CÁC bài TOÁN điển HÌNH bồi DƯỠNG học SINH GIỎI

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1
CÁC BÀI TOÁN ĐIỂN HÌNH BỒI DƯỠNG HỌC SINH GIỎI
§ 1. CẤU TẠO SỐ TỰ NHIÊN
Bài 1:



Hd:

abc

Ta có: b = c


 suy ra c = 3.
+  
Th

2 + 2; 7 = 3

2 + 1.

Bài 2:
 
c 2000.
Hd:

10,,,0;0,  dcbaaabcd

-
abcd
= a + b + c + d hay 2000  (a + b + c + d) =
abcd
.

ab
= 19.

 1981 = 1 + 9 + 8 + 1 = 19.


Bài 3:
Tìm B   
A = B + C + 51.

Hd:

ab
,
0;0 , 10a a b  
.

51 cbaab
hay
519  ca


519  ca




b6

= 66.


Bài 4:
Tìm 


Hd:

)10,;0(,  baaab


ab
= (a  b) 15 +2
Hay b  16 = a  5 + 2
 
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


2
 
+ Vì a  5 + 2

0 nên b


0.

b = 2 thì a = 6.
 2)  15 + 2 = 62.


Bài 5:



Hd:

ab
, ( 0

a, b < 10, a

0).
Ta có
ab
= 5  (a + b) + 12, a + b > 12.
 a = 4  b + 12.
+ Vì 4  : 5  a chia h cho 4 , suy ra 
 


Bài 6:



Hd:

abc


( ) 11abc a b c   
(theo bài ra)

100 10 11 11 11a b c a b c         



89 10a b c   

11 10a b c   
)

89 1, 89 198a cb a cb abc      



Bài 7:
 


Hd:

- 

- 

- 
10, 14, 9.

55  10 = 45
là 104  14 = 90.

5544
-

104
-

144
-

9



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3
 9 = 135



Bài 8:
n 


Hd:
 208 =





Bài 9:



Hd:

ab
, ( 0

a, b < 10, a

0).
Ta có
ab
= (a  b)  28 + 1.
 
ab


 b = 1 thì
ab


 b = 2 thì
ab


 b = 3 thì
ab
 b = 8  5 = 3.
Bài 10:


Hd:

abc
, ( 0

a, b, c < 10, a

0).
Theo bài ra ta có:
abc
= (a + b + c)  20.

Khi  ó 10
ab
= 20 (a+b)
Suy ra
ab
=2(a+b) suy ra 10a+b=2a+2b
 a = b suy ra a = 1, b = 8.
 20.

Bài 11:
có 3 

Hd:

abc
, ( 0

a, b, c < 10, a

0).
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


4
Theo bài ra ta có:
abc
= 5  a  b  c.  
5abc

5.


5 25ab
.


 

Bài 12:
có 3 
.

Hd:

abc
, ( 0

a, b, c < 10, a

0).
Theo bài ra ta có:
cab - abc = 765

 11  c = 85 + b + 10  a
Vì 85 + b + 10  a  95  11  c  95  c = 9
 14 = b + 10  a  a = 1, b = 4.
 

Bài 13:
có 3 
.

Hd:

abc
, ( 0

a, b, c < 10, a


0).
Theo bài ra ta có:
abc = 7 bc


a 100 = 6 bc  

a 50 = 3 bc  
  a = 3,
bc = 50

 

Bài 14:
có 3 
.

Hd:

abc
, ( 0

a, b, c < 10, a

0).
Theo bài ra ta có:
cba - abc = 693

 99  (c  a) = 693

 c  a = 693 : 99 = 7
 
Bài 15:

.

Hd:
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


5

abc5
, ( 0

a, b, c < 10, a

0).
Theo bài ra ta có:
abc5 - 5abc = 531


abc 10 + 5 - ( 5000 + abc) = 531


abc = 614

Bài 16:

.

Hd:

abcd
, ( 0

a, b, c, d < 10, a

0).
Theo bài ra ta có:
abcd - ab = 4455


cd = 99 ( 45 - ab )

( 45 - ab ) = 0, ( 45 - ab ) = 1


( 45 - ab ) = 0:


( 45 - ab ) = 1:

Bài 17:

.

Hd:

abcd

, ( 0

a, b, c, d < 10, a

0).
Theo bài ra ta có:
abcd 4 = dcba

  3 thì tích
abcd 4 


1bcd 4 = dcb1


2bcd 4 = dcb2
 4  
 

2bc3 4 > 3cb2
là vô lý

2bc8 4 = 8cb2
 390  b + 30 = 60  c
 39  b + 3 = 6  c  b = 1, c = 7


Bài 18:

.

Hd:


Ab
, ( 0

b < 10, A > 0).
Theo bài ra ta có:
Ab 7 = A0b

 b  6 = A  5  6  b = A  5  b = 5 (Vì A > 0)  A =
 

Bài 19:

.
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


6

Hd:
 

Abc
, ( 0

b, c < 10, A > 0).
Theo bài ra ta có:

Abc 6 = A0bc


bc 5 = A 80 5  

bc = A 80

bc = 80
(Vì A > 0)  A =


§ 2. DÃY SỐ CÁCH ĐỀU
Bài 1:



Hd:
06  2) : 2 + 1 = 1003.
 1)  2 + 2 = 380
98 có 4 + [(98  10) : 2 + 1]  

 


Bài 2:




Hd:

 11) : 2 + 1]  
có [(175  101) : 2 + 1] x 3 = 1


 , 175.




175 = (11 + 175) 83 : 2 = 7719

Bài 3:



 
Hd:
không

 4) : 4 + 1 = 502.
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


7
b) [(96  12) : 4 + 1] × 2 = 44 
c  2  22 × 2 



Bài 4:




Hd:
 11) : 3 + 1] × 2 = 60 .
 có (998  101) : 3 + 1] × 3 = 900.
Vì 60 < 166 < 900 
  6
998.
Ta có: 106 : 
 


(11 + 398) × 100 : 2 = 20450.

Bài 5:



Hd:
 1) : 2 + 1 = 1005.
 1)  2 + 1 = 459
 0.

Bài 6:



Hd:
rong dãy 10, 1 


 
+ Ta có: 13 : 3 = 
 
108


b 10) : 2 + 1 = 65
(10 + 138)  65 : 2 = 4810.

Bài 7:
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


8

 



Hd:
 101) : 1 + 1 = 1905.
 1) × 1 + 101 = 175.
 × 3 + 1006 × 4 = 8721.
Vì có: 116 < 899  3 
Ta o
 (39  1)  

Bài 8:





Hd:
a) [(96  11) : 5 + 1]  2 + [(996  101) : 5 + 1]  3] + 1  4 = 18  2 + 180  3 + 1 
4 = 580.
Ta có 18  2 < 124 < 180  
996.
  18  
996.
Ta c
 1)  

b 1)  5 + 11 = 1021.
 203 : 2 = 104748.

Bài 9:




Hd:
 2) : 3 + 1 = 670.
  1)  3 + 2 = 296.
 98 có [(98  11) : 3 + 1]  2 = 60 

 
Ta c
 1)  
.

Bài 10:
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


9


 

Hd:
97 có 3 + [(97  13) : 4 + 1]  
 101) : 4 + 1]  

 


, 9 1)  

b 1)  4 + 1 = 797.
 200 : 2 = 79800.
Bài 11:

a)  ?


Hd:
 1)  3] + 5 = 620
 102 = 62500 + 1250 = 63750


 5) : 3 + 1 = 2
 11) : 3 + 1 = 30
  111) : 3 + 1 = 330
Ta có 2  1 + 30  2 < 135 < 330   
.
   30  2 - 2 
.

 1)  


Bài 12:

Hd:
S 
= [(99  100) : 2  (9  10) : 2] + [(99  100) : 2  (9  10) : 2 : 100]
= 4905 + 49, 05
= 4954, 05
Bài 13:
 2 + 3  - 1000 + 1001
Hd:
S = 1 + (3  2) + (5 -  1000)

Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


10
= 1 + [(1001  2) : 1 + 1] : 2 = 501
Bài 14:


1
3
,
2
3
3
, 7,
1
10
3

a) ?
?  


Hd:
a) 
10
3


10 1 20081
(2009 - 1) + =
3 3 3




10 1
(2007 - 1) + = 669

33







Bài 15:



 1, 2  2, 3 -  7, 8  8, 9

Hd:
a) Ta có:


10  x + 145 = 155
x = 1
b) Ta có:
 1, 2  2, 3 -  7, 8  8, 9
= (2, 1  1, 2) + (3, 2   7, 8) + (9, 8  8, 9)
= 1, 1  8 = 8, 8


§ 3. TOÁN VỀ TUỔI

Bài 1:
 cháu.


Hd:
 là: 8  
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


11
 1 = 1, 4 


1, 4 
i cháu lúc 





 1) 
 8 = 24 )
Bài 2:
Tính

Hd:
 là: 5  1 = 4  
 

 
 






 1)  4 = 8 
 5 = 40 )

Bài 3:


 
5
9
 
Hd:
 
 
 
 4 =








T:

 :


:

:
:
:
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


12


 

Bài 4:



Hd:
: 2, 2  
 2  

 : 






 1)  

 
 
 


Bài 5:



Hd:

 
.




(4  1)  1 
: 
: 8  
Bài 6:



Hd:
 



25

6

Tu nay:
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


13

 
 
Bài 7:


Hd:
: 3  
 
 
 

 1)  
 
  
Bài 8:
 
?

Hd:
 là: 2,5   
   
.

 




 1)  2 = 12 
 2,5 = 3

Bài 9:


Hd:
Theo bài ra ta có:










 27 : 3  




27
6


Tu:
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


14
Bài 10:


Hd:
Theo bài ra ta có:










 20 : (3 + 2) 
  20  
Bài 11:



Hd:
Theo bài ra ta có:











 15 : (1 + 2) 
  15  
Bài 12:

Bì
3
5

Hd:
Theo bài ra ta có:
 
3
5
 sau này

52
- 1 =
33
 sau này
1  sau này


21
1 - =
33





20




15
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15







 1) × 
 1) × 

Bài 13:


5
3

Hd:
Theo bài ra ta có:
 
3
5


52
- 1 =
33
nh sau này
nh sau này

21
1 - =
33




 1) × 3 = 
 1) × 

Bài 14:


Hd:
Theo bài ra ta có:
× 
× 
 





14




12
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


16
 

con  20 : (3 - 1) 1 = 10 (tu

 là: 30  
Bài 15:



Hd:
Theo bài ra ta có:









con  20 : (3 - 1) 
  10 × 4 = 40 
Bài 16:



Hd:
Theo bài ra ta có:


2 × 


1

2


 × 1 = 
- 20 
  50 - i)
Bài 17:


Hd:
Theo bài ra ta có:
 em sau này






2
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


17





 7 : (2 - 1) 
  

Bài 18:


1
3

Hd:
Theo bài ra ta có:

 










 25 : 5  

Bài 19:


3
5

Hd:
Theo bài ra ta có:


5
3



52
- 1 =
33
à

21
1 - =
33







  4 : 1  




7





25




4
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


18
Bài 20:


Hd:
Theo bài ra ta có:

 





× 
 2 × 
  12 : 2 
  
Bài 21:



Hd:
Theo bài ra ta có:

 

 100% - - 
× 
- 
× 


§ 4. TOÁN CHUYỂN ĐỘNG ĐỀU

Bài 1:
  
ì

Hd:


 30 = 156 (km).


242)(
5
2
265:156
h
h 
phút.

7
h
+ 2
h
24 = 9
h


)(10230
5
2
230 km
.
30 km
156 km
C
B
A


Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


19

Bài 2:



Hd:

Do trên cùn
:  
km/h. Ta có  sau:



 dài là 60  2  2 = 240 (km).


Bài 3:



Hd:

2
3


:
:
 
c xe máy là 60  
 2 = 120 (km).


 là 60  1,2 = 70 (km).
Bài 4:
5 
7 

6 
Hd:
là
55 11
45 9


: 
 
11
9
 



Quãn (2 : 2)  9 = 495 (km).
 tô ph
Bài 5:
2 giờ
20 km/h

:


Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


20




km?
Hd:




1


2


1

 2 = 10 km

1
 là:
130 + 10 = 140 (km).



Bài 6:
 
là 

hai ng
Hd:


- 
 2,75 = 110 (km)


-  

 10 = 100 (km).


 = 80 (km).
Bài 7:



Hd:
 

+  là:


200
75

168
75
28
23
56
25
8

3

g AB).

28 23 5
28 28 28


A
v
1


v
2
= v
1
+5km
B
C
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21
+ Vì
5
28

25 : 5  28 = 140 (km).


)/(42
3
10
:140 hkm
.

)/(50
3
14
:140 hkm
.
Bài 8:
         
  

Hd:
Sau  1, 5 - 10  1, 5 = 18  15 = 3 (km).


 
7  1 = 25 (km).

Bài 9:


Hd:

1
3
khúc sông và m

1
5


1 1 1
( ) : 2
3 5 15

(khúc sông)

1
1 : 15
15


Vì 50m/ph = 3km/h nên khúc sông dài là 3  15 = 45(km).
Bài 10:


Hd:

52 giây tàu  
(210 m).
  210 m là:
52  8 = 42 (giây).
: 210 : 42 = 5(m/s) (= 18km/h)
 10 = 40 (m).
Bài 11:
 
 giây. Tính 

.
Hd:


3 s
3 s
75 m
30 m
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676


22



10  3 = 30 (m)

30 m. là:
(75  30) : 3 = 15(m/s) = 54( km/h)
Bài 12:



Hd:



 

  m là: 47  9 = 38 (s)

 9 = 45 (m)
Bài 13:




Hd:





 

12
21
t v 20
= =
t v 30


2
 t
1


2
3



Bài 14:



9 s
9 s
9 m
47 s
181 m
v
1
=30 km
? km
20 km
30 km
v
2
=20 km
A
B
C
D
Biên Soạn : Lê Thị Thu Hiền – 0987.615.676

23
Hd:








12
21
t v 60
= =
t v 40



12
200
t + t =
60

1
, t
2

 520 km.

Bài 15:



Hd:
Theo bài ra ta có: + v
1
= 1, 5  v
2

+ v
1
+ v
2
=
1
12


1
1
v
20


2
1
v
30







Bài 16:


4
5


1
8
 


Hd:
Theo bài ra ta có: + v
ra
=
4
5
 v
vào

+ v
vào
- v
ra

=
1
40


vào
=
1
40
 5 =
1
8



Bài 17:
B
? km
100 km
A
C
D
100 km
E
t
1
, v
1
=40km/h
t

2
, v
2
=60km/h
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24
 


Hd:
vòi là: v
1
+ v
2
=
1
3


12
2 =
33




21
1 - =

33


11
: 3 =
39


1 1 2
- =
3 9 9

Bài 18:
kim 
kim phút trùng nhau? 
t







kim phút trùng nhau là:

1 1 1
: [1 - ] =
12 12 11

kim phút trùng nhau là:

11
+ 1 = 1
11 11

Bài 19:
kim 
kim  nhau?









Hd:

h

f
, ta
có: v
h
=
1
12
vòng/h, v

f
= 1 vòng/h

1
12
vòng

Hd:

h

f
, ta
có: v
h
=
1
12
vòng/h, v
f
= 1 vòng/h
ng là
1
12
vòng

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25

 2 kim vuông góc là
1
4
vòng





kim 
trùng nhau là:

1 1 3
: [1 - ] =
4 12 11

kim phút vuông g


3 1 4
+ =
11 11 11


kim nhau là:

44
+ 1 = 1
11 11


Bài 20:



Hd:






17 × 2 = 34 (km)
Sau 2 g
34 - 10 = 24 (km)

24 : 


Bài 21:

 
5
6


1
2


Hd:
B
1/12
A
D
C
1/4
E
10 km

A
B
C
A

B

Ô tô
Ca nô
2 × 17 = 34 km

13 km

A
B
A

B
48 km
35 km