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Problems
Contests
National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2013 Bangladesh Mathematical Olympiad
2013 Bangladesh Mathematical Olympiad
Part of
Bangladesh Mathematical Olympiad
Subcontests
(10)
10
1
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Ordered sequence of elements
Higher Secondary P10
X
X
X
is a set of
n
n
n
elements.
P
m
(
X
)
P_m(X)
P
m
(
X
)
is the set of all
m
m
m
element subsets (i.e. subsets that contain exactly
m
m
m
elements) of
X
X
X
. Suppose
P
m
(
X
)
P_m(X)
P
m
(
X
)
has
k
k
k
elements. Prove that the elements of
P
m
(
X
)
P_m(X)
P
m
(
X
)
can be ordered in a sequence
A
1
,
A
2
,
.
.
.
A
i
,
.
.
.
A
k
A_1, A_2,...A_i,...A_k
A
1
,
A
2
,
...
A
i
,
...
A
k
such that it satisfies the two conditions: (A) each element of
P
m
(
X
)
P_m(X)
P
m
(
X
)
occurs exactly once in the sequence, (B) for any
i
i
i
such that
0
<
i
<
k
0<i<k
0
<
i
<
k
, the size of the set
A
i
∩
A
i
+
1
A_i \cap A_{i+1}
A
i
∩
A
i
+
1
is
m
−
1
m-1
m
−
1
.
3
1
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Find the area $0$
Higher Secondary P3Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a regular hexagon with
A
B
=
7
AB=7
A
B
=
7
.
M
M
M
is the midpoint of
D
E
DE
D
E
.
A
C
AC
A
C
and
B
F
BF
BF
intersect at
P
P
P
,
A
C
AC
A
C
and
B
M
BM
BM
intersect at
Q
Q
Q
,
A
M
AM
A
M
and
B
F
BF
BF
intersect at
R
R
R
. Find the value of
[
A
P
B
]
+
[
B
Q
C
]
+
[
A
R
F
]
−
[
P
Q
M
R
]
[APB]+[BQC]+[ARF]-[PQMR]
[
A
PB
]
+
[
BQC
]
+
[
A
RF
]
−
[
PQMR
]
. Here
[
X
]
[X]
[
X
]
denotes the area of polygon
X
X
X
.
7
1
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Largest awesome prime finding
Higher Secondary P7If there exists a prime number
p
p
p
such that
p
+
2
q
p+2q
p
+
2
q
is prime for all positive integer
q
q
q
smaller than
p
p
p
, then
p
p
p
is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.
4
1
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Continued fraction or fairy sequence
Higher Secondary P4If the fraction
a
b
\dfrac{a}{b}
b
a
is greater than
31
17
\dfrac{31}{17}
17
31
in the least amount while
b
<
17
b<17
b
<
17
, find
a
b
\dfrac{a}{b}
b
a
.
1
1
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Degenerate Pentagon
Higher Secondary P1A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
,
A
B
=
A
E
AB=AE
A
B
=
A
E
,
B
C
=
D
E
BC=DE
BC
=
D
E
,
P
P
P
and
Q
Q
Q
are midpoints of
A
E
AE
A
E
and
A
B
AB
A
B
respectively.
P
Q
∣
∣
C
D
PQ||CD
PQ
∣∣
C
D
,
B
D
BD
B
D
is perpendicular to both
A
B
AB
A
B
and
D
E
DE
D
E
. Prove that
A
B
C
D
E
ABCDE
A
BC
D
E
is a degenerate pentagon.
5
1
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Integer divided by prime number
Higher Secondary P5Let
x
>
1
x>1
x
>
1
be an integer such that for any two positive integers
a
a
a
and
b
b
b
, if
x
x
x
divides
a
b
ab
ab
then
x
x
x
either divides
a
a
a
or divides
b
b
b
. Find with proof the number of positive integers that divide
x
x
x
.
2
1
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Functional nice equation and 0
Higher Secondary P2Let
g
g
g
be a function from the set of ordered pairs of real numbers to the same set such that
g
(
x
,
y
)
=
−
g
(
y
,
x
)
g(x, y)=-g(y, x)
g
(
x
,
y
)
=
−
g
(
y
,
x
)
for all real numbers
x
x
x
and
y
y
y
. Find a real number
r
r
r
such that
g
(
x
,
x
)
=
r
g(x, x)=r
g
(
x
,
x
)
=
r
for all real numbers
x
x
x
.
6
1
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Road to go $n$ cities.
There are
n
n
n
cities in a country. Between any two cities there is at most one road. Suppose that the total number of roads is
n
.
n.
n
.
Prove that there is a city such that starting from there it is possible to come back to it without ever travelling the same road twice.
9
1
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Find the angle!
Six points
A
,
B
,
C
,
D
,
E
,
F
A, B, C, D, E, F
A
,
B
,
C
,
D
,
E
,
F
are chosen on a circle anticlockwise. None of
A
B
,
C
D
,
E
F
AB, CD, EF
A
B
,
C
D
,
EF
is a diameter. Extended
A
B
AB
A
B
and
D
C
DC
D
C
meet at
Z
,
C
D
Z, CD
Z
,
C
D
and
F
E
FE
FE
at
X
,
E
F
X, EF
X
,
EF
and
B
A
BA
B
A
at
Y
.
A
C
Y. AC
Y
.
A
C
and
B
F
BF
BF
meets at
P
,
C
E
P, CE
P
,
CE
and
B
D
BD
B
D
at
Q
Q
Q
and
A
E
AE
A
E
and
D
F
DF
D
F
at
R
.
R.
R
.
If
O
O
O
is the point of intersection of
Y
Q
YQ
Y
Q
and
Z
R
,
ZR,
ZR
,
find the
∠
X
O
P
.
\angle XOP.
∠
XOP
.
8
1
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Find the angle.
△
A
B
C
\triangle ABC
△
A
BC
is an acute angled triangle. Perpendiculars drawn from its vertices on the opposite sides are
A
D
AD
A
D
,
B
E
BE
BE
and
C
F
CF
CF
. The line parallel to
D
F
DF
D
F
through
E
E
E
meets
B
C
BC
BC
at
Y
Y
Y
and
B
A
BA
B
A
at
X
X
X
.
D
F
DF
D
F
and
C
A
CA
C
A
meet at
Z
Z
Z
. Circumcircle of
X
Y
Z
XYZ
X
Y
Z
meets
A
C
AC
A
C
at
S
S
S
. Given,
∠
B
=
3
3
∘
.
\angle B=33 ^\circ.
∠
B
=
3
3
∘
.
find the angle
∠
F
S
D
\angle FSD
∠
FS
D
with proof.