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National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2003 Greece Junior Math Olympiad
2003 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
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Number Theory
Find all positive integers which can be written in the form
(
m
n
+
1
)
/
(
m
+
n
)
(mn+1)/(m+n)
(
mn
+
1
)
/
(
m
+
n
)
, where
m
,
n
m,n
m
,
n
are positive integers.
2
1
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Combinatorics
Find all four-digit natural numbers
x
y
z
w
‾
\overline{xyzw}
x
yz
w
with the property that their sum plus the sum of their digits equals
2003
2003
2003
.
1
1
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Easy Number Theory
Find all positive integers
n
n
n
for which number
A
=
n
3
−
n
2
+
n
−
1
A = n^3-n^2+n-1
A
=
n
3
−
n
2
+
n
−
1
is prime
3
1
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rhombus wanted, starting with an isosceles (Greece Junior 2003)
Let
A
B
C
ABC
A
BC
be an isosceles triangle (
A
B
=
A
C
AB=AC
A
B
=
A
C
). The altitude
A
H
AH
A
H
and the perpendiculare bisector
(
e
)
(e)
(
e
)
of side
A
B
AB
A
B
intersect at point
M
M
M
. The perpendicular on line
(
e
)
(e)
(
e
)
passing through
M
M
M
intersects
B
C
BC
BC
at point
D
D
D
. If the circumscribed circle of the triangle
B
M
D
BMD
BM
D
intersects line
(
e
)
(e)
(
e
)
at point
S
S
S
, the prove that: a)
B
S
/
/
A
M
BS // AM
BS
//
A
M
. b) quadrilateral
A
M
B
S
AMBS
A
MBS
is rhombus.