Tuyển tập 69 bài toán khó trong đề thi THPT quốc gia môn toán

Câu Lạc Bộ Yêu Toán Lý
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



2016 y  7 1  9x 2  1  2017  3x  1 y  1  2016 6x




9
3
 6x   20   9y 
 16   2016
y
xy




Biên Soạn: Nguyễn Minh Thành – Nguyễn Tuấn Thành
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Nguyễn Minh Thành – Sư Phạm Toán, Nguyễn Tuấn Thành – CS

1


Câu Lạc Bộ Yêu Toán Lý

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Phương trình – bất phương trình
1) (NMT)

2x 2  3x  1  3 x 3  1  x  1  x  1

7  2x  x 2  2 

2) (NMT)

1

3) (Mathlinks)

x 8
2

 2x  2 

4) (NMT)
5) (NMT)



1
1

 5x  
2

x
x


1
8x 2  1



x

1
3

x2

x  1   3x  9  2x  3  4x 2  10x  4

x 3  6x 2  18x  10  2  x  1 2x  1

4x 2  5x  4   x  1 6x  1   x  3 2x 2  12x  1

6) (NMT)

1  x 

7) (Mathlinks)

1  x  1  x  1  x  1 



x2 1  2



2

8) (NMT)

6 x 2  5  12 3 x 2  3x  2  3x 2  x  32

9) (NMT)

3x  2  2 2x 2  6x  21   x  6  2  x  2 2x  5

4x 3  1  3x 4   x 2  1 x 2 1

10) (NMT)



x 2  2  x 2x  x  4

11) (NMT)

2x  4 x  1  1

12) (NMT)

13) (NMT)

x



3





4  x2



x  x 2 1  2  x 2  x  x



2

x 3  3x  13  x  1  x 2  2x  3  2x 2  1

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Nguyễn Minh Thành – Sư Phạm Toán, Nguyễn Tuấn Thành – CS

2

Câu Lạc Bộ Yêu Toán Lý
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14) (NMT)

x 2  x 2  1  x 4  x 2  1  2x  4x 2  2x  1  2x  1





15) (NMT)

x3  x  2 x  2  x3  4  2  x  4

16) (NMT)

6x 2  7x  4   x  2  3x  2  7  x  1 3 3x  2

2x 3  x 2  1  2  x  1 x 2  1. x 2  x

17) (NMT)

18) (NMT)

3 3 2x 2  x  6 4x  5  2x 2  3x  16

2x 4  2x 3  3x 2  1  1  x 4  2x 3

19) (NMT)





x  2x 3  x 2  4x  2  x 2  3x  4

20) (NMT)

2

21) (NMT)

2  9x  2 2x 2  x  3 

22) (NMT)

x  1  x 2  2x  1  3 x 3  2x 2  5x  2

23)

24) (NTT)

x2
2  x 4  x 2  1  1



4





x 1



2x  1  1

x  2x  1  2

1
x 1

1  2x  x 2  1  2x  x 2  2  x  1  2x 2  4x  1
4

Hệ phương trình
1)

2) (NMT)

 x  3y  xy   y  12  1  2 y  2x

2 x  y  4  4 2  y  9y 2  16

 x 2  xy  7y 2  y 2  xy  7x 2  6 xy



3
3
2
2
18 2  x  y   12 3 3  x  y  x  4x  2   17y  55

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Nguyễn Minh Thành – Sư Phạm Toán, Nguyễn Tuấn Thành – CS

3


Câu Lạc Bộ Yêu Toán Lý
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3) (NMT)

4)





x 2   y  5 x 2  1  1  y  3


 4x 2  1  x  1  y  3




x2 1  y  4



12x 2 y  3xy  3xy3  2y  2


2
y
2
 xy  xy  3  2x x  2 y  1  2 x


5) (NMT)



 x  2y  3
1
 1 
7xy
 x  y  2  2 

x  y 1 
y 1



4
xy
 x  y  4 x  1  y  1  x  y  x  1  6

6) (NMT)


x y  x  2  y 1
 y  x  2 1 
x x


3
3
2
 y  x  2  2 x  3x  4  3x  2x  x

7)

8) (NTT)

9)





 y
x
y4




2
x  y y  x 2  x  y2 


 y  x  1  32  x  2y  1 2y  2









 y  3 1  y 1  x  5


 x 1  x2  y 1  1






y2  x  2  1

 x 2  1 y 4  1  2xy 2  y3  1


 2
4
4
 xy  3xy  2   xy  x  2y   1

10) (NMT)

 xy  4y  4  4  x  y  1 y  1

 xy  y  1  2 1  x

11) (NMT)

  x  y 2  5x  4y  13  x  2  2 y  1


 3 y  5  y  2  1  x 2

12) (NMT)
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Nguyễn Minh Thành – Sư Phạm Toán, Nguyễn Tuấn Thành – CS

4


Câu Lạc Bộ Yêu Toán Lý
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



2x 3  y  x  y  3  12  2 x  3  y  1

 2
 x  1 y  1   y 2  2y  2  x  3  2  x  y  1 x  y  1

13)

14) (NTT)

 x  1 x  1  2y  y  x  1
 2 2
2
2
 y x  3x  4  4 xy  3x  4  x  17
4
2 2
2
4
2

4x  4x y  4x  6xy  y  5y  4  0
 4
2 2
2
4
2


4x  4x y  2x  8xy  y  2y  3  0

15) (NMT)

 y  1  2  2x  1 x  y   x  y  2

5x  4  2 3x  2  y  1

16) (NMT)

 x  2y  2y  1  2


 x  2y  1  2 2xy  x  4  x  2y  1  8y  x  1

17) (NMT)

 2
xy2
x 1

 x  2x 
2
y 1
y2  1

 x y  3  y  1 x  4  2  x 3 y  1  2x  x 2
 


 


18)





 xy  2  y x 2  2
 2
2
2
 y  2  x  1 x  2x  3  2x  4x

19) (NMT)

 x  y 1  y 1
1
3
 3
x 1
x 1  x  1

3

2
3 3
2 x  y  1  y  2x  3y  x  3x  3y  4

20) (NMT)

 x  x 2  2  x  y  xy 2  4x  x  1  16


2
 x 6x  3y  4 2xy  8 2x  4x 2  y 2   16 2x x  y x  1


21) (NMT)

2y 2   x  12  4y x  2  7


1
x  y3

2

x

1

y

1

1
x


2

y

7

x

y














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Nguyễn Minh Thành – Sư Phạm Toán, Nguyễn Tuấn Thành – CS

5

Câu Lạc Bộ Yêu Toán Lý
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

22) (NMT)

 

 xy  3 x  y  2  x  4 x  3

 2 1 
1
 x   1 xy  1  1 
x 
x




23)

 x 2  y3  y 2  2 3 x 4  3 x 2  2y y  1 x  3 x


3
 x 4  x 3  x 2  1  x  y  1  1

24)

 4 7  x  12  y 2  y 2  1  x  6


2
 y 2  3x   y 2  2y 3x  1  x  2


25)

4x 2  x 2  1  1  x 2  y3  3y  2 


2
 x 2  y 2   1  x 2  2y




y 1







26) (NMT)







2016 y  7 1  9x 2  1  2017  3x  1 y  1  2016 6x




9
3
 6x   20   9y 
 16   2016
y
xy




Hình học phẳng Oxy
1) (NMT)

 21 13 
K ; 
 10 10 
y40

(I;R).

M(0;4)

2x  y  8  0 .

B  2; 2 

2)

H 1;3  H  A 

M  2;0 

3)

3x  y  2  0

x  2y  2  0

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Nguyễn Minh Thành – Sư Phạm Toán, Nguyễn Tuấn Thành – CS

6


Câu Lạc Bộ Yêu Toán Lý
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4)

D  2;1 , H 1;3 

5)

x 1

 17 59 
I ; 
9 9 

4x  7y  57  0

yA  yD .

D  3;1

6)

N  6; 4 
2x  y  1  0

x  5y  0
7)

 25 19 
 13 19 
E  ;   HC  A 'B', F  ;   CI  AB . M  2;3
 7 7
7 7
xA  xB .
3x  y  3  0 ,

8)

 55 24 
EC  FB  I  ; 
 23 23 

x  5y  10  0

Bất đẳng thức
1) (NMT)

x, y

P

2) (NMT) Cho x, y

x, y  1;3 .

4 5  x  y  1
x
y 1


x 3  x  y  2x   3  x  2y  2  y  1  6y  2
25
xy  1 .



 x  2y  3 x  y
1
P   x  y  2 2 
 1 

 2y  4
y 1
x 1
 x  y 1 



x 1  y 1



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Nguyễn Minh Thành – Sư Phạm Toán, Nguyễn Tuấn Thành – CS

7


Câu Lạc Bộ Yêu Toán Lý
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x 2  y2  z2  5 .

3) (NMT) Cho x, y, z

P   x  z

z
3x 2  4y 2  8z 2  8 1
y 1

 z 
2
2
x y
16z
2
4 8
4z 2  41  9xy(2z  3) .

4) (NMT) Cho x, y, z
P

x
x 2  9yz



y
y 2  9zx



1
(z 2  5)

2 10

x 2  y 2  z 2  2x .

5) (NMT) Cho x, y, z

x 2  y 2  xz 

2z  x  y 

 x 
P
 2
 15 
2
2
2
 x  1 y  1 x   y  xz  x  y  xz  1  x  1 
a 2  b 2  bc  c 2 .

6) (NMT) Cho a, b, c

P  a 2  2a 





4c 1  ab  1  abc
a

b


bc ca
bc
x  z, y  z .

7) Cho x, y, z
P

8) Cho x, y, z

x
y
3 z
z

 .

2
yz xz 2 xy
z  xy  yz  xz
x, y, z  1; x 2  y 2  z 2  6xy  2  x  y  z  .

x 1
y 1
xy
P




y  z 1 x  z 1  z 

9) Cho a, b

2

2

a 2  b 2  1  3b .

 1  b2 
b2
1
P4



a 1 a  b  1 a2 

2

10) Cho a, b, c  1 .
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Nguyễn Minh Thành – Sư Phạm Toán, Nguyễn Tuấn Thành – CS

8

Câu Lạc Bộ Yêu Toán Lý
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F

11)

1
2
3


 6 1  abc(abc  1)
6
3
1  a 1  b 1  c2

a, b, c sao cho a  b  c .

 3 1

a  2c 
2
2

b
c

P 2
 2 2  2

2
a b b c 
c






2

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Nguyễn Minh Thành – Sư Phạm Toán, Nguyễn Tuấn Thành – CS

9