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Đề thi Olympic Toán học TMO năm 2005

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AITMO PROBLEMS – Team Contest


1.. Refer to the diagram. Quadrilateral ABCD with diagonals BD and
AC bounds the rhombus EFGH as shown in the figure. If BD k


AC = ,


||



EF AC, and FG BD|| , find Area


Area


ABCD
EFGH .
Solution:


Let =λ


AB
AE


, then EH BD
BD


EH
BA


BE = λ =λ =λ


,
,


1 . Similarly, EF =

(

1−

λ

)

AC , but EH=EF, so
that

λ

BD=

(

1−

λ

)

AC and


λλ

=


= 1
AC
BD


k , then


kH


1


=


λ .


Hence

(

) (

)



k
k
k
k
EH
BD
EF
AC
EH
EF
BD
AC
S
S


EFGH
ABCD
2
1
1
1
1
1
1
2
1
1
1
1
2
1
2
1
sin
sin
2
1
2
+
=
+













+
+
=













=

=





=

λ


λ


ϑ


ϑ


.


2. Prove or disprove: 100 consecutive positive integers can be placed around a circle so that the product
of any two adjacent numbers is a perfect square.


Solution:


3. How many triples (a, b, c) of positive integers are there such that a, b and c are primes and 2 2
ab =c?


Solution:


The unique ordered triple is (3, 2, 5).


4. Find the smallest positive integer k such that !k ends with 500 zeros.
[Note: k!=k k

(

−1

)(

k−2 ... 3 2 1

) ( )( )( )

].


Solution:
k=2005.


5. Let x = a + b – c, y = a + c – b and z = b + c – a, where a, b and c are prime numbers. Given that x2 = y
and

(

zy

)

is the square of a prime number, find the value of the product abc.


Solution:



abc=3*23*29=2001


6. The real numbers x1 , x2 , x3 , x4 , x5 , x6 are arbitrarily chosen within the interval (0,1).


Prove that

(

1 2

)(

2 3

)(

3 4

)(

4 5

)(

5 6

)(

6 1

)



1
16


xx xx xx xx xx xx ≤ .


Solution:


7. The number 222+1 has exactly one prime factor greater than 1000. Find it.


Solution:


(

2 1

)

2

(

2 1 2

)(

2 1 2

)

2113*1985 2113*397*5


1


222+ = 11+ 2− 12 = 11+ + 6 11+ − 6 = =



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8. We can assign one of the integers 1, 2, 3, …, 7 (with no repetitions) to each
of the seven regions in the diagram so that numbers in adjacent regions
(having a common edge) differ by 2 or more.


How many different values are possible for region g?



Solution:


7.


9. Three motorists A, B, and C often travel on a certain highway, and each motorist always travels at a
constant speed. A is the fastest of the three and C is the slowest. One day when the three travel in the
same direction, B overtakes C. Five minutes later, A overtakes C. In another three minutes, A overtakes


B.


On another occasion when they again travel in the same direction, A overtakes B first. Nine minutes
later, A overtakes C.


When will B overtake C?


Solution:


15 minutes.


10. Divide the diagram into pentominoes so that the sum of
the digits within each part is 10. The pentomino shapes
are shown below. They can be rotated or reflected. Each
must be used exactly once in the problem.





Solution:


3 2 1 3 2 2 2 3 1 1



1 1 5 2 1 2 3 1 4 1


3 2 1 2 4 2 3 2 1 2


2 1 1 2 2 1 1 5 2 1


4 3 2 4 2 1 2 2 1 1


2 1 1 2 1 2 1 1 4 2


Example


a b c


d e f





×