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Problems
Contests
National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2019 Bangladesh Mathematical Olympiad
2019 Bangladesh Mathematical Olympiad
Part of
Bangladesh Mathematical Olympiad
Subcontests
(10)
1
1
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BdMO National Higher Secondary 2019/1
Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.
2
1
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BdMO National Higher Secondary 2019/2
Prove that,if
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers,
a
b
c
+
b
c
a
+
c
a
b
≥
2
a
+
2
b
−
2
c
\dfrac{a}{bc}+ \dfrac{b}{ca}+\dfrac{c}{ab}\geq \dfrac{2}{a}+\dfrac{2}{b}-\dfrac{2}{c}
b
c
a
+
c
a
b
+
ab
c
≥
a
2
+
b
2
−
c
2
3
1
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BdMO National Higher Secondary 2019/3
Let
α
\alpha
α
and
ω
\omega
ω
be two circles such that
ω
\omega
ω
goes through the center of
α
\alpha
α
.
ω
\omega
ω
intersects
α
\alpha
α
at
A
A
A
and
B
B
B
.Let
P
P
P
any point on the circumference
ω
\omega
ω
.The lines
P
A
PA
P
A
and
P
B
PB
PB
intersects
α
\alpha
α
again at
E
E
E
and
F
F
F
respectively.Prove that
A
B
=
E
F
AB=EF
A
B
=
EF
.
4
1
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BdMO National Higher Secondary 2019/4
A
A
A
is a positive real number.
n
n
n
is positive integer number.Find the set of possible values of the infinite sum
x
0
n
+
x
1
n
+
x
2
n
+
.
.
.
x_0^n+x_1^n+x_2^n+...
x
0
n
+
x
1
n
+
x
2
n
+
...
where
x
0
,
x
1
,
x
2
.
.
.
x_0,x_1,x_2...
x
0
,
x
1
,
x
2
...
are all positive real numbers so that the infinite series
x
0
+
x
1
+
x
2
+
.
.
.
x_0+x_1+x_2+...
x
0
+
x
1
+
x
2
+
...
has sum
A
A
A
.
5
1
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BdMO National Higher Secondary 2019/5
Prove that for all positive integers
n
n
n
we can find a permutation of {
1
,
2
,
.
.
.
,
n
1,2,...,n
1
,
2
,
...
,
n
} such that the average of two numbers doesn't appear in-between them.For example {
1
,
3
,
2
,
4
1,3,2,4
1
,
3
,
2
,
4
}works,but {
1
,
4
,
2
,
3
1,4,2,3
1
,
4
,
2
,
3
} doesn't because
2
2
2
is between
1
1
1
and
3
3
3
.
6
1
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BdMO National Higher Secondary 2019/6
When a function
f
(
x
)
f(x)
f
(
x
)
is differentiated
n
n
n
times ,the function we get id denoted
f
n
(
x
)
f^n(x)
f
n
(
x
)
.If
f
(
x
)
=
e
x
x
f(x)=\dfrac {e^x}{x}
f
(
x
)
=
x
e
x
.Find the value of
lim
n
→
∞
f
2
n
(
1
)
(
2
n
)
!
\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}
n
→
∞
lim
(
2
n
)!
f
2
n
(
1
)
7
1
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BdMO National Higher Secondary 2019/7
Given three cocentric circles
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
,
ω
3
\omega_3
ω
3
with radius
r
1
,
r
2
,
r
3
r_1,r_2,r_3
r
1
,
r
2
,
r
3
such that
r
1
+
r
3
≥
2
r
2
r_1+r_3\geq {2r_2}
r
1
+
r
3
≥
2
r
2
.Constrat a line that intersects
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
,
ω
3
\omega_3
ω
3
at
A
,
B
,
C
A,B,C
A
,
B
,
C
respectively such that
A
B
=
B
C
AB=BC
A
B
=
BC
.
8
1
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BdMO National Higher Secondary 2019/8
The set of natural numbers
N
\mathbb{N}
N
are partitioned into a finite number of subsets.Prove that there exists a subset of
S
S
S
so that for any natural numbers
n
n
n
,there are infinitely many multiples of
n
n
n
in
S
S
S
.
9
1
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BdMO National Higher Secondary 2019/9
Let
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral.The internal angle bisectors of
∠
B
A
C
\angle {BAC}
∠
B
A
C
and
∠
B
D
C
\angle {BDC}
∠
B
D
C
meets at
P
P
P
.
∠
A
P
B
=
∠
C
P
D
\angle {APB}=\angle {CPD}
∠
A
PB
=
∠
CP
D
.Prove that
A
B
+
B
D
=
A
C
+
C
D
AB+BD=AC+CD
A
B
+
B
D
=
A
C
+
C
D
.
10
1
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Hard one?
Given
2020
∗
2020
2020*2020
2020
∗
2020
chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other. Warrior is a special chess piece which can move either
3
3
3
steps forward and one step sideward and
2
2
2
step forward and
2
2
2
step sideward in any direction.