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Balkan MO
2009 Balkan MO
2009 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
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Functional equation
Denote by
S
S
S
the set of all positive integers. Find all functions
f
:
S
→
S
f: S \rightarrow S
f
:
S
→
S
such that f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2 for all
m
,
n
∈
S
m,n \in S
m
,
n
∈
S
.Bulgaria
3
1
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Broken line with a centre of symmetry in a rectangle
A
9
×
12
9 \times 12
9
×
12
rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres
C
1
,
C
2
,
…
,
C
96
C_1,C_2,\ldots ,C_{96}
C
1
,
C
2
,
…
,
C
96
in such way that the following to conditions are both fulfilled i) the distances
C
1
C
2
,
…
,
C
95
C
96
,
C
96
C
1
C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}
C
1
C
2
,
…
,
C
95
C
96
,
C
96
C
1
are all equal to
13
\sqrt {13}
13
, ii) the closed broken line
C
1
C
2
…
C
96
C
1
C_1C_2\ldots C_{96}C_1
C
1
C
2
…
C
96
C
1
has a centre of symmetry?Bulgaria
2
1
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Nice Symmedian property
Let
M
N
MN
MN
be a line parallel to the side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
, with
M
M
M
on the side
A
B
AB
A
B
and
N
N
N
on the side
A
C
AC
A
C
. The lines
B
N
BN
BN
and
C
M
CM
CM
meet at point
P
P
P
. The circumcircles of triangles
B
M
P
BMP
BMP
and
C
N
P
CNP
CNP
meet at two distinct points
P
P
P
and
Q
Q
Q
. Prove that
∠
B
A
Q
=
∠
C
A
P
\angle BAQ = \angle CAP
∠
B
A
Q
=
∠
C
A
P
.Liubomir Chiriac, Moldova
1
1
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Equation in naturals
Solve the equation 3^x \minus{} 5^y \equal{} z^2. in positive integers.Greece