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Problems
Contests
National and Regional Contests
Bangladesh Contests
Bangladesh Mathematical Olympiad
2022 Bangladesh Mathematical Olympiad
2022 Bangladesh Mathematical Olympiad
Part of
Bangladesh Mathematical Olympiad
Subcontests
(8)
8
1
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2 NT problems in 1
Solve the following problems - A) Find any
158
158
158
consecutive integers such that the sum of digits for any of the numbers is not divisible by
17.
17.
17.
B) Prove that, among any
159
159
159
consecutive integers there will always be at least one integer whose sum of digits is divisible by
17.
17.
17.
7
1
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A city where the names of people are either A or B and 10 letters long
Sabbir noticed one day that everyone in the city of BdMO has a distinct word of length
10
10
10
, where each letter is either
A
A
A
or
B
B
B
. Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word
A
B
B
ABB
A
BB
is located consecutively, then these letters can be changed to
B
B
A
BBA
BB
A
or if
B
B
A
BBA
BB
A
is located somewhere in the word consecutively, then these letters can be changed to
A
B
B
ABB
A
BB
(if wanted, the word can be kept as it is, without making this change.) For example
A
A
B
B
A
AABBA
AA
BB
A
can be transformed into
A
A
A
B
B
AAABB
AAA
BB
(the opposite is also possible.) Now Sabbir made a team of
N
N
N
citizens where no one is friends with anyone. What is the highest value of
N
.
N.
N
.
6
1
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Sum of remainder pairs greater than 2017=N
About
5
5
5
years ago, Joydip was researching on the number
2017
2017
2017
. He understood that
2017
2017
2017
is a prime number. Then he took two integers
a
,
b
a,b
a
,
b
such that
0
<
a
,
b
<
2017
0<a,b <2017
0
<
a
,
b
<
2017
and
a
+
b
≠
2017.
a+b\neq 2017.
a
+
b
=
2017.
He created two sequences
A
1
,
A
2
,
…
,
A
2016
A_1,A_2,\dots ,A_{2016}
A
1
,
A
2
,
…
,
A
2016
and
B
1
,
B
2
,
…
,
B
2016
B_1,B_2,\dots, B_{2016}
B
1
,
B
2
,
…
,
B
2016
where
A
k
A_k
A
k
is the remainder upon dividing
a
k
ak
ak
by
2017
2017
2017
, and
B
k
B_k
B
k
is the remainder upon dividing
b
k
bk
bk
by
2017.
2017.
2017.
Among the numbers
A
1
+
B
1
,
A
2
+
B
2
,
…
A
2016
+
B
2016
A_1+B_1,A_2+B_2,\dots A_{2016}+B_{2016}
A
1
+
B
1
,
A
2
+
B
2
,
…
A
2016
+
B
2016
count of those that are greater than
2017
2017
2017
is
N
N
N
. Prove that
N
=
1008.
N=1008.
N
=
1008.
5
1
Hide problems
Four points lie on the same circle
In an acute triangle
△
A
B
C
\triangle ABC
△
A
BC
, the midpoint of
B
C
BC
BC
is
M
M
M
. Perpendicular lines
B
E
BE
BE
and
C
F
CF
CF
are drawn respectively on
A
C
AC
A
C
from
B
B
B
and on
A
B
AB
A
B
from
C
C
C
such that
E
E
E
and
F
F
F
lie on
A
C
AC
A
C
and
A
B
AB
A
B
respectively. The midpoint of
E
F
EF
EF
is
N
.
N.
N
.
M
N
MN
MN
intersects
A
B
AB
A
B
at
K
.
K.
K
.
Prove that, the four points
B
,
K
,
E
,
M
B,K,E,M
B
,
K
,
E
,
M
lie on the same circle.
4
1
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Easy combinatorics about finding minimum integer
Pratyya and Payel have a number each,
n
n
n
and
m
m
m
respectively, where
n
>
m
.
n>m.
n
>
m
.
Everyday, Pratyya multiplies his number by
2
2
2
and then subtracts
2
2
2
from it, and Payel multiplies his number by
2
2
2
and then add
2
2
2
to it. In other words, on the first day their numbers will be
(
2
n
−
2
)
(2n-2)
(
2
n
−
2
)
and
(
2
m
+
2
)
(2m+2)
(
2
m
+
2
)
respectively. Find minimum integer
x
x
x
with proof such that if
n
−
m
≥
x
,
n-m\geq x,
n
−
m
≥
x
,
then Pratyya's number will be larger than Payel's number everyday.
3
1
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a,b,c in one of two disjoint sets such that ab=c
Prove that if the numbers
3
,
4
,
5
,
…
,
3
5
3,4,5, \dots ,3^5
3
,
4
,
5
,
…
,
3
5
are partitioned into two disjoint sets, then in one of the sets the number
a
,
b
,
c
a,b,c
a
,
b
,
c
can be found such that
a
b
=
c
.
ab=c.
ab
=
c
.
(
a
,
b
,
c
a,b,c
a
,
b
,
c
may not be pairwise distinct)
2
1
Hide problems
Find value of <MAN
In
△
A
B
C
,
∠
B
A
C
\triangle ABC, \angle BAC
△
A
BC
,
∠
B
A
C
is a right angle.
B
P
BP
BP
and
C
Q
CQ
CQ
are bisectors of
∠
B
\angle B
∠
B
and
∠
C
\angle C
∠
C
respectively, which intersect
A
C
AC
A
C
and
A
B
AB
A
B
at
P
P
P
and
Q
Q
Q
respectively. Two perpendicular segments
P
M
PM
PM
and
Q
N
QN
QN
are drawn on
B
C
BC
BC
from
P
P
P
and
Q
Q
Q
respectively. Find the value of
∠
M
A
N
\angle MAN
∠
M
A
N
with proof.
1
1
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All solutions of floor function equation
Find all solutions for real
x
x
x
,
⌊
x
⌋
3
−
7
⌊
x
+
1
3
⌋
=
−
13.
\lfloor x\rfloor^3 -7 \lfloor x+\frac{1}{3} \rfloor=-13.
⌊
x
⌋
3
−
7
⌊
x
+
3
1
⌋
=
−
13.