MathDB
Problems
Contests
National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2023 Bangladesh Mathematical Olympiad
2023 Bangladesh Mathematical Olympiad
Part of
Bangladesh Mathematical Olympiad
Subcontests
(10)
P10
2
Hide problems
Colouring the grid for points
Joy has a square board of size
n
×
n
n \times n
n
×
n
. At every step, he colours a cell of the board. He cannot colour any cell more than once. He also counts points while colouring the cells. At first, he has
0
0
0
points. Every step, after colouring a cell
c
c
c
, he takes the largest possible set
S
S
S
that creates a "
+
+
+
" sign where all cells are coloured and
c
c
c
lies in the centre. Then, he gets the size of set
S
S
S
as points. After colouring the whole
n
×
n
n \times n
n
×
n
board, what is the maximum possible amount of points he can get?
Polynomial with 2023 roots.
Let all possible
2023
2023
2023
-degree real polynomials:
P
(
x
)
=
x
2023
+
a
1
x
2022
+
a
2
x
2021
+
⋯
+
a
2022
x
+
a
2023
P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}
P
(
x
)
=
x
2023
+
a
1
x
2022
+
a
2
x
2021
+
⋯
+
a
2022
x
+
a
2023
, where
P
(
0
)
+
P
(
1
)
=
0
P(0)+P(1)=0
P
(
0
)
+
P
(
1
)
=
0
, and the polynomial has 2023 real roots
r
1
,
r
2
,
⋯
r
2023
r_1, r_2,\cdots r_{2023}
r
1
,
r
2
,
⋯
r
2023
[not necessarily distinct] so that
0
≤
r
1
,
r
2
,
⋯
r
2023
≤
1
0\leq r_1,r_2,\cdots r_{2023}\leq1
0
≤
r
1
,
r
2
,
⋯
r
2023
≤
1
. What is the maximum value of
r
1
⋅
r
2
⋯
r
2023
?
r_1 \cdot r_2 \cdots r_{2023}?
r
1
⋅
r
2
⋯
r
2023
?
P9
2
Hide problems
Big regular polygons
Let
A
1
A
2
…
A
2
n
A_1A_2\dots A_{2n}
A
1
A
2
…
A
2
n
be a regular
2
n
2n
2
n
-gon inscribed in circle
ω
\omega
ω
. Let
P
P
P
be any point on the circle
ω
\omega
ω
. Let
H
1
,
H
2
,
…
,
H
n
H_1,H_2,\dots, H_n
H
1
,
H
2
,
…
,
H
n
be orthocenters of triangles
P
A
1
A
2
,
P
A
3
A
4
,
…
,
P
A
2
n
−
1
A
2
n
PA_1A_2, PA_3A_4,\dots, PA_{2n-1}A_{2n}
P
A
1
A
2
,
P
A
3
A
4
,
…
,
P
A
2
n
−
1
A
2
n
respectively. Prove that
H
1
H
2
…
H
n
H_1H_2\dots H_n
H
1
H
2
…
H
n
is a regular
n
n
n
-gon.
Cute Geometry Problem
Let
Δ
A
B
C
\Delta ABC
Δ
A
BC
be an acute angled triangle.
D
D
D
is a point on side
B
C
BC
BC
such that
A
D
AD
A
D
bisects angle
∠
B
A
C
\angle BAC
∠
B
A
C
. A line
l
l
l
is tangent to the circumcircles of triangles
A
D
B
ADB
A
D
B
and
A
D
C
ADC
A
D
C
at point
K
K
K
and
L
L
L
, respectively. Let
M
M
M
,
N
N
N
and
P
P
P
be its midpoints of
B
D
BD
B
D
,
D
C
DC
D
C
and
K
L
KL
K
L
, respectively. Prove that
l
l
l
is tangent to the circumcircle of
Δ
M
N
P
\Delta MNP
Δ
MNP
.
P8
1
Hide problems
Overlapping intervals in the number lines
We are given
n
n
n
intervals
[
l
1
,
r
1
]
,
[
l
2
,
r
2
]
,
[
l
3
,
r
3
]
,
…
,
[
l
n
,
r
n
]
[l_1,r_1],[l_2,r_2],[l_3,r_3],\dots, [l_n,r_n]
[
l
1
,
r
1
]
,
[
l
2
,
r
2
]
,
[
l
3
,
r
3
]
,
…
,
[
l
n
,
r
n
]
in the number line. We can divide the intervals into two sets such that no two intervals in the same set have overlaps. Prove that there are at most
n
−
1
n-1
n
−
1
pairs of overlapping intervals.
P7
2
Hide problems
3^m*2^n representation
Prove that every positive integer can be represented in the form
3
m
1
⋅
2
n
1
+
3
m
2
⋅
2
n
2
+
⋯
+
3
m
k
⋅
2
n
k
3^{m_1}\cdot 2^{n_1}+3^{m_2}\cdot 2^{n_2} + \dots + 3^{m_k}\cdot 2^{n_k}
3
m
1
⋅
2
n
1
+
3
m
2
⋅
2
n
2
+
⋯
+
3
m
k
⋅
2
n
k
where
m
1
>
m
2
>
⋯
>
m
k
≥
0
m_1 > m_2 > \dots > m_k \geq 0
m
1
>
m
2
>
⋯
>
m
k
≥
0
and
0
≤
n
1
<
n
2
<
⋯
<
n
k
0 \leq n_1 < n_2 < \dots < n_k
0
≤
n
1
<
n
2
<
⋯
<
n
k
are integers.
Reflection of Perpendicular Bisector
Let
Δ
A
B
C
\Delta ABC
Δ
A
BC
be an acute triangle and
ω
\omega
ω
be its circumcircle. Perpendicular from
A
A
A
to
B
C
BC
BC
intersects
B
C
BC
BC
at
D
D
D
and
ω
\omega
ω
at
K
K
K
. Circle through
A
A
A
,
D
D
D
and tangent to
B
C
BC
BC
at
D
D
D
intersect
ω
\omega
ω
at
E
E
E
.
A
E
AE
A
E
intersects
B
C
BC
BC
at
T
T
T
.
T
K
TK
T
K
intersects
ω
\omega
ω
at
S
S
S
. Assume,
S
D
SD
S
D
intersects
ω
\omega
ω
at
X
X
X
. Prove that
X
X
X
is the reflection of
A
A
A
with respect to the perpendicular bisector of
B
C
BC
BC
.
P6
1
Hide problems
Double parallel lines
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute angle triangle and
ω
\omega
ω
be its circumcircle. Let
N
N
N
be a point on arc
A
C
AC
A
C
not containing
B
B
B
and
S
S
S
be a point on line
A
B
AB
A
B
. The line tangent to
ω
\omega
ω
at
N
N
N
intersects
B
C
BC
BC
at
T
T
T
,
N
S
NS
NS
intersects
ω
\omega
ω
at
K
K
K
. Assume that
∠
N
T
C
=
∠
K
S
B
\angle NTC = \angle KSB
∠
NTC
=
∠
K
SB
. Prove that
C
K
∥
A
N
∥
T
S
CK\parallel AN \parallel TS
C
K
∥
A
N
∥
TS
.
P5
2
Hide problems
Find the minimum
Let
m
m
m
,
n
n
n
and
p
p
p
are real numbers such that
(
m
+
n
+
p
)
(
1
m
+
1
n
+
1
p
)
=
1
\left(m+n+p\right)\left(\frac 1m + \frac 1n + \frac1p\right) =1
(
m
+
n
+
p
)
(
m
1
+
n
1
+
p
1
)
=
1
. Find all possible values of
1
(
m
+
n
+
p
)
2023
−
1
m
2023
−
1
n
2023
−
1
p
2023
.
\frac 1{(m+n+p)^{2023}} -\frac 1{m^{2023}} -\frac 1{n^{2023}} -\frac 1{p^{2023}}.
(
m
+
n
+
p
)
2023
1
−
m
2023
1
−
n
2023
1
−
p
2023
1
.
Integrable Function
Consider an integrable function
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that
a
f
(
a
)
+
b
f
(
b
)
=
0
af(a)+bf(b)=0
a
f
(
a
)
+
b
f
(
b
)
=
0
when
a
b
=
1
ab=1
ab
=
1
. Find the value of the following integration:
∫
0
∞
f
(
x
)
d
x
\int_{0}^{\infty} f(x) \,dx
∫
0
∞
f
(
x
)
d
x
P4
2
Hide problems
Balls and buckets
2023
2023
2023
balls are divided into several buckets such that no bucket contains more than
99
99
99
balls. We can remove balls from any bucket or remove an entire bucket, as many times as we want. Prove that we can remove them in such a way that each of the remaining buckets will have an equal number of balls and the total number of remaining balls will be at least
100
100
100
.
Orthocenter of variable triangles
Let
A
B
C
D
ABCD
A
BC
D
be an isosceles trapezium inscribed in circle
ω
\omega
ω
, such that
A
B
∣
∣
C
D
AB||CD
A
B
∣∣
C
D
. Let
P
P
P
be a point on the circle
ω
\omega
ω
. Let
H
1
H_1
H
1
and
H
2
H_2
H
2
be the orthocenters of triangles
P
A
D
PAD
P
A
D
and
P
B
C
PBC
PBC
respectively. Prove that the length of
H
1
H
2
H_1H_2
H
1
H
2
remains constant, when
P
P
P
varies on the circle.
P3
2
Hide problems
Weird diophantine equation
Solve the equation for the positive integers:
(
x
+
2
y
)
2
+
2
x
+
5
y
+
9
=
(
y
+
z
)
2
(x+2y)^2+2x+5y+9=(y+z)^2
(
x
+
2
y
)
2
+
2
x
+
5
y
+
9
=
(
y
+
z
)
2
Smallest Positive Integer
For any positive integer
n
n
n
, define
f
(
n
)
f(n)
f
(
n
)
to be the smallest positive integer that does not divide
n
n
n
. For example,
f
(
1
)
=
2
f(1)=2
f
(
1
)
=
2
,
f
(
6
)
=
4
f(6)=4
f
(
6
)
=
4
. Prove that for any positive integer
n
n
n
, either
f
(
f
(
n
)
)
f(f(n))
f
(
f
(
n
))
or
f
(
f
(
f
(
n
)
)
)
f(f(f(n)))
f
(
f
(
f
(
n
)))
must be equal to
2
2
2
.
P2
2
Hide problems
Touching semicircles
Let the points
A
,
B
,
C
A,B,C
A
,
B
,
C
lie on a line in this order.
A
B
AB
A
B
is the diameter of semicircle
ω
1
\omega_1
ω
1
,
A
C
AC
A
C
is the diameter of semicircle
ω
2
\omega_2
ω
2
. Assume both
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
lie on the same side of
A
C
AC
A
C
.
D
D
D
is a point on
ω
2
\omega_2
ω
2
such that
B
D
⊥
A
C
BD\perp AC
B
D
⊥
A
C
. A circle centered at
B
B
B
with radius
B
D
BD
B
D
intersects
ω
1
\omega_1
ω
1
at
E
E
E
.
F
F
F
is on
A
C
AC
A
C
such that
E
F
⊥
A
C
EF\perp AC
EF
⊥
A
C
. Prove that
B
C
=
B
F
BC=BF
BC
=
BF
.
Three elements of a set
Let {
a
1
,
a
2
,
⋯
,
a
n
a_1, a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
} be a set of
n
n
n
real numbers whos sym equals S. It is known that each number in the set is less than
S
n
−
1
\frac{S}{n-1}
n
−
1
S
. Prove that for any three numbers
a
i
a_i
a
i
,
a
j
a_j
a
j
and
a
k
a_k
a
k
in the set,
a
i
+
a
j
>
a
k
a_i+a_j>a_k
a
i
+
a
j
>
a
k
.
P1
2
Hide problems
Diophantine factorial
Find all possible non-negative integer solution
(
x
,
y
)
(x,y)
(
x
,
y
)
of the following equation-
x
!
+
2
y
=
(
x
+
1
)
!
x! + 2^y =(x+1)!
x
!
+
2
y
=
(
x
+
1
)!
Note:
x
!
=
x
⋅
(
x
−
1
)
!
x!=x \cdot (x-1)!
x
!
=
x
⋅
(
x
−
1
)!
and
0
!
=
1
0!=1
0
!
=
1
. For example,
5
!
=
5
×
4
×
3
×
2
×
1
=
120
5! = 5\times 4\times 3\times 2\times 1 = 120
5
!
=
5
×
4
×
3
×
2
×
1
=
120
.
Diophantine Factorial
Find all possible non-negative integer solution (
x
,
x,
x
,
y
y
y
) of the following equation-
x
!
+
2
y
=
z
!
x!+2^y=z!
x
!
+
2
y
=
z
!
Note:
x
!
=
x
⋅
(
x
−
1
)
!
x!=x\cdot(x-1)!
x
!
=
x
⋅
(
x
−
1
)!
and
0
!
=
1
0!=1
0
!
=
1
. For example,
5
!
=
5
×
4
×
3
×
2
×
1
=
120
5!=5\times4\times3\times2\times1=120
5
!
=
5
×
4
×
3
×
2
×
1
=
120
.