8
- - : , , 1 : (- ) : (): ( ) : Name (In English): Registration No: [ ।] 49 100 । । ? After Sachin Tendulkar had scored 100 runs from 49 balls, he got out in the very next ball. What is the average run he scored in each ball he palyed? 18 34 ? Age of Boltu’s father is 18 times of the age of Boltu. Boltu is 34 years younger than his father. What is the age of Boltu’s father? 1 7 ? LCM of the numbers from 1 to 7 is product of two consecutive numbers. What is the sum of those two numbers? C P K C , P K CK = 2 ? In the given diagram, C is the centre of the largest circle, P is the centre of the medium circle and K is the centre of the small circle. If CK = 2, what is the radius of the largest circle? 9 15 09:30 10:45 , 10:48 ? Orthi left for class when it was 9 o’clock in her watch. After 15 minutes, her watch stopped. Reaching the class, she saw that it was 09:30 in Shoiley’s watch. Their class is supposed to be finished at 10:45, though at that time it was 10:48 in Shoiey’s watch. How much time did it take Orthi to reach her class from her home? x y , 2 2 + + + + = y x y x y ? x and y are two numbers and none of them is zero. If 2 2 + + + + = y x y x , what is the value of y ? 12 ? In the given diagram, both circles have the same centre. Area of the larger circle is 4 times of the smaller circle. The area of the shaded region is 12, what is the area of the larger circle?

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Page 1: 2012 Chittagong Question

���-����� ����� - �� ��� � �� ���� ��� ���� � �� � �� � � ����

��� �: �������� � �� � � ���� � �

�� ������ ����, �� ����� �����, �����। 1

�������� : ����� (�� -� � ��) ��: � ��� �� �� �

�(������): � ��(���� ���):

Name (In English): Registration No:

[�� ������ �� � ���� ��� ���� ���। ����� ��� ���� ���� ������ ���� ��� ��� �� ��� ��� ���। ��� ����� ����� �� !��� ��"�#। ������ �� �� ���� ��� ��� ���।]

� ���� ��� � $% � !&'(���� 49 ��� 100 ��� ����#�। ������ ���� �� *�& ��" !����। ��+� !��� ,�& ��� ��

��� �� ���� ? After Sachin Tendulkar had scored 100 runs from 49 balls, he got out in the very next ball. What is the average run he scored in each ball he palyed?

� �-(� ����� ��� �-(� ����� 18 ./। *��� �-( ��� ����� !%�" 34 �#��� !#�&। �-(� ����� ��� ? Age of Boltu’s father is 18 times of the age of Boltu. Boltu is 34 years younger than his father. What is the age of Boltu’s father?

� 1 !��� 7 �0 1 �����.���� ���. 2& 3�� ������ ./4�। 5 �����2&� !0��4� ��? LCM of the numbers from 1 to 7 is product of two consecutive numbers. What is the sum of those two numbers?

CP K

���$� %� C ���� �� ���&� !�6, P ��7�� ���&� !�6 *� K !#�& ���&� !�6। CK = 2 ��� �� ���&� �����8 ��? In the given diagram, C is the centre of the largest circle, P is the centre of the medium circle and K is the centre of the small circle. If CK = 2, what is the radius of the largest circle?

� 9� ���� :�� ��" 9�(���� ���� 9 &�" ;���� ��� !�� �"। 0�<"�� ��� 15 ��& ��� ��� :� �= ��" 0�"। ;��� !�>+�# !� ?$� � :��� ��" !��� 09:30 । �@� ��" 9�(���� <��� ;�� 10:45 ���& !$A �<"�� ��, �B ;�� !$A ���� ��" ?$� � :��� ��" #� 10:48 ��&। ���� !��� ;��� !0�� 9� � �� ��& ��" !���#�? Orthi left for class when it was 9 o’clock in her watch. After 15 minutes, her watch stopped. Reaching the class, she saw that it was 09:30 in Shoiley’s watch. Their class is supposed to be finished at 10:45, though at that time it was 10:48 in Shoiey’s watch. How much time did it take Orthi to reach her class from her home?

� x ��� y 2& �����, ���� ��8� ����� ���� $C�� �"।

22++++

====y

xyx

��� y �� �� ?

x and y are two numbers and none of them is zero. If 2

2++++

====y

xyx

, what is the value of y ?

���$� %� 2& ����� !�6 ���। �� ���&� !D4� !#�& ���&� !D4��� %��./। #�� ����� 9�$&(�(� !D4� 12 �� ��� ��� �� ���&� !D4� ��? In the given diagram, both circles have the same centre. Area of the larger circle is 4 times of the smaller circle. The area of the shaded region is 12, what is the area of the larger circle?

Page 2: 2012 Chittagong Question

���-����� ����� - �� ��� � �� ���� ��� ���� � �� � �� � � ����

��� �: �������� � �� � � ���� � �

�� ������ ����, �� ����� �����, �����। 2

� ���� ��� � ��& �� 9���� !��� ������ ���� ������� 9��& ��� ��" ��� !�& ��& !�>�� ����� �"। ����

��.��� ����� ����� ����? If the leftmost digit of a three digit even number is omitted, the number becomes a prime. What is the number of such three digit numbers?

� ��& ;���� #������ !&���� !�<"� ��E। ;���� ,�� #�&�� 1 !��� ��& !&���� !�<"� ��"�#। ���� 2���� !�<"� ��"�# 2 !��� !&����, ��� ���� ������ !�<"� ��"�# 3 !��� !&����, ����� ������� !&���� !�<"� ����। ��& 9�� �$� ����� !��� !&���� !��"�# ��� !$A #�&� !��� �F� ��? In a class, students are given tokens. The first boy will get a token with number 1, the next two boys will get tokens with number 2, the next three with number 3 and so on. What is the roll number of the last student who will get a token with a single digit number?

�� ��G� ���# 2& ��2� ��H *�#। ,�� ���H ��& ����� I(��" ��� ��H !��� �����&� J./ !�� ��" *��। J� " ���H ��& ����� I(��" ��� 5 ������ ���� 2 !0�� ��" 0�"। ���(� ���� ��E���� ,�� ���H ��& ����� I(��" ����, !0 �����&� !�� ���� !�&�� ��" !� J� " ���H I(��" ����। ���� !� !0 �����& !���� !�&�� 9�8 � ��� *���& ���H I(��" ����। !���� !��� !�� �<"� �����& ,�� ���H I(����� ������ ����। ������ ��� ��� �� " ��H&�� 0� 34 �����& I(��" !�<"� �" ����� �� !���" *���? Nontey has two magic boxes. If he inserts a number in the first, it is doubled. Any number inserted in the second box is increased by 2. Ratul inserted an arbitrary number in the first box. Whatever he got as output was inserted in the second box. The output from the second box was divided by 2 and the result was inserted in a third magic box that Ratul had. The output was equal to the number inserted in the first box. If Ratul inserts the number 34 in the third box, what will be the output?

Page 3: 2012 Chittagong Question

���-����� ����� - �� ��� � �� ���� ��� ���� � �� � �� � � ����

��� �: �������� � �� � � ���� � �

�� ������ ����, �� ����� �����, �����। 3

��������: ��� �� (�� -� � ��) ��: � ��� �� �� �

�(������): � ��(���� ���):

Name (In English): Registration No:

[�� ������ �� � ���� ��� ���� ���। ����� ��� ���� ���� ������ ���� ��� ��� �� ��� ��� ���। ��� ����� ����� �� !��� ��"�#। ������ �� �� ���� ��� ��� ���।] � ���� ��� � $% � !&'(��� 49 ��� 100 ��� ����#�। ������ ���� �� *�& ��" !����। ��+� !��� ,�& ��� ��

��� �� ���� ? After Sachin Tendulkar had scored 100 runs from 49 balls, he got out in the very next ball. What is the average run he scored in each ball he palyed?

� 420 �����& ���& 3�� �C/ ������ ���.। 5 �����.���� !0��4� ��? The number 420 is the LCM of seven consecutive integers. What is their sum?

� 9� ���� :�� ��" 9�(���� ���� 9 &�" ;���� ��� !�� �"। 0�<"�� ��� 15 ��& ��� ��� :� �= ��� 0�"। ;��� !�>+�# !� ?$� � :��� ��" !��� 09:30 । �@� ��" 9�(���� <��� ;�� 10:45 ���& !$A �<"�� ��, �B ;�� !$A ���� ��" ?$� � :��� ��" #� 10:41 ��&। ���� !��� ;��� !0�� 9� � �� ��& ��" !���#�? Orthi left for class when it was 9 o’clock in her watch. After 15 minutes, her watch stopped. Reaching the class, she saw that it was 09:30 in Shoiley’s watch. Their class is supposed to be finished at 10:45, though at that time it was 10:41 in Shoiey’s watch. How much time did it take Orthi to reach her class from her home?

� ��& K(��� ����� ��L D(M�� ��L� J./ ��� ����� !��/ D(M�� !���/� ��./। K(��� !D4� 3

29

��� �� ����� ��L� ��� ��? The largest side of a triangle is twice the smallest and the largest angle is thrice the

smallest. The area of the triangle is 329

. What is the length of the largest side?

� ��G� ���# 2& ��2� ��H *�#। ,�� ���H ��& ����� I(��" ��� ��H !��� �����&� J./ !�� ��" *��। J� " ���H ��& ����� I(��" ��� 5 ������ ���� 2 !0�� ��" 0�"। ���(� ���� ��E���� J� " ���H ��& ����� I(��" ����, !0 �����&� !�� ���� !�&�� ��" !� ,�� ���H I(��" ����। ���� !� !0 �����& !���� !�&�� 9�8 � ��� *���& ���H I(��" ����। !���� !��� !�� �<"� �����& ,�� ���H I(����� ������ ����। ���(��� ���# ���� ��� " ��H&�� 0� 2012 �����& I(��" !�<"� �" ����� �� !���" *���? Nontey has two magic boxes. If he inserts a number in the first, it is doubled. Any number inserted in the second box is increased by 2. Ratul inserted an arbitrary number in the second box. Whatever he got as output was inserted in the first box. The output from the first box was divided by 2 and the result was inserted in a third magic box that Ratul had. The output was equal to the number inserted in the first box. If Ratul inserts the number 2012 in the third box, what will be the output?

228

168

130

====++++====++++====++++

abca

cabc

bcab

���$� �� ��/.��� !��� a + b + c �� ��� !�� ����। Find the value of a + b + c from the given set of equations.

Page 4: 2012 Chittagong Question

���-����� ����� - �� ��� � �� ���� ��� ���� � �� � �� � � ����

��� �: �������� � �� � � ���� � �

�� ������ ����, �� ����� �����, �����। 4

� ��& !������ ��"��� �/��� !��%�� !����। ,��� 2�� �/��� !��%�� !����, ��� 2�� �/���

�� ���� � ��� � !�$ !��%�� ��� ������। !�����& 6 �� 8�� %���। !������ ����D ���� ������ ��� ���? In a seminar two mathematicians will give lectures each day. However, no two of them will give another lecture on the same day. The seminar will run for 6 days. What is the minimum number of mathematicians to attend the seminar?

� ��& ;���� #������ !&���� !�<"� ��E। ;���� ,�� #�&�� 1 !��� ��& !&���� !�<"� ��"�#। ���� 2���� !�<"� ��"�# 3 !��� !&����, ��� ���� ������ !�<"� ��"�# 5 !��� !&����, ����� ������� !&���� !�<"� ����। ��& 9�� �$� ����� !��� !&���� !��"�# ��� !$A #�&� !��� �F� ��? In a class, students are given tokens. The first boy will get a token with number 1, the next two boys will get tokens with number 3, the next three with number 5 and so on. What is the roll number of the last student who will get a token with a single digit number?

� ABC �����/ K(�� ∠∠∠∠B = 900 ��� AB ≠≠≠≠ BC । D, E �N(J" BC !� ��� F, G �N(J" AB !� ���O� ��� !0���� BD < BE ��� BF < BG । FQ � GP, AB �� !� ��� DQ � PE, BC ��

��� �F। QP, AC !� X �N(�� !#� ���। 0� ba====++++

PQPX

AXAC

�" !0���� a ��� b 2& !�>��

����� ����� a + b �� �� ? In the right triangle ABC with ∠∠∠∠B = 900, AB ≠≠≠≠ BC. D, E trisect BC and F, G trisect AB where BD < BE and BF < BG. FQ and GP are perpendicular to AB and DQ and PE are

perpendicular to BC. The line QP meets AC at X. If ba====++++

PQPX

AXAC

where a and b

are primes, find a + b

�� ��& ����� 2& ��� ��P��� �FK��� !#� ���। ��& ��� �� 2& 9��$� ?�: � �" x ��� x + 5, 9�� ���

�� 2& 9��$� ?�: � �" x + 1 ��� 6। ���&� !�6 !��� ���J�"� !#��N(� �C�Q��ba

*���� !��� 0�",

!0���� a ��� b 2& !�>�� �����। a + b �� �� ? Two chords meet inside a circle perpendicular on each other. Lengths of the parts of one chord are x and x + 5, those of the other are x + 1 and 6. The distance of the chords’

meeting point from the centre of the circle is ba

, where a and b are primes. Find the

value of a + b.

Page 5: 2012 Chittagong Question

���-����� ����� - �� ��� � �� ���� ��� ���� � �� � �� � � ����

��� �: �������� � �� � � ���� � �

�� ������ ����, �� ����� �����, �����। 5

��������: ��������(�-�� � ��) ��: � ��� �� �� �

�(������): � ��(���� ���):

Name (In English): Registration No:

[�� ������ �� � ���� ��� ���� ���। "���� #�� ���� ���� ������ ���� ��� ��� �� ��� ��� ���। ��� ����� ����� �� !��� ��"�#। ������ �� �� ���� ��� ��� ���।] � ���� ��� � $% � !&'(���� 49 ��� 100 ��� ����#�। ������ ���� �� *�& ��" !����। ��+� !��� ,�& ��� ��

��� �� ���� ? After Sachin Tendulkar had scored 100 runs from 49 balls, he got out in the very next ball. What is the average run he scored in each ball he palyed?

� ABCD ���� %�(K( �� AD = CD ��� K(� ABC ����L। ∠∠∠∠DAC �� ��� �/ " ��। In the cyclic quadrilateral ABCD, AD = CD and ∆∆∆∆ABC is equilateral. Find ∠∠∠∠DAC.

182

90

230

====++++====++++====++++

abca

cabc

bcab

���$� �� ��/.��� !��� a + b + c �� ��� !�� ����। Find the value of a + b + c from the given set of equations.

� 4������ ���& ��& �F� ��R�� ���$� 9���। 5 ��R� ����� !� ����� ���� !�#�� ��E���� ��+&�� ����। ���� ����� !� x �����&�� ��R� ��+&���। ���� !� �� ���� � ��� ���� �� ��� *��� y ������$��

��R� ��+&���। !� !����� !0 !� ��� ���� !��� ��� z �����&�� �C��Q 9���� ���#। yx

z++++

�� ��� �S ���

? Forhad’s cottage lies by a long road. He can walk forward or backward along that road. One morning, he walks for x kilometers. At this point he might (or might not) change his direction. Next he walks for another y kilometers. He is now at a distance of z kilometers

from his house. What is the maximum value of yx

z++++

?

� 2& �C/ ����� a ��� b �� ��� 3503log2log 20122012 ====++++ ba । a + b �� �� ?

a and b are integers so that 3503log2log 20122012 ====++++ ba . What is the value of a + b?

� ��& K(��� ����� ��L D(M�� ��L� J./ ��� ����� !��/ D(M�� !���/� ��./। K(��� !D4� 332 ��� �� ����� ��L� ��� ��? The largest side of a triangle is twice the smallest and the largest angle is thrice the

smallest. The area of the triangle is 332 . What is the length of the largest side?

� �� 9���� ��� ��.��� !����� ����� *�# 0���� 9��.���� !0��4� !���? How many three digit odd numbers are there so that the sum of their digits is even?

� ��& ����� 2& ��� ��P��� �FK��� !#� ���। ��& ��� �� 2& 9��$� ?�: � �" x ��� x + 5, 9�� ���

�� 2& 9��$� ?�: � �" x + 1 ��� 6। ���&� !�6 !��� ���J�"� !#��N(� �C�Q��ba

*���� !��� %��,

!0���� a ��� b 2& !�>�� �����। a + b �� �� ? Two chords meet inside a circle perpendicular on each other. Lengths of the parts of one chord are x and x + 5, those of the other are x + 1 and 6. The distance of the chords’

meeting point from the centre of the circle is ba

, where a and b are primes. Find the

value of a + b.

Page 6: 2012 Chittagong Question

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��� �: �������� � �� � � ���� � �

�� ������ ����, �� ����� �����, �����। 6

� ���� ��� �

��& 8��� ���K��� ��T�"� ��� ���� !0� 20121 ====a ��� 1−−−−

====n

n an

a ��।

!2... 20321 yaaaa x ××××====×××××××××××××××× ��� x + y �� ��� �/ " ��।

Consider a series with 20121 ====a and 1−−−−

====n

n an

a . !2... 20321 yaaaa x ××××====×××××××××××××××× , Find the

value of x + y.

��

%�, ∠∠∠∠ABC = 600, ∠∠∠∠DCB = 300, AD||BC ��� AP⊥⊥⊥⊥BC । ABCD ��� APQD �� !���&�� ��� ,

!D4�< ����। DQCsin ∠∠∠∠ �� �� aab −−−− ��� a + b �� �� �/ " ��। �"�� b ��� a 8��U� �C/ �����। In this diagram, ∠∠∠∠ABC = 600, ∠∠∠∠DCB = 300, AD||BC, AP⊥⊥⊥⊥BC. Both area and perimeter of ABCD and APQD are equal. The value of

DQCsin ∠∠∠∠ is of the form aab −−−− , b and a are integers. What is the value of a + b?

Page 7: 2012 Chittagong Question

���-����� ����� - �� ��� � �� ���� ��� ���� � �� � �� � � ����

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�� ������ ����, �� ����� �����, �����। 7

��������: ����� �������� (��� !-����-�"#����) ��: � ��� �� �� �

�(������): � ��(���� ���):

Name (In English): Registration No:

[�� ������ �� � ���� ��� ���� ���। ����� ��� ���� ���� ������ ���� ��� ��� �� ��� ��� ���। ��� ����� ����� �� !��� ��"�#। ������ �� �� ���� ��� ��� ���।] � ���� ��� � $% � !&'(���� 49 ��� 100 ��� ����#�। ������ ���� �� *�& ��" !����। ��+� !��� ,�& ��� ��

��� �� ���� ? After Sachin Tendulkar had scored 100 runs from 49 balls, he got out in the very next ball. What is the average run he scored in each ball he palyed?

� ��$ �� 9���� !��� ������ ���� ������� 9��& ��� ��" ��� !�& ��& !�>�� ����� �"। ���� ��.��� ����� ����� ����? If the leftmost digit of a three digit even number is omitted, the number becomes a prime. What is the number of such three digit numbers?

� ABCD ���� %�(K( �� AD = CD ��� K(� ABC ����L। ∆∆∆∆ADC �� !���&�� 323 ++++ ��� ∆∆∆∆ABC �� ��� �� �/ " ��। In the cyclic quadrilateral ABCD, AD = CD and ∆∆∆∆ABC is equilateral. Perimieter of

∆∆∆∆ADC is 323 ++++ . What is the perimeter of ∆∆∆∆ABC?

� 2& �C/ ����� a ��� b �� ��� 201273log3log 19711971 ====++++ ba । a + b �� �� ?

a and b are integers so that 201273log3log 19711971 ====++++ ba . What is the value of a + b?

� 2& #V�� .& #(�� !��� !�� !0 ���� !0��4� ����D 10. ��� " *���& .& !#��� ��� 5 ��& ������ !0��4� ����D 15 ��� ��� �W����� ��? Two dies have been thrown and the sum of the numbers appearing in them is at least 10. If a third die is thrown, what is the probability that the sum will be at least 15?

� 0 < x < 1 �� ��� )1log()1(log)( xxxxxf −−−−−−−−++++==== 4��$�&� �� �X ��� ��? �"�� ���������

K� 2. )1log()1(log)( xxxxxf −−−−−−−−++++==== , what is the minimum value of this function for

10 <<<<<<<< x ? Here, the base of logarithm is 2.

� %�� 9���� ��� ��.��� !��� ����� *�# 0���� 9��.���� !0��4� !�����? How many four digit even numbers are there so that the sum of their digits is odd?

� ��$ � � (��) }0{[,0[: −−−−→→→→∞∞∞∞ Rf �� ���

)()()()(

)(1

yfxfyfxf

yfxf

++++====

++++ । f (0) = 1 ���

f (2012) �� �� ?

For an injective function }0{[,0[: −−−−→→→→∞∞∞∞ Rf the relation )()(

)()()(

1yfxf

yfxfyf

xf++++

====

++++

holds. Find the value of f (2012) if f (0) = 1.

Page 8: 2012 Chittagong Question

���-����� ����� - �� ��� � �� ���� ��� ���� � �� � �� � � ����

��� �: �������� � �� � � ���� � �

�� ������ ����, �� ����� �����, �����। 8

��& 8��� ���K��� ��T�"� ��� ���� !0� 20121 ====a ��� 1−−−−

====n

n an

a ��।

!2... 20321 yaaaa x ××××====×××××××××××××××× ��� x + y �� ��� �/ " ��।

Consider a series with 20121 ====a and 1−−−−

====n

n an

a . !2... 20321 yaaaa x ××××====×××××××××××××××× , Find the

value of x + y.

��

%�, ∠∠∠∠ABC = 600, ∠∠∠∠DCB = 300, AD||BC ��� AP⊥⊥⊥⊥BC । ABCD ��� APQD �� !���&�� ��� , !D4�<

��� । DQCsin ∠∠∠∠ �� �� aab −−−− ��� a + b �� �� �/ " ��। �"�� b ��� a 8��U� �C/ �����। In this diagram, ∠∠∠∠ABC = 600, ∠∠∠∠DCB = 300, AD||BC, AP⊥⊥⊥⊥BC. Both area and perimeter of ABCD and APQD are equal. The value of

DQCsin ∠∠∠∠ is of the form aab −−−− , b and a are integers. What is the value of a + b?