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International Contests
Junior Balkan MO
2001 Junior Balkan MO
2001 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
1
1
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solve a^3+b^3+c^3=2001 in positive integers
Solve the equation
a
3
+
b
3
+
c
3
=
2001
a^3+b^3+c^3=2001
a
3
+
b
3
+
c
3
=
2001
in positive integers. Mircea Becheanu, Romania
3
1
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sum of areas of the triangles DEF and DEG is at most twice
Let
A
B
C
ABC
A
BC
be an equilateral triangle and
D
D
D
,
E
E
E
points on the sides
[
A
B
]
[AB]
[
A
B
]
and
[
A
C
]
[AC]
[
A
C
]
respectively. If
D
F
DF
D
F
,
E
F
EF
EF
(with
F
∈
A
E
F\in AE
F
∈
A
E
,
G
∈
A
D
G\in AD
G
∈
A
D
) are the interior angle bisectors of the angles of the triangle
A
D
E
ADE
A
D
E
, prove that the sum of the areas of the triangles
D
E
F
DEF
D
EF
and
D
E
G
DEG
D
EG
is at most equal with the area of the triangle
A
B
C
ABC
A
BC
. When does the equality hold? Greece
2
1
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Show that for X <> C on line CL, <XAC = <XBC
Let
A
B
C
ABC
A
BC
be a triangle with
∠
C
=
9
0
∘
\angle C = 90^\circ
∠
C
=
9
0
∘
and
C
A
≠
C
B
CA \neq CB
C
A
=
CB
. Let
C
H
CH
C
H
be an altitude and
C
L
CL
C
L
be an interior angle bisector. Show that for
X
≠
C
X \neq C
X
=
C
on the line
C
L
CL
C
L
, we have
∠
X
A
C
≠
∠
X
B
C
\angle XAC \neq \angle XBC
∠
X
A
C
=
∠
XBC
. Also show that for
Y
≠
C
Y \neq C
Y
=
C
on the line
C
H
CH
C
H
we have
∠
Y
A
C
≠
∠
Y
B
C
\angle YAC \neq \angle YBC
∠
Y
A
C
=
∠
Y
BC
. Bulgaria
4
1
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1415 vertices
Let
N
N
N
be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of
N
N
N
which form a triangle of area smaller than 1.