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International Contests
Balkan MO
2000 Balkan MO
2000 Balkan MO
Part of
Balkan MO
Subcontests
(4)
3
1
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Rectangles that can be cut from a 50x90 rectangle
How many
1
×
10
2
1 \times 10\sqrt 2
1
×
10
2
rectangles can be cut from a
50
×
90
50\times 90
50
×
90
rectangle using cuts parallel to its edges?
4
1
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Average of elements is a perfect power
Show that for any
n
n
n
we can find a set
X
X
X
of
n
n
n
distinct integers greater than 1, such that the average of the elements of any subset of
X
X
X
is a square, cube or higher power.
2
1
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Balkan geo 2000
Let
A
B
C
ABC
A
BC
be an acute-angled triangle and
D
D
D
the midpoint of
B
C
BC
BC
. Let
E
E
E
be a point on segment
A
D
AD
A
D
and
M
M
M
its projection on
B
C
BC
BC
. If
N
N
N
and
P
P
P
are the projections of
M
M
M
on
A
B
AB
A
B
and
A
C
AC
A
C
then the interior angule bisectors of
∠
N
M
P
\angle NMP
∠
NMP
and
∠
N
E
P
\angle NEP
∠
NEP
are parallel.
1
1
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Find all functions
Find all functions
f
:
R
→
R
f: \mathbb R \to \mathbb R
f
:
R
→
R
such that
f
(
x
f
(
x
)
+
f
(
y
)
)
=
f
2
(
x
)
+
y
f( xf(x) + f(y) ) = f^2(x) + y
f
(
x
f
(
x
)
+
f
(
y
))
=
f
2
(
x
)
+
y
for all
x
,
y
∈
R
x,y\in \mathbb R
x
,
y
∈
R
.