MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan BMO TST
2016 Azerbaijan BMO TST
2016 Azerbaijan BMO TST
Part of
Azerbaijan BMO TST
Subcontests
(4)
3
3
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combinatorics
k
k
k
is a positive integer.
A
A
A
company has a special method to sell clocks. Every customer can reason with two customers after he has bought a clock himself
;
;
;
it's not allowed to reason with an agreed person. These new customers can reason with other two persons and it goes like this.. If both of the customers agreed by a person could play a role (it can be directly or not) in buying clocks by at least
k
k
k
customers, this person gets a present. Prove that, if
n
n
n
persons have bought clocks, then at most
n
k
+
2
\frac{n}{k+2}
k
+
2
n
presents have been accepted.
Balkan TSTp4.3
a
,
b
a,b
a
,
b
are positive integers and
(
a
!
+
b
!
)
∣
a
!
b
!
(a!+b!)|a!b!
(
a
!
+
b
!)
∣
a
!
b
!
.Prove that
3
a
≥
2
b
+
2
3a\ge 2b+2
3
a
≥
2
b
+
2
.
Balkan TSTp3.3
There are some checkers in
n
⋅
n
n\cdot n
n
⋅
n
size chess board.Known that for all numbers
1
≤
i
,
j
≤
n
1\le i,j\le n
1
≤
i
,
j
≤
n
if checkwork in the intersection of
i
i
i
th row and
j
j
j
th column is empty,so the number of checkers that are in this row and column is at least
n
n
n
.Prove that there are at least
n
2
2
\frac{n^2}{2}
2
n
2
checkers in chess board.
2
3
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Balkan TSTp3.2
İn triangle
A
B
C
ABC
A
BC
the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
intersects the side
B
C
BC
BC
at the point
D
D
D
.The circle
ω
\omega
ω
passes through
A
A
A
and tangent to the side
B
C
BC
BC
at
D
D
D
.
A
C
AC
A
C
and
ω
\omega
ω
intersects at
M
M
M
second time ,
B
M
BM
BM
and
ω
\omega
ω
intersects at
P
P
P
second time. Prove that point
P
P
P
lies on median of triangle
A
B
D
ABD
A
B
D
.
No perfect square
Set
A
A
A
consists of natural numbers such that these numbers can be expressed as
2
x
2
+
3
y
2
,
2x^2+3y^2,
2
x
2
+
3
y
2
,
where
x
x
x
and
y
y
y
are integers.
(
x
2
+
y
2
≠
0
)
(x^2+y^2\not=0)
(
x
2
+
y
2
=
0
)
a
)
a)
a
)
Prove that there is no perfect square in the set
A
.
A.
A
.
b
)
b)
b
)
Prove that multiple of odd number of elements of the set
A
A
A
cannot be a perfect square.
Balkan TSTp4.2
There are
100
100
100
students who praticipate at exam.Also there are
25
25
25
members of jury.Each student is checked by one jury.Known that every student likes
10
10
10
jury
a
)
a)
a
)
Prove that we can select
7
7
7
jury such that any student likes at least one jury.
b
)
b)
b
)
Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most
10
10
10
students.
1
3
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Geometry
A line is called
g
o
o
d
good
g
oo
d
if it bisects perimeter and area of a figure at the same time.Prove that:a) all of the good lines in a triangle concur. b) all of the good lines in a regular polygon concur too.
Balkan TSTp3.1
Find all
n
n
n
natural numbers such that for each of them there exist
p
,
q
p , q
p
,
q
primes such that these terms satisfy.
1.
1.
1.
p
+
2
=
q
p+2=q
p
+
2
=
q
2.
2.
2.
2
n
+
p
2^n+p
2
n
+
p
and
2
n
+
q
2^n+q
2
n
+
q
are primes.
Balkan TSTp4.1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be nonnegative real numbers.Prove that
3
(
a
2
+
b
2
+
c
2
)
≥
(
a
+
b
+
c
)
(
a
b
+
b
c
+
c
a
)
+
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
≥
(
a
+
b
+
c
)
2
3(a^2+b^2+c^2)\ge (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2\ge (a+b+c)^2
3
(
a
2
+
b
2
+
c
2
)
≥
(
a
+
b
+
c
)
(
ab
+
b
c
+
c
a
)
+
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
≥
(
a
+
b
+
c
)
2
.
4
3
Hide problems
Functional Equation
Find all functions
f
:
N
→
N
f:\mathbb{N}\to\mathbb{N}
f
:
N
→
N
such that
f
(
f
(
n
)
)
=
n
+
2015
f(f(n))=n+2015
f
(
f
(
n
))
=
n
+
2015
where
n
∈
N
.
n\in \mathbb{N}.
n
∈
N
.
Balkan TSTp3.4
For all numbers
n
≥
1
n\ge 1
n
≥
1
does there exist infinite positive numbers sequence
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
such that
x
n
+
2
=
x
n
+
1
−
x
n
x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_n}
x
n
+
2
=
x
n
+
1
−
x
n
Balkan TSTp4.4
Let
A
B
C
ABC
A
BC
be an acute triangle and let
M
M
M
be the midpoint of
A
C
AC
A
C
. A circle
ω
\omega
ω
passing through
B
B
B
and
M
M
M
meets the sides
A
B
AB
A
B
and
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
respectively. Let
T
T
T
be the point such that
B
P
T
Q
BPTQ
BPTQ
is a parallelogram. Suppose that
T
T
T
lies on the circumcircle of
A
B
C
ABC
A
BC
. Determine all possible values of
B
T
B
M
\frac{BT}{BM}
BM
BT
.