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International Contests
Junior Balkan MO
2017 Junior Balkan MO
2017 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
4
1
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Find the number of distinct
Consider a regular 2n-gon
P
P
P
,
A
1
,
A
2
,
⋯
,
A
2
n
A_1,A_2,\cdots ,A_{2n}
A
1
,
A
2
,
⋯
,
A
2
n
in the plane ,where
n
n
n
is a positive integer . We say that a point
S
S
S
on one of the sides of
P
P
P
can be seen from a point
E
E
E
that is external to
P
P
P
, if the line segment
S
E
SE
SE
contains no other points that lie on the sides of
P
P
P
except
S
S
S
.We color the sides of
P
P
P
in 3 different colors (ignore the vertices of
P
P
P
,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover ,from every point in the plane external to
P
P
P
, points of most 2 different colors on
P
P
P
can be seen .Find the number of distinct such colorings of
P
P
P
(two colorings are considered distinct if at least one of sides is colored differently).Proposed by Viktor Simjanoski, MacedoniaJBMO 2017, Q4
1
1
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Determine all the sets of six consecutive positive integers
Determine all the sets of six consecutive positive integers such that the product of some two of them . added to the product of some other two of them is equal to the product of the remaining two numbers.
3
1
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A,M,X are collinear
Let
A
B
C
ABC
A
BC
be an acute triangle such that
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
,with circumcircle
Γ
\Gamma
Γ
and circumcenter
O
O
O
. Let
M
M
M
be the midpoint of
B
C
BC
BC
and
D
D
D
be a point on
Γ
\Gamma
Γ
such that
A
D
⊥
B
C
AD \perp BC
A
D
⊥
BC
. let
T
T
T
be a point such that
B
D
C
T
BDCT
B
D
CT
is a parallelogram and
Q
Q
Q
a point on the same side of
B
C
BC
BC
as
A
A
A
such that
∠
B
Q
M
=
∠
B
C
A
\angle{BQM}=\angle{BCA}
∠
BQM
=
∠
BC
A
and
∠
C
Q
M
=
∠
C
B
A
\angle{CQM}=\angle{CBA}
∠
CQM
=
∠
CB
A
. Let the line
A
O
AO
A
O
intersect
Γ
\Gamma
Γ
at
E
E
E
(
E
≠
A
)
(E\neq A)
(
E
=
A
)
and let the circumcircle of
△
E
T
Q
\triangle ETQ
△
ETQ
intersect
Γ
\Gamma
Γ
at point
X
≠
E
X\neq E
X
=
E
. Prove that the point
A
,
M
A,M
A
,
M
and
X
X
X
are collinear.
2
1
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Simple inequality
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive integers such that
x
≠
y
≠
z
≠
x
x\neq y\neq z \neq x
x
=
y
=
z
=
x
.Prove that
(
x
+
y
+
z
)
(
x
y
+
y
z
+
z
x
−
2
)
≥
9
x
y
z
.
(x+y+z)(xy+yz+zx-2)\geq 9xyz.
(
x
+
y
+
z
)
(
x
y
+
yz
+
z
x
−
2
)
≥
9
x
yz
.
When does the equality hold?Proposed by Dorlir Ahmeti, Albania