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Contests
International Contests
Junior Balkan MO
2006 Junior Balkan MO
2006 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
4
1
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Board
Consider a
2
n
×
2
n
2n \times 2n
2
n
×
2
n
board. From the
i
i
i
th line we remove the central
2
(
i
−
1
)
2(i-1)
2
(
i
−
1
)
unit squares. What is the maximal number of rectangles
2
×
1
2 \times 1
2
×
1
and
1
×
2
1 \times 2
1
×
2
that can be placed on the obtained figure without overlapping or getting outside the board?
3
1
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Sum of divisors is 2n
We call a number perfect if the sum of its positive integer divisors(including
1
1
1
and
n
n
n
) equals
2
n
2n
2
n
. Determine all perfect numbers
n
n
n
for which
n
−
1
n-1
n
−
1
and
n
+
1
n+1
n
+
1
are prime numbers.
2
1
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Compute angle
The triangle
A
B
C
ABC
A
BC
is isosceles with
A
B
=
A
C
AB=AC
A
B
=
A
C
, and
∠
B
A
C
<
6
0
∘
\angle{BAC}<60^{\circ}
∠
B
A
C
<
6
0
∘
. The points
D
D
D
and
E
E
E
are chosen on the side
A
C
AC
A
C
such that,
E
B
=
E
D
EB=ED
EB
=
E
D
, and
∠
A
B
D
≡
∠
C
B
E
\angle{ABD}\equiv\angle{CBE}
∠
A
B
D
≡
∠
CBE
. Denote by
O
O
O
the intersection point between the internal bisectors of the angles
∠
B
D
C
\angle{BDC}
∠
B
D
C
and
∠
A
C
B
\angle{ACB}
∠
A
CB
. Compute
∠
C
O
D
\angle{COD}
∠
CO
D
.
1
1
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Divisibility
If
n
>
4
n>4
n
>
4
is a composite number, then
2
n
2n
2
n
divides
(
n
−
1
)
!
(n-1)!
(
n
−
1
)!
.