MathDB
Problems
Contests
National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2024 Bangladesh Mathematical Olympiad
2024 Bangladesh Mathematical Olympiad
Part of
Bangladesh Mathematical Olympiad
Subcontests
(10)
P9
2
Hide problems
Observe the beauty
Let
A
B
C
ABC
A
BC
be a triangle and
M
M
M
be the midpoint of side
B
C
BC
BC
. The perpendicular bisector of
B
C
BC
BC
intersects the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at points
K
K
K
and
L
L
L
(
K
K
K
and
A
A
A
lie on the opposite sides of
B
C
BC
BC
). A circle passing through
L
L
L
and
M
M
M
intersects
A
K
AK
A
K
at points
P
P
P
and
Q
Q
Q
(
P
P
P
lies on the line segment
A
Q
AQ
A
Q
).
L
Q
LQ
L
Q
intersects the circumcircle of
△
K
M
Q
\triangle KMQ
△
K
MQ
again at
R
R
R
. Prove that
B
P
C
R
BPCR
BPCR
is a cyclic quadrilateral.
Product divisible by factorial
Find all pairs of positive integers
(
k
,
m
)
(k, m)
(
k
,
m
)
such that for any positive integer
n
n
n
, the product
(
n
+
m
)
(
n
+
2
m
)
⋯
(
n
+
k
m
)
(n+m)(n+2m)\cdots(n+km)
(
n
+
m
)
(
n
+
2
m
)
⋯
(
n
+
km
)
is divisible by
k
!
k!
k
!
.
P3
1
Hide problems
Find the value
Let
a
a
a
and
b
b
b
be real numbers such that
a
a
2
−
5
=
b
5
−
b
2
=
a
b
a
2
b
2
−
5
\frac{a}{a^2-5} = \frac{b}{5-b^2} = \frac{ab}{a^2b^2-5}
a
2
−
5
a
=
5
−
b
2
b
=
a
2
b
2
−
5
ab
where
a
+
b
≠
0
a+b \neq 0
a
+
b
=
0
.
a
4
+
b
4
=
a^4 + b^4 =
a
4
+
b
4
=
?
P4
1
Hide problems
Non-trivial sequence, do you exist?
Let
a
1
,
a
2
,
…
,
a
11
a_1, a_2, \ldots, a_{11}
a
1
,
a
2
,
…
,
a
11
be integers. Prove that there exist numbers
b
1
,
b
2
,
…
,
b
11
b_1, b_2, \ldots, b_{11}
b
1
,
b
2
,
…
,
b
11
such that [*]
b
i
b_i
b
i
is equal to
−
1
,
0
-1,0
−
1
,
0
or
1
1
1
for all
i
∈
{
1
,
2
,
…
,
11
}
i \in \{1, 2,\dots, 11\}
i
∈
{
1
,
2
,
…
,
11
}
. [*] all numbers can't be zero at a time. [*] the number
N
=
a
1
b
1
+
a
2
b
2
+
…
+
a
11
b
11
N=a_1b_1+a_2b_2+\ldots+a_{11}b_{11}
N
=
a
1
b
1
+
a
2
b
2
+
…
+
a
11
b
11
is divisible by
2024
2024
2024
.
P8
2
Hide problems
Bosonti n-point
A set consisting of
n
n
n
points of a plane is called a bosonti
n
n
n
-point if any three of its points are located in vertices of an isosceles triangle. Find all positive integers
n
n
n
for which there exists a bosonti
n
n
n
-point.
Too many ways
Let
k
k
k
be a positive integer. Show that there exist infinitely many positive integers
n
n
n
such that
n
n
−
1
n
−
1
\frac{n^n-1}{n-1}
n
−
1
n
n
−
1
has at least
k
k
k
distinct prime divisors.Proposed by Adnan Sadik
P5
2
Hide problems
Equal circumradii
Consider
△
X
P
Q
\triangle XPQ
△
XPQ
and
△
Y
P
Q
\triangle YPQ
△
Y
PQ
such that
X
X
X
and
Y
Y
Y
are on the opposite sides of
P
Q
PQ
PQ
and the circumradius of
△
X
P
Q
\triangle XPQ
△
XPQ
and the circumradius of
△
Y
P
Q
\triangle YPQ
△
Y
PQ
are the same.
I
I
I
and
J
J
J
are the incenters of
△
X
P
Q
\triangle XPQ
△
XPQ
and
△
Y
P
Q
\triangle YPQ
△
Y
PQ
respectively. Let
M
M
M
be the midpoint of
P
Q
PQ
PQ
. Suppose
I
,
M
,
J
I, M, J
I
,
M
,
J
are collinear. Prove that
X
P
Y
Q
XPYQ
XP
Y
Q
is a parallelogram.
Spot the symmetry
Let
I
I
I
be the incenter of
△
A
B
C
\triangle ABC
△
A
BC
and
P
P
P
be a point such that
P
I
PI
P
I
is perpendicular to
B
C
BC
BC
and
P
A
PA
P
A
is parallel to
B
C
BC
BC
. Let the line parallel to
B
C
BC
BC
, which is tangent to the incircle of
△
A
B
C
\triangle ABC
△
A
BC
, intersect
A
B
AB
A
B
and
A
C
AC
A
C
at points
Q
Q
Q
and
R
R
R
respectively. Prove that
∠
B
P
Q
=
∠
C
P
R
\angle BPQ = \angle CPR
∠
BPQ
=
∠
CPR
.
P2
1
Hide problems
Four points on a circle
In a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
, the diagonals intersect at
E
E
E
.
F
F
F
and
G
G
G
are on chord
A
C
AC
A
C
and chord
B
D
BD
B
D
respectively such that
A
F
=
B
E
AF = BE
A
F
=
BE
and
D
G
=
C
E
DG = CE
D
G
=
CE
. Prove that,
A
,
G
,
F
,
D
A, G, F, D
A
,
G
,
F
,
D
lie on the same circle.
P1
2
Hide problems
Easy diophantine equation
Find all non-negative integers
x
,
y
x, y
x
,
y
such that
x
3
y
+
x
+
y
=
x
y
+
2
x
y
2
x^3y+x+y=xy+2xy^2
x
3
y
+
x
+
y
=
x
y
+
2
x
y
2
Too random mod
Find all prime numbers
p
p
p
and
q
q
q
such that
p
3
−
3
q
=
10.
p^3-3^q=10.
p
3
−
3
q
=
10.
Proposed by Md. Fuad Al Alam
P10
1
Hide problems
Keep it disconnected
Juty and Azgor plays the following game on a
(
2
n
+
1
)
×
(
2
n
+
1
)
(2n+1) \times (2n+1)
(
2
n
+
1
)
×
(
2
n
+
1
)
board with Juty moving first. Initially all cells are colored white. On Juty's turn, she colors a white cell green and on Azgor's turn, he colors a white cell red. The game ends after they color all the cells of the board. Juty wins if all the green cells are connected, i.e. given any two green cells, there is at least one chain of neighbouring green cells connecting them (we call two cells neighboring if they share at least one corner), otherwise Azgor wins. Determine which player has a winning strategy.Proposed by Atonu Roy Chowdhury
P6
2
Hide problems
Minimum possible value
Let
a
1
,
a
2
,
…
,
a
2024
a_1, a_2, \ldots, a_{2024}
a
1
,
a
2
,
…
,
a
2024
be a permutation of
1
,
2
,
…
,
2024
1, 2, \ldots, 2024
1
,
2
,
…
,
2024
. Find the minimum possible value of
∑
i
=
1
2023
[
(
a
i
+
a
i
+
1
)
(
1
a
i
+
1
a
i
+
1
)
+
1
a
i
a
i
+
1
]
\sum_{i=1} ^{2023} \Big[(a_i+a_{i+1})\Big(\frac{1}{a_i}+\frac{1}{a_{i+1}}\Big)+\frac{1}{a_ia_{i+1}}\Big]
i
=
1
∑
2023
[
(
a
i
+
a
i
+
1
)
(
a
i
1
+
a
i
+
1
1
)
+
a
i
a
i
+
1
1
]
Proposed by Md. Ashraful Islam Fahim
Polynomial with a constructor sequence
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
for which there exists a sequence
a
1
,
a
2
,
a
3
,
…
a_1, a_2, a_3, \ldots
a
1
,
a
2
,
a
3
,
…
of real numbers such that
a
m
+
a
n
=
P
(
m
n
)
a_m + a_n = P(mn)
a
m
+
a
n
=
P
(
mn
)
for any positive integer
m
m
m
and
n
n
n
.
P7
2
Hide problems
Bash? Anyone??
Let
A
B
C
D
ABCD
A
BC
D
be a square.
E
E
E
and
F
F
F
lie on sides
A
B
AB
A
B
and
B
C
BC
BC
, respectively, such that
B
E
=
B
F
BE = BF
BE
=
BF
. The line perpendicular to
C
E
CE
CE
, which passes through
B
B
B
, intersects
C
E
CE
CE
and
A
D
AD
A
D
at points
G
G
G
and
H
H
H
, respectively. The lines
F
H
FH
F
H
and
C
E
CE
CE
intersect at point
P
P
P
and the lines
G
F
GF
GF
and
C
D
CD
C
D
intersect at point
Q
Q
Q
. Prove that the line
D
P
DP
D
P
is perpendicular to the line
B
Q
BQ
BQ
.
Ceiling in FE
Find all functions
f
:
N
→
N
f:\mathbb{N} \to \mathbb{N}
f
:
N
→
N
such that
f
(
⌈
f
(
m
)
n
⌉
)
=
⌈
m
f
(
n
)
⌉
f\left(\Big \lceil \frac{f(m)}{n} \Big \rceil\right)=\Big \lceil \frac{m}{f(n)} \Big \rceil
f
(
⌈
n
f
(
m
)
⌉
)
=
⌈
f
(
n
)
m
⌉
for all
m
,
n
∈
N
m,n \in \mathbb{N}
m
,
n
∈
N
.Proposed by Md. Ashraful Islam Fahim