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International Contests
Balkan MO
2014 Balkan MO
2014 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
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A regular hexagon divided into equilateral triangles
Let
n
n
n
be a positive integer. A regular hexagon with side length
n
n
n
is divided into equilateral triangles with side length
1
1
1
by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles.UK - Sahl Khan
2
1
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2014=a³+2b³/c³+2d³
A special number is a positive integer
n
n
n
for which there exists positive integers
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
with
n
=
a
3
+
2
b
3
c
3
+
2
d
3
.
n = \frac {a^3 + 2b^3} {c^3 + 2d^3}.
n
=
c
3
+
2
d
3
a
3
+
2
b
3
.
Prove thati) there are infinitely many special numbers; ii)
2014
2014
2014
is not a special number.Romania
3
1
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Trapezium inscribed in a circle
Let
A
B
C
D
ABCD
A
BC
D
be a trapezium inscribed in a circle
Γ
\Gamma
Γ
with diameter
A
B
AB
A
B
. Let
E
E
E
be the intersection point of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. The circle with center
B
B
B
and radius
B
E
BE
BE
meets
Γ
\Gamma
Γ
at the points
K
K
K
and
L
L
L
(where
K
K
K
is on the same side of
A
B
AB
A
B
as
C
C
C
). The line perpendicular to
B
D
BD
B
D
at
E
E
E
intersects
C
D
CD
C
D
at
M
M
M
. Prove that
K
M
KM
K
M
is perpendicular to
D
L
DL
D
L
.Greece - Silouanos Brazitikos
1
1
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xy+yz+xz=3xyz
Let
x
,
y
x,y
x
,
y
and
z
z
z
be positive real numbers such that
x
y
+
y
z
+
x
z
=
3
x
y
z
xy+yz+xz=3xyz
x
y
+
yz
+
x
z
=
3
x
yz
. Prove that
x
2
y
+
y
2
z
+
z
2
x
≥
2
(
x
+
y
+
z
)
−
3
x^2y+y^2z+z^2x \ge 2(x+y+z)-3
x
2
y
+
y
2
z
+
z
2
x
≥
2
(
x
+
y
+
z
)
−
3
and determine when equality holds.UK - David Monk