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International Contests
Junior Balkan MO
2010 Junior Balkan MO
2010 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
3
1
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JBMO 2010. Problem 3.
Let
A
L
AL
A
L
and
B
K
BK
B
K
be angle bisectors in the non-isosceles triangle
A
B
C
ABC
A
BC
(
L
L
L
lies on the side
B
C
BC
BC
,
K
K
K
lies on the side
A
C
AC
A
C
). The perpendicular bisector of
B
K
BK
B
K
intersects the line
A
L
AL
A
L
at point
M
M
M
. Point
N
N
N
lies on the line
B
K
BK
B
K
such that
L
N
LN
L
N
is parallel to
M
K
MK
M
K
. Prove that
L
N
=
N
A
LN = NA
L
N
=
N
A
.
2
1
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JBMO 2010. Problem 2.
Find all integers
n
n
n
,
n
≥
1
n \ge 1
n
≥
1
, such that
n
⋅
2
n
+
1
+
1
n \cdot 2^{n+1}+1
n
⋅
2
n
+
1
+
1
is a perfect square.
1
1
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JBMO 2010. Problem 1.
The real numbers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
satisfy simultaneously the equations
a
b
c
−
d
=
1
,
b
c
d
−
a
=
2
,
c
d
a
−
b
=
3
,
d
a
b
−
c
=
−
6.
abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.
ab
c
−
d
=
1
,
b
c
d
−
a
=
2
,
c
d
a
−
b
=
3
,
d
ab
−
c
=
−
6.
Prove that
a
+
b
+
c
+
d
≠
0
a + b + c + d \not = 0
a
+
b
+
c
+
d
=
0
.
4
1
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JBMO 2010. Problem 4.
A
9
×
7
9\times 7
9
×
7
rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with
9
0
∘
90^\circ
9
0
∘
) and square tiles composed by four unit squares. Let
n
≥
0
n\ge 0
n
≥
0
be the number of the
2
×
2
2 \times 2
2
×
2
tiles which can be used in such a tiling. Find all the values of
n
n
n
.