MathDB
Problems
Contests
National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2018 Bangladesh Mathematical Olympiad
2018 Bangladesh Mathematical Olympiad
Part of
Bangladesh Mathematical Olympiad
Subcontests
(8)
5
1
Hide problems
Four circles and quadrilateral
Four circles are drawn with the sides of quadrilateral
A
B
C
D
ABCD
A
BC
D
as diameters. The two circles passing through
A
A
A
meet again at
E
E
E
. The two circles passing through
B
B
B
meet again at
F
F
F
. The two circles passing through
C
C
C
meet again at
G
G
G
. The two circles passing through
D
D
D
meet again at
H
H
H
. Suppose,
E
,
F
,
G
,
H
E, F, G,H
E
,
F
,
G
,
H
are all distinct. Is the quadrilateral
E
F
G
H
EFGH
EFG
H
similar to
A
B
C
D
ABCD
A
BC
D
? Show with proof.
2
1
Hide problems
Diameter,tangent & intersection point
BdMO National 2018 Higher Secondary P2
A
B
AB
A
B
is a diameter of a circle and
A
D
AD
A
D
&
B
C
BC
BC
are two tangents of that circle.
A
C
AC
A
C
&
B
D
BD
B
D
intersect on a point of the circle.
A
D
=
a
AD=a
A
D
=
a
&
B
C
=
b
BC=b
BC
=
b
.If
a
≠
b
a\neq b
a
=
b
then
A
B
=
?
AB=?
A
B
=
?
3
1
Hide problems
Fair dice & Impossible 18+
BdMO National 2018 Higher Secondary P3Nazia rolls four fair six-sided dice. She doesn’t see the results. Her friend Faria tells her that the product of the numbers is
144
144
144
. Faria also says the sum of the dice,
S
S
S
satisfies
14
≤
S
≤
18
14\leq S\leq 18
14
≤
S
≤
18
. Nazia tells Faria that
S
S
S
cannot be one of the numbers in the set {
14
,
15
,
16
,
17
,
18
14,15,16,17,18
14
,
15
,
16
,
17
,
18
} if the product is
144
144
144
. Which number in the range {
14
,
15
,
16
,
17
,
18
14,15,16,17,18
14
,
15
,
16
,
17
,
18
} is an impossible value for
S
S
S
?
7
1
Hide problems
Evaluate the value of the trignometric function
Evaluate
∫
0
π
/
2
cos
4
x
+
sin
x
cos
3
x
+
sin
2
x
cos
2
x
+
sin
3
x
cos
x
sin
4
x
+
cos
4
x
+
2
s
i
n
x
cos
3
x
+
2
sin
2
x
cos
2
x
+
2
sin
3
x
cos
x
d
x
\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx
∫
0
π
/2
s
i
n
4
x
+
c
o
s
4
x
+
2
s
in
x
c
o
s
3
x
+
2
s
i
n
2
x
c
o
s
2
x
+
2
s
i
n
3
x
c
o
s
x
c
o
s
4
x
+
s
i
n
x
c
o
s
3
x
+
s
i
n
2
x
c
o
s
2
x
+
s
i
n
3
x
c
o
s
x
d
x
1
1
Hide problems
2018 Bangladesh National Olympiad
Solve:
x
2
(
2
−
x
)
2
=
1
+
2
(
1
−
x
)
2
x^2(2-x)^2=1+2(1-x)^2
x
2
(
2
−
x
)
2
=
1
+
2
(
1
−
x
)
2
Where
x
x
x
is real number.
8
1
Hide problems
a hard problem
a tournament is playing between n persons. Everybody plays with everybody one time. There is no draw here. A number
k
k
k
is called
n
n
n
good if there is any tournament such that in that tournament they have any player in the tournament that has lost all of
k
k
k
's. prove that 1.
n
n
n
is greater than or equal to
2
k
+
1
−
1
2^{k+1}-1
2
k
+
1
−
1
2.Find all
n
n
n
such that
2
2
2
is a n-good
6
1
Hide problems
Equality problem
Find all the pairs of integers
(
m
,
n
)
(m,n)
(
m
,
n
)
satisfying the equality
3
(
m
2
+
n
2
)
−
7
(
m
+
n
)
=
−
4
3(m^2+n^2)-7(m+n)=-4
3
(
m
2
+
n
2
)
−
7
(
m
+
n
)
=
−
4
4
1
Hide problems
Intersecting 200 points
Yukihira is counting the minimum number of lines
m
m
m
, that can be drawn on the plane so that they intersect in exactly
200
200
200
distinct points.What is
m
m
m
?