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International Contests
Junior Balkan MO
2007 Junior Balkan MO
2007 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
4
1
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prime number implies non-square number
Prove that if
p
p
p
is a prime number, then
7
p
+
3
p
−
4
7p+3^{p}-4
7
p
+
3
p
−
4
is not a perfect square.
3
1
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50 points in plane
Given are
50
50
50
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least
130
130
130
scalene triangles with vertices of that color.
2
1
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balkan "slicing" problem
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
D
A
C
=
∠
B
D
C
=
3
6
∘
\angle{DAC}= \angle{BDC}= 36^\circ
∠
D
A
C
=
∠
B
D
C
=
3
6
∘
,
∠
C
B
D
=
1
8
∘
\angle{CBD}= 18^\circ
∠
CB
D
=
1
8
∘
and
∠
B
A
C
=
7
2
∘
\angle{BAC}= 72^\circ
∠
B
A
C
=
7
2
∘
. The diagonals and intersect at point
P
P
P
. Determine the measure of
∠
A
P
D
\angle{APD}
∠
A
P
D
.
1
1
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equation with no real solution
Let
a
a
a
be positive real number such that
a
3
=
6
(
a
+
1
)
a^{3}=6(a+1)
a
3
=
6
(
a
+
1
)
. Prove that the equation
x
2
+
a
x
+
a
2
−
6
=
0
x^{2}+ax+a^{2}-6=0
x
2
+
a
x
+
a
2
−
6
=
0
has no real solution.