a a a a a a aa aa a a a a a

a a a a a a a a a a a a a

a a a a a aa a a a a a

a

11 612

10a a

Tng ng vi cc gi tr ny, ta tm c bn nghim ca h cho l

47 4 29 31 61 2 61 28 13 61( , , ) ( 2, 3, ), ( 4, , ), ( , , ),15 3 15 10 15 15

61 31 2 61 28 39 61( , , )10 15 15

x y z

Nhn xt. Vic t n ph 2, 3a x b y lm cho h cho n gin i kh nhiu nhng cc lin h phc tp gia cc bin th vn cn. Bi ton y c th c gii theo mt cch nhn cc phng trnh cho mt i lng thch hp ri cng li nhng r rng iu ny khng phi d dng thc hin c. Vic dng php th tuy phc tp nhng li rt t nhin v cng may mn l phng trnh cui khng c cha cn g na. y tnh ton kh nng v cng khng d dng m t tin bin i biu thc nhn c sau php th khi cha chc g n c nghim p m nh gi.

Bi 55.

Tm , ,x y z tha mn h

2 2

2 2

2 2

2 ( ) 11 2 2 2

(3 1) 2 ( 1)

z x y x yy z xy zx yzy x x x

( thi chn i tuyn trng H KHTN H Ni, vng 3)

Li gii. T phng trnh th ba ca h, ta c

2 3 2 3

2 2 2

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

x x x x x x x xy x y x yx x x

.

t tan , ( , ) cos 02 2

x . Ta c 3

2

tan 3 tantan tan 3 tan3tan 1

y y

.

67

T phng trnh th nht ca h, ta c 2 2 21 (2 tan tan 3 ). tan 3 1 2 tan . tan 3 tan 3 12( ) 2 tan 3 2 tan 3

tan 3 cot 3 1 sin 3 cos3 1tan tan ( ) tan2 2 cos3 sin 3 sin 6

x yz x y

T phng trnh th hai ca h, ta c 2 2 2 2 2 2

2 2

22 2

2 2

22

2 2 2 1 ( ) 11(tan 3 tan tan tan ) 1 tan

sin 6sin 3 1 1 2sin 3 1 1( tan ) ( tan )cos3 2sin 3 cos3 cos 2sin 3 cos3 coscos6 1 cos 6 cos sin 6 sin( tan ) (sin 6 cos

x y z xy zx yz x y z x x

2 2

22

1)sin 6 cos cos

cos5 1( ) cos5 cos sin 6 cos cos5 cos( 6 )sin 6 cos cos 2

2cos5 cos( 6 ) 5 ( 6 ) 2 , 22 2 22 11 2

2cos5 cos( 6 ) 5 ( 6 ) 2 , 22 2 22 11 2

kk k

kk k

, k

Do ( , )2 2

nn hai h nghim 2 , 2

k k khng tha mn.

Vi hai h nghim 2 22 11

k , ta tm c tt c 10 gi tr tha mn l

3 5 7 9, , , ,22 22 22 22 22

.

Vy h phng trnh cho c cc nghim l

1 3 5 7 9( , , ) (tan , tan 3 tan , tan ), , , , ,sin 6 22 22 22 22 22

x y z

.

Nhn xt. Cch dng lng gic y c l l con ng duy nht gii bi ny bi v vi cc nghim nh trn th khng th c cch i s no m tm ra c. tng quan trng nht ny xut pht t biu thc ca x y hon ton ging h s ca khai trin tan 3 . Do , bi ny tuy bin i phc tp nhng tng cng kh t nhin!


Documents

5.pthpt_2