MathDB
Problems
Contests
National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2021 Bangladesh Mathematical Olympiad
2021 Bangladesh Mathematical Olympiad
Part of
Bangladesh Mathematical Olympiad
Subcontests
(12)
Problem 10
1
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a√b-c=length of side of octagon, find a+b+c
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
A_1A_2A_3A_4A_5A_6A_7A_8
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
is a regular octagon. Let
P
P
P
be a point inside the octagon such that the distances from
P
P
P
to
A
1
A
2
,
A
2
A
3
A_1A_2, A_2A_3
A
1
A
2
,
A
2
A
3
and
A
3
A
4
A_3A_4
A
3
A
4
are
24
,
26
24, 26
24
,
26
and
27
27
27
respectively. The length of
A
1
A
2
A_1A_2
A
1
A
2
can be written as
a
b
−
c
a \sqrt{b} -c
a
b
−
c
, where
a
,
b
a,b
a
,
b
and
c
c
c
are positive integers and
b
b
b
is not divisible by any square number other than
1
1
1
. What is the value of
(
a
+
b
+
c
)
(a+b+c)
(
a
+
b
+
c
)
?
Problem 12
1
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Smallest value of adjective function f(25)
A function
g
:
Z
→
Z
g: \mathbb{Z} \to \mathbb{Z}
g
:
Z
→
Z
is called adjective if
g
(
m
)
+
g
(
n
)
>
m
a
x
(
m
2
,
n
2
)
g(m)+g(n)>max(m^2,n^2)
g
(
m
)
+
g
(
n
)
>
ma
x
(
m
2
,
n
2
)
for any pair of integers
m
m
m
and
n
n
n
. Let
f
f
f
be an adjective function such that the value of
f
(
1
)
+
f
(
2
)
+
⋯
+
f
(
30
)
f(1)+f(2)+\dots+f(30)
f
(
1
)
+
f
(
2
)
+
⋯
+
f
(
30
)
is minimized. Find the smallest possible value of
f
(
25
)
f(25)
f
(
25
)
.
Problem 11
1
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Number of integers (a,b,m,n) for holding Conditions
How many quadruples of positive integers
(
a
,
b
,
m
,
n
)
(a,b,m,n)
(
a
,
b
,
m
,
n
)
are there such that all of the following statements hold? 1.
a
,
b
<
5000
a,b<5000
a
,
b
<
5000
2.
m
,
n
<
22
m,n<22
m
,
n
<
22
3.
g
c
d
(
m
,
n
)
=
1
gcd(m,n)=1
g
c
d
(
m
,
n
)
=
1
4.
(
a
2
+
b
2
)
m
=
(
a
b
)
n
(a^2+b^2)^m=(ab)^n
(
a
2
+
b
2
)
m
=
(
ab
)
n
Problem 9
1
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Nice integers smaller than 3000
A positive integer
n
n
n
is called nice if it has at least
3
3
3
proper divisors and it is equal to the sum of its three largest proper divisors. For example,
6
6
6
is nice because its largest proper divisors are
3
,
2
,
1
3,2,1
3
,
2
,
1
and
6
=
3
+
2
+
1
6=3+2+1
6
=
3
+
2
+
1
. Find the number of nice integers not greater than
3000
3000
3000
.
Problem 8
1
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Sum of all possible values of winning strategy in a game
Shakur and Tiham are playing a game. Initially, Shakur picks a positive integer not greater than
1000
1000
1000
. Then Tiham picks a positive integer strictly smaller than that.Then they keep on doing this taking turns to pick progressively smaller and smaller positive integers until some one picks
1
1
1
. After that, all the numbers that have been picked so far are added up. The person picking the number
1
1
1
wins if and only if this sum is a perfect square. Otherwise, the other player wins. What is the sum of all possible values of
n
n
n
such that if Shakur starts with the number
n
n
n
, he has a winning strategy?
Problem 7
1
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Calculate number of 1-runs in a binary string expression
A binary string is a word containing only
0
0
0
s and
1
1
1
s. In a binary string, a
1
−
1-
1
−
run is a non extendable substring containing only
1
1
1
s. Given a positive integer
n
n
n
, let
B
(
n
)
B(n)
B
(
n
)
be the number of
1
−
1-
1
−
runs in the binary representation of
n
n
n
. For example,
B
(
107
)
=
3
B(107)=3
B
(
107
)
=
3
since
107
107
107
in binary is
1101011
1101011
1101011
which has exactly three
1
−
1-
1
−
runs. What is the following expression equal to?
B
(
1
)
+
B
(
2
)
+
B
(
3
)
+
⋯
+
B
(
255
)
B(1)+B(2)+B(3)+ \dots + B(255)
B
(
1
)
+
B
(
2
)
+
B
(
3
)
+
⋯
+
B
(
255
)
Problem 6
1
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Find a+b where a/b is the length QA
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. The external bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
meets the line
B
C
BC
BC
at point
N
N
N
. Let
M
M
M
be the midpoint of
B
C
BC
BC
.
P
P
P
and
Q
Q
Q
are two points on line
A
N
AN
A
N
such that,
∠
P
M
N
=
∠
M
Q
N
=
9
0
∘
\angle PMN=\angle MQN=90^{\circ}
∠
PMN
=
∠
MQN
=
9
0
∘
. If
P
N
=
5
PN=5
PN
=
5
and
B
C
=
3
BC=3
BC
=
3
, then the length
Q
A
QA
Q
A
can be expressed as
a
b
\frac{a}{b}
b
a
where
a
a
a
and
b
b
b
are co-prime positive integers. What is the value of
(
a
+
b
)
(a+b)
(
a
+
b
)
?
Problem 5
1
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Ways to score 42 with 3 rolls of a 20 sided dice
How many ways can you roll three 20-sided dice such that the sum of the three rolls is exactly
42
42
42
? Here the order of the rolls matter. (Note that a 20-sided die is is very much like a regular 6-sided die other than the fact that it has
20
20
20
faces instead of
6
6
6
)
Problem 4
1
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Sum of all possible values of P(10)
P
(
x
)
P(x)
P
(
x
)
is a polynomial in
x
x
x
with non-negative integer coefficients. If
P
(
1
)
=
5
P(1)=5
P
(
1
)
=
5
and
P
(
P
(
1
)
)
=
177
P(P(1))=177
P
(
P
(
1
))
=
177
, what is the sum of all possible values of
P
(
10
)
P(10)
P
(
10
)
?
Problem 3
1
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Find the value of m+n of m/n=∠ACB
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
. Points
E
E
E
and
F
F
F
are on segments
A
C
AC
A
C
and
B
C
BC
BC
respectively such that,
A
E
=
A
I
AE=AI
A
E
=
A
I
and
B
F
=
B
I
BF=BI
BF
=
B
I
. If
E
F
EF
EF
is the perpendicular bisector of
C
I
CI
C
I
, then
∠
A
C
B
\angle{ACB}
∠
A
CB
in degrees can be written as
m
n
\frac{m}{n}
n
m
where
m
m
m
and
n
n
n
are co-prime positive integers. Find the value of
m
+
n
m+n
m
+
n
.
Problem 2
1
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Find the value of 10n where n is the minimum of an expression
Let
u
,
v
u, v
u
,
v
be real numbers. The minimum value of
u
2
+
v
2
+
(
u
−
1
)
2
+
v
2
+
u
2
+
(
v
−
1
)
2
+
(
u
−
1
)
2
+
(
v
−
1
)
2
\sqrt{u^2+v^2} +\sqrt{(u-1)^2+v^2}+\sqrt {u^2+ (v-1)^2}+ \sqrt{(u-1)^2+(v-1)^2}
u
2
+
v
2
+
(
u
−
1
)
2
+
v
2
+
u
2
+
(
v
−
1
)
2
+
(
u
−
1
)
2
+
(
v
−
1
)
2
can be written as
n
\sqrt{n}
n
. Find the value of
10
n
10n
10
n
.
Problem 1
1
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Find the value of the function A(T(2021))
For a positive integer
n
n
n
, let
A
(
n
)
A(n)
A
(
n
)
be the equal to the remainder when
n
n
n
is divided by
11
11
11
and let
T
(
n
)
=
A
(
1
)
+
A
(
2
)
+
A
(
3
)
+
⋯
+
A
(
n
)
T(n)=A(1)+A(2)+A(3)+ \dots + A(n)
T
(
n
)
=
A
(
1
)
+
A
(
2
)
+
A
(
3
)
+
⋯
+
A
(
n
)
. Find the value of
A
(
T
(
2021
)
)
A(T(2021))
A
(
T
(
2021
))