Hệ phương trình từ các kì thi học sinh giỏi

8/3/2019 H phng trnh t cc k thi hc sinh gii

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PHNG TRNH, H PHNG TRNH(Trong cc k thi chn hc sinh gii cc trng, cc tnh)

Phn 1. Mt s bi ton c

1. (THPT Chu Vn An, H Ni)3 2 3

3 2 3

3 2 3

x x 2x 2y 1

y y 2y 2z 1

z z 2z 2x 1

2. (THPT chuyn L Qu n)

2

2 2 1

xy

xy 2x y 6 y 2x 0

x y 1 8

3. (H Ni Armsterdam, chn i tuyn lp 10)Gii h phng trnh

2 2 2 2

x y x y 2

x y x y 4

4. (Bc Giang, 08-09)

2

3 3

x x y 9

x(y x ) 7

5. (THPT chuyn L Qu n, B Ra Vng Tu, 08-09)

2 2 2 8 2 18x y z yz 2zx 3xy

x y z

6. (H Ni TST, 02-03)

Gii h phng trnh sau:

3 3

2

3

2

cos cos cos

sin sin sin

x y z

x y z

7. (H Ni, 03-04)

Gii cc phng trnh sau:

1. 2cosx+sin19x-5 2 sin 21 3 2 sin10x x

2.5 3

32 40 10 3 0 x x x 8. (H Ni, 04-05)

Gii phng trnh:)(493)(

sincos 22232

22

xx

xx

9. (H Ni, 05-06)Gii h ph-ng trnh:

2

1 y

x y 1 x 2

2x y 1 (1 2x y)(1 12x 2005y ) 0

2004 2.2005 2006 0


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10. (H Ni, 06-07)

Gii cc phng trnh sau:

1. 5 315 11 28 1 3 x x x

2. 2 24 1 1 2 2 1 x x x x

11. (H Ni TST, 06-07)

0yx

yx2y

2yx

y2xx

22

22

12. (H Ni, 08-09)

Gii phng trnh: 3 322 1 1 1 1 5 x x x x

13. (Chuyn H Long, Qung Ninh)

||1

||1log||3||

||1

||1log||3||

3

3

x

xxy

y

yyx

14. (Nam nh, 02-03)

a. Gii h phng trnh:x y 1 2x y

2x y x y

x.2 3y.2 2

2x.2 3y.8 1

b. Tm m phng trnh sau v nghim: (4m - 3) 3x +(3m - 4) x1 1 - m

15. (Nam nh, 01-02)

Tm tt c cc cp s (x;y) tha mn:

xy

yx

sin

12000sin

2000sin

1sin

sin

1sin

16. (Nam nh 08-09)2

2 2 2 3 4

4 3 2

z 2xyz 1

3x y 3y x 1 x y

z zy 4y 4y 6y z

17. (Ngh An TST, 06-07)Gii phng trnh(a 4)x + ax = 2(a 2)x vi hng s a > 4.

18. (Ngh An, bng B, 2008)

1)]80016093(8

cos[ 2 xxx

19. (Qung Nam, 07-08)Gii h phng trnh


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2 2 2

3 2

x y 2x y 0

2x 3x 6y 12x 13 0

20. (Hi Dng)Gii h phng trnh

2 y

2 x

x x 1 2006

y y 1 2006

21. (Olympic ng bng sng Cu Long, ln XV)

Gii h phng trnh4 3 2

4 3 2

4 3 2

x y 4x 6x y 4x y 0

y z 4y 6y z 4y z 0

z x 4z 6z x 4z x 0

21. (Bnh Phc, 08-09, vng 2)Gii phng trnh

24 1 x 1 3x 2 1 x 1 x

23. (Hi Phng, chn HSG khng chuyn, 08-09)Cho h phng trnh sau

2cos x x

y tan y 1

Chng minh rng h cho c duy nht nghim tho mn 0 < x < y


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3

3

3

3

x y 2y 2

y z 2z 2

z t 2t 2

t x 2x 2

28. (K s ti nng Bch khoa 2006)Phng trnh x3 ax2 + 4 = 0, (a l tham s) c bao nhiu nghim?

29. (K s ti nng Bch khoa 2007)

Cho phng trnh: 3

1 x x x 1 x m (1), m l tham s

a. Gii phng trnh (1) khi m =1b. Tm m (1) c nghim

30. (Thanh Ho TST, 08-09)

Gii bt phng trnh2x 4 2 x 23 (x 4).3 1

31. (ng Nai, 08-09)Gii phng trnh

1 2sin x 1 2cos x 1 3

32. (Thi Bnh 03-04)Gii v bin lun phng trnh

2 2x 2ax 2 2x 4ax a 2 2

3 3 x 2ax a

33. (Thi Bnh 04-05)

a. Gii h phng trnh

3 2

3 2

3 2

2z 1 x x x

2y 1 z z z

2x 1 y y y

b. Gii phng tr nh vi 0 < a < 1

34. (Thi Bnh 05-06)

a. Gii cc phng trnh

x x x x

1. 5sin x cos2x 2cos x 0

2.2007 2006 2005 2004

b. Tm m h sau c nghim

35. (Thi Bnh 06-07)a. Gii phng trnh

cos 2x 1 21 1

2 22 cos2x log 3cos2x 1

b. Tm gi tr nh nht ca a h c nghim


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36. (Olympic Min Trung v Ty Nguyn)

37. (Qung Nam 06 07)

a. Gii cc phng trnh sau trn tp s thc :a)

2

3

1 2 2

1

5

59 log ( 2 3) 3 log (2 1 2) 0

x x x x x x

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

b. Gii h phng trnh

2 2

2 2

2 2 1 28

x y x y

x y

38. (Hu)Gii h phng trnh:

2 2 2

2 2 2

x + log y = ylog 3 + log x

xlog 72 + log x = 2y + log y

39. (Hu 00 01)Gii phng trnh

2x 2x 1 2x 116 2.4 0

40. (Hu 06-07, vng 2)Gii h phng trnh

2 26x y 5xy 7x 3y 2 0

x yln x 2 ln y 2

3

41. (Hu 08-09)a. (Dnh cho hc sinh chuyn)Tm cc cp s thc ;x y sao cho:

2 4 32

8

x y

xy

b. (Dnh cho hc sinh khng chuyn)

Gii h phng trnh8 8 8

2

log 3log log

3log log

4y

xy x y

xx

y

42. (H Tnh TST, 07-08)

Gii h3 3

2 2

x - 8x = y + 2x

x - 3y = 6

43. (Cn Th TST, 08-09)a. (vng 1) Gii phng trnh sau trn R

2

2 2

y xy 1 0

x y 2x 2y 1 0


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b. (vng 2) Tm tt c cc nghim thc ca phng trnh2 2x 5x 10 60 24x 5x

44. (Tp. H Ch Minh, 04-05, vng 1)

2

2

2

33 2 2

4x 2 2y 15a.

4y 2 2x 15

b. 9x 12x 2 3x 8

c. x 4x 5x 6 7x 9x 4

45. (Tp.HCM, 03-04, vng 1)

Gii h phng trnh sau

46. (Tp.HCM, 02-03, vng 2)

a. Gii h phng trnh

b. Gii phng trnh

47. (Tp.HCM, 01-02, vng 1)

1. Gii cc phng trnh

a.

b.

2. ( ngh) Gii h phng trnh

48. (Tp.HCM, 96-97, vng 2)

49. (Thanh Ho, 01-02)

1. Gii phng trnh

1x

xlog cos x sin x log cos x cos 2x 0

2. Gii phng trnh sau vi x 0;2

:

1 2x 22x 1 x 2x 1 1 1

4 4 x4 x

50. (Thanh Ho, 02-03)

51. (Thanh Ho, 03-04, bng B)1. Gii cc phng trnh

2

7 3

a. sin x sin x sin x cos x 1

b.log x log ( x 2)

2. Xc nh s nghim x2

0; ca phng trnhsin x cos x2 2


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52. (Thanh Ho, 05-06, bng B)Gii h phng trnh

2logloglog

2logloglog

2logloglog

16164

993

442

xyz

zxy

zyx

53. (Thanh Ho, 07-08)1. Gii phng trnh

2. Gii h phng trnh 3 2

3 9 4

x y 2xy

x y 2xy

54. (H Tnh, TST, 09-10)4 3 4x 20 7 x x 9

55. (Thanh Ho, 05-06, ngh)

1.x

x

x

x

x

1

2

122

22

2 211

2. sin xcos xlog cos xlog sin x(1 cos x) (1 sin x)

3.3cosx cos x

(7 + 5 2) - (17 + 12 2) = cos3x .

4.2 4 23

3x - 3x + 1 = - x + x + 1 .

5. cosx + cos2x + cos3x + cos4x = -2

1

6. 23log5log3 53 xxxx

7. 3 1tgx (sin x + 2cos x)=5(sin x +3cos x).

8. log22

x + x.log7(x + 3)= log2x [2

x+ 2.log7(x + 3)]

9. Gii ph-ng trnh (sin)x + (tan)

x = ()x

(vi x l tham s, 0 < x

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