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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2014 Greece Junior Math Olympiad
2014 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
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Combinatorics
We color the numbers
1
,
2
,
3
,
.
.
.
.
,
20
1, 2, 3,....,20
1
,
2
,
3
,
....
,
20
with two colors white and black in such a way that both colors are used. Find the number of ways, we can perform this coloring if the product of white numbers and the product of black numbers have greatest common divisor equal to
1
1
1
.
3
1
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Algebra system
Solve in real number the system
x
3
=
z
y
−
2
y
z
,
y
3
=
x
z
−
2
z
x
,
z
3
=
y
x
−
2
x
y
x^3=\frac{z}{y}-\frac{2y}{z}, y^3=\frac{x}{z}-\frac{2z}{x}, z^3=\frac{y}{x}-\frac{2x}{y}
x
3
=
y
z
−
z
2
y
,
y
3
=
z
x
−
x
2
z
,
z
3
=
x
y
−
y
2
x
2
1
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Number Theory
Let
p
p
p
prime and
m
m
m
a positive integer. Determine all pairs
(
p
,
m
)
( p,m)
(
p
,
m
)
satisfying the equation:
p
(
p
+
m
)
+
p
=
(
m
+
1
)
3
p(p+m)+p=(m+1)^3
p
(
p
+
m
)
+
p
=
(
m
+
1
)
3
1
1
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triangle ABC, # BCDE, BE//AM, BE=AM/2, midpoint (Greece Junior 2014)
Let
A
B
C
ABC
A
BC
be a triangle and let
M
M
M
be the midpoint
B
C
BC
BC
. On the exterior of the triangle, consider the parallelogram
B
C
D
E
BCDE
BC
D
E
such that
B
E
/
/
A
M
BE//AM
BE
//
A
M
and
B
E
=
A
M
/
2
BE=AM/2
BE
=
A
M
/2
. Prove that line
E
M
EM
EM
passes through the midpoint of segment
A
D
AD
A
D
.