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8/3/2019 H phng trnh t cc k thi hc sinh gii
1/15 Page 1 of 15
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
8/3/2019 H phng trnh t cc k thi hc sinh gii
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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
8/3/2019 H phng trnh t cc k thi hc sinh gii
3/15 Page 3 of 15
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
8/3/2019 H phng trnh t cc k thi hc sinh gii
4/15 Page 4 of 15
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
8/3/2019 H phng trnh t cc k thi hc sinh gii
5/15 Page 5 of 15
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
8/3/2019 H phng trnh t cc k thi hc sinh gii
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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
8/3/2019 H phng trnh t cc k thi hc sinh gii
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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