MathDB
Problems
Contests
National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2011 Bangladesh Mathematical Olympiad
2011 Bangladesh Mathematical Olympiad
Part of
Bangladesh Mathematical Olympiad
Subcontests
(1)
HS
1
Hide problems
Bangladesh National Mathematical Olympiad (BdMO) 2011
Higher Secondary: 2011Time: 4 HoursProblem 1: Prove that for any non-negative integer
n
n
n
the numbers
1
,
2
,
3
,
.
.
.
,
4
n
1, 2, 3, ..., 4n
1
,
2
,
3
,
...
,
4
n
can be divided in tow mutually exclusive classes with equal number of members so that the sum of numbers of each class is equal. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=709Problem 2: In the first round of a chess tournament, each player plays against every other player exactly once. A player gets
3
,
1
3, 1
3
,
1
or
−
1
-1
−
1
points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is
90
90
90
. How many players were there in the tournament? http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=708Problem 3:
E
E
E
is the midpoint of side
B
C
BC
BC
of rectangle
A
B
C
D
ABCD
A
BC
D
.
A
A
A
point
X
X
X
is chosen on
B
E
BE
BE
.
D
X
DX
D
X
meets extended
A
B
AB
A
B
at
P
P
P
. Find the position of
X
X
X
so that the sum of the areas of
△
B
P
X
\triangle BPX
△
BPX
and
△
D
X
C
\triangle DXC
△
D
XC
is maximum with proof. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=683Problem 4: Which one is larger 2011! or,
(
1006
)
2011
(1006)^{2011}
(
1006
)
2011
? Justify your answer. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=707Problem 5: In a scalene triangle
A
B
C
ABC
A
BC
with
∠
A
=
9
0
∘
\angle A = 90^{\circ}
∠
A
=
9
0
∘
, the tangent line at
A
A
A
to its circumcircle meets line
B
C
BC
BC
at
M
M
M
and the incircle touches
A
C
AC
A
C
at
S
S
S
and
A
B
AB
A
B
at
R
R
R
. The lines
R
S
RS
RS
and
B
C
BC
BC
intersect at
N
N
N
while the lines
A
M
AM
A
M
and
S
R
SR
SR
intersect at
U
U
U
. Prove that the triangle
U
M
N
UMN
U
MN
is isosceles. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=706Problem 6:
p
p
p
is a prime and sum of the numbers from
1
1
1
to
p
p
p
is divisible by all primes less or equal to
p
p
p
. Find the value of
p
p
p
with proof. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=693Problem 7: Consider a group of
n
>
1
n > 1
n
>
1
people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If
A
A
A
and
B
B
B
are friends/enemies then we count it as
1
1
1
friendship/enmity. It is observed that the number of friendships and number of enmities are equal in the group. Find all possible values of
n
n
n
. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=694Problem 8:
A
B
C
ABC
A
BC
is a right angled triangle with
∠
A
=
9
0
∘
\angle A = 90^{\circ}
∠
A
=
9
0
∘
and
D
D
D
be the midpoint of
B
C
BC
BC
. A point
F
F
F
is chosen on
A
B
AB
A
B
.
C
A
CA
C
A
and
D
F
DF
D
F
meet at
G
G
G
and
G
B
∥
A
D
GB \parallel AD
GB
∥
A
D
.
C
F
CF
CF
and
A
D
AD
A
D
meet at
O
O
O
and
A
F
=
F
O
AF = FO
A
F
=
FO
.
G
O
GO
GO
meets
B
C
BC
BC
at
R
R
R
. Find the sides of
A
B
C
ABC
A
BC
if the area of
G
D
R
GDR
G
D
R
is
2
15
\dfrac{2}{\sqrt{15}}
15
2
http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=704Problem 9: The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of
123
123
123
is
123123
123123
123123
). Find a positive integer (if any) whose repeat is a perfect square. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=703Problem 10: Consider a square grid with
n
n
n
rows and
n
n
n
columns, where
n
n
n
is odd (similar to a chessboard). Among the
n
2
n^2
n
2
squares of the grid,
p
p
p
are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers
(
p
,
q
,
n
)
(p, q, n)
(
p
,
q
,
n
)
so that the number of white squares is
q
2
q^2
q
2
. http://matholympiad.org.bd/forum/viewtopic.php?f=13&t=702The problems of the Junior categories are available in [url=http://matholympiad.org.bd/forum/]BdMO Online forum: http://matholympiad.org.bd/forum/viewtopic.php?f=25&t=678