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Problems
Contests
National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2020 Bangladesh Mathematical Olympiad National
2020 Bangladesh Mathematical Olympiad National
Part of
Bangladesh Mathematical Olympiad
Subcontests
(10)
Problem 10
1
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Find the minimum value of collinear points
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral.
O
O
O
is the intersection of
A
C
AC
A
C
and
B
D
BD
B
D
.
A
O
=
3
AO=3
A
O
=
3
,
B
O
=
4
BO=4
BO
=
4
,
C
O
=
5
CO=5
CO
=
5
,
D
O
=
6
DO=6
D
O
=
6
.
X
X
X
and
Y
Y
Y
are points in segment
A
B
AB
A
B
and
C
D
CD
C
D
respectively, such that
X
,
O
,
Y
X,O,Y
X
,
O
,
Y
are collinear. The minimum of
X
B
X
A
+
Y
C
Y
D
\frac{XB}{XA}+\frac{YC}{YD}
X
A
XB
+
Y
D
Y
C
can be written as
a
c
b
\frac{a\sqrt{c}}{b}
b
a
c
, where
a
b
\frac{a}{b}
b
a
is in lowest term and
c
c
c
is not divisible by any square number greater then
1
1
1
. What is the value of
10
a
+
b
+
c
10a+b+c
10
a
+
b
+
c
?
Problem 9
1
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The special set
Bristy wants to build a special set
A
A
A
. She starts with
A
=
{
0
,
42
}
A=\{0, 42\}
A
=
{
0
,
42
}
. At any step, she can add an integer
x
x
x
to the set
A
A
A
if it is a root of a polynomial which uses the already existing integers in
A
A
A
as coefficients. She keeps doing this, adding more and more numbers to
A
A
A
. After she eventually runs out of numbers to add to
A
A
A
, how many numbers will be in
A
A
A
?
Problem 8
1
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Kawaii Permutations
We call a permutation of the numbers
1
1
1
,
2
2
2
,
3
3
3
,
…
\dots
…
,
n
n
n
'kawaii' if there is exactly one number that is greater than its position. For example:
1
1
1
,
4
4
4
,
3
3
3
,
2
2
2
is a kawaii permutation (when
n
=
4
n=4
n
=
4
) because only the number
4
4
4
is greater than its position
2
2
2
. How many kawaii permutations are there if
n
=
14
n=14
n
=
14
?
Problem 7
1
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Find the area of a subset of a complex plane
f
f
f
is a function on the set of complex numbers such that
f
(
z
)
=
1
/
(
z
∗
)
f(z)=1/(z*)
f
(
z
)
=
1/
(
z
∗
)
, where
z
∗
z*
z
∗
is the complex conjugate of
z
z
z
.
S
S
S
is the set of complex numbers
z
z
z
such that the real part of
f
(
z
)
f(z)
f
(
z
)
lies between
1
/
2020
1/2020
1/2020
and
1
/
2018
1/2018
1/2018
. If
S
S
S
is treated as a subset of the complex plane, the area of
S
S
S
can be expressed as
m
×
π
m× \pi
m
×
π
where
m
m
m
is an integer. What is the value of
m
m
m
?
Problem 6
1
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Minimum value of an one to one function
f
f
f
is a one-to-one function from the set of positive integers to itself such that
f
(
x
y
)
=
f
(
x
)
×
f
(
y
)
f(xy) = f(x) × f(y)
f
(
x
y
)
=
f
(
x
)
×
f
(
y
)
Find the minimum possible value of
f
(
2020
)
f(2020)
f
(
2020
)
.
Problem 5
1
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Evaluation of the value of equal segments
In triangle
A
B
C
ABC
A
BC
,
A
B
=
52
AB = 52
A
B
=
52
,
B
C
=
34
BC = 34
BC
=
34
and
C
A
=
50
CA = 50
C
A
=
50
. We split
B
C
BC
BC
into
n
n
n
equal segments by placing
n
−
1
n-1
n
−
1
new points. Among these points are the feet of the altitude, median and angle bisector from
A
A
A
. What is the smallest possible value of
n
n
n
?
Problem 4
1
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Maximum number having the same slope
56
56
56
lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly
594
594
594
points, what is the maximum number of them that could have the same slope?
Problem 3
1
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Finding the value of 1/a
Let
R
R
R
be the set of all rectangles centered at the origin and with perimeter
1
1
1
(the center of a rectangle is the intersection point of its two diagonals). Let
S
S
S
be a region that contains all of the rectangles in
R
R
R
(region
A
A
A
contains region
B
B
B
, if
B
B
B
is completely inside of
A
A
A
). The minimum possible area of
S
S
S
has the form
π
a
\pi a
πa
, where
a
a
a
is a real number. Find
1
/
a
1/a
1/
a
.
Problem 2
1
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n raised to the power n, as a perfect square
How many integers
n
n
n
are there subject to the constraint that
1
≤
n
≤
2020
1 \leq n \leq 2020
1
≤
n
≤
2020
and
n
n
n^n
n
n
is a perfect square?
Problem 1
1
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Probability relating to a 24-sided dice
Lazim rolls two
24
24
24
-sided dice. From the two rolls, Lazim selects the die with the highest number.
N
N
N
is an integer not greater than
24
24
24
. What is the largest possible value for
N
N
N
such that there is a more than
50
50
50
% chance that the die Lazim selects is larger than or equal to
N
N
N
?