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International Contests
Junior Balkan MO
2000 Junior Balkan MO
2000 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
2
1
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n^2+3^n is a perfect square
Find all positive integers
n
≥
1
n\geq 1
n
≥
1
such that
n
2
+
3
n
n^2+3^n
n
2
+
3
n
is the square of an integer. Bulgaria
1
1
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Prove that x+y=10
Let
x
x
x
and
y
y
y
be positive reals such that
x
3
+
y
3
+
(
x
+
y
)
3
+
30
x
y
=
2000.
x^3 + y^3 + (x + y)^3 + 30xy = 2000.
x
3
+
y
3
+
(
x
+
y
)
3
+
30
x
y
=
2000.
Show that
x
+
y
=
10
x + y = 10
x
+
y
=
10
.
3
1
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intersection lies on the altitude
A half-circle of diameter
E
F
EF
EF
is placed on the side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
and it is tangent to the sides
A
B
AB
A
B
and
A
C
AC
A
C
in the points
Q
Q
Q
and
P
P
P
respectively. Prove that the intersection point
K
K
K
between the lines
E
P
EP
EP
and
F
Q
FQ
FQ
lies on the altitude from
A
A
A
of the triangle
A
B
C
ABC
A
BC
. Albania
4
1
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Very hard(at least for me)
At a tennis tournament there were
2
n
2n
2
n
boys and
n
n
n
girls participating. Every player played every other player. The boys won
7
5
\frac 75
5
7
times as many matches as the girls. It is knowns that there were no draws. Find
n
n
n
. Serbia