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Contests
National and Regional Contests
Albania Contests
Albania-Balkan MO TST
2017 BMO TST
2017 BMO TST
Part of
Albania-Balkan MO TST
Subcontests
(5)
4
1
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Albania BMO TST Problem 4
The incircle of
△
A
0
B
0
C
0
\triangle A_{0}B_{0}C_{0}
△
A
0
B
0
C
0
, meets legs
B
0
C
0
B_{0}C_{0}
B
0
C
0
,
C
0
A
0
C_{0}A_{0}
C
0
A
0
,
A
0
B
0
A_{0}B_{0}
A
0
B
0
, respectively on points
A
A
A
,
B
B
B
,
C
C
C
, and the incircle of
△
A
B
C
\triangle ABC
△
A
BC
, with center
I
I
I
, meets legs
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
, on points
A
1
A_{1}
A
1
,
B
1
B_{1}
B
1
,
C
1
C_{1}
C
1
, respectively. We write with
σ
(
A
B
C
)
\sigma (ABC)
σ
(
A
BC
)
, and
σ
(
A
1
B
1
C
1
)
\sigma (A_{1}B_{1}C_{1})
σ
(
A
1
B
1
C
1
)
the areas of
△
A
B
C
\triangle ABC
△
A
BC
, and
△
A
1
B
1
C
1
\triangle A_{1}B_{1}C_{1}
△
A
1
B
1
C
1
respectively. Prove that if
σ
(
A
B
C
)
=
2
σ
(
A
1
B
1
C
1
)
\sigma (ABC)=2 \sigma (A_{1}B_{1}C_{1})
σ
(
A
BC
)
=
2
σ
(
A
1
B
1
C
1
)
, then lines
A
A
0
AA_{0}
A
A
0
,
B
B
0
BB_{0}
B
B
0
,
C
C
0
CC_{0}
C
C
0
are concurrent.
5
1
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Albania BMO TST Problem 5
Given a set
A
A
A
which contains
n
n
n
elements. For any two distinct subsets
A
1
A_{1}
A
1
,
A
2
A_{2}
A
2
of the given set
A
A
A
, we fix the number of elements of
A
1
∩
A
2
A_1 \cap A_2
A
1
∩
A
2
. Find the sum of all the numbers obtained in the described way.
3
1
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Albania BMO TST Problem 3
Find all functions
f
:
R
+
→
R
+
f : \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}
f
:
R
+
→
R
+
such that :
f
(
x
)
f
(
y
)
f
(
z
)
=
9
f
(
z
+
x
y
f
(
z
)
)
f(x)f(y)f(z)=9f(z+xyf(z))
f
(
x
)
f
(
y
)
f
(
z
)
=
9
f
(
z
+
x
y
f
(
z
))
, where
x
x
x
,
y
y
y
,
z
z
z
, are three positive real numbers.
2
1
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Albania BMO TST Problem 2
Given a random positive integer
N
N
N
. Prove that there exist infinitely many positive integers
M
M
M
whose none of its digits is
0
0
0
and such that the sum of the digits of
N
⋅
M
N \cdot M
N
⋅
M
is same as sum of digits
M
M
M
.
1
1
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Albania BMO TST Problem 1
Given
n
n
n
numbers different from
0
0
0
, (
n
∈
N
n \in \mathbb{N}
n
∈
N
) which are arranged randomly. We do the following operation: Choose some consecutive numbers in the given order and change their sign (i.e.
x
→
−
x
x \rightarrow -x
x
→
−
x
). What is the minimum number of operations needed, in order to make all the numbers positive for any given initial configuration of the
n
n
n
numbers?