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Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2015 Greece Junior Math Olympiad
2015 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
3
1
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Combinatorics
It is possible to place the
2014
2014
2014
points in the plane so that we can construct
100
6
2
1006^2
100
6
2
parralelograms with vertices among these points, so that the parralelograms have area 1?
2
1
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Number Theory
Determine all pairs of non-negative integers
(
m
,
n
)
(m, n)
(
m
,
n
)
with m ≥n, such that
(
m
+
n
)
3
(m+n)^3
(
m
+
n
)
3
divides
2
n
(
3
m
2
+
n
2
)
+
8
2n(3m^2+n^2)+8
2
n
(
3
m
2
+
n
2
)
+
8
1
1
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Easy Quadric equation
Find all values of the real parameter
a
a
a
, so that the equation
x
2
+
(
a
−
2
)
x
−
(
a
−
1
)
(
2
a
−
3
)
=
0
x^2+(a-2)x-(a-1)(2a-3)=0
x
2
+
(
a
−
2
)
x
−
(
a
−
1
)
(
2
a
−
3
)
=
0
has two real roots, so that the one is the square of the other.
4
1
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AB=AC iff CD=BC/2, perpendicular on tangent (Greece Junior 2015)
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
≤
A
C
AB\le AC
A
B
≤
A
C
and let
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
be it's circumscribed circle (with center
O
O
O
and radius
R
R
R
). The perpendicular from vertex
A
A
A
on the tangent of the circle passing through point
C
C
C
, intersect it at point
D
D
D
. a) If the triangle
A
B
C
ABC
A
BC
is isosceles with
A
B
=
A
C
AB=AC
A
B
=
A
C
, prove that
C
D
=
B
C
/
2
CD=BC/2
C
D
=
BC
/2
. b) If
C
D
=
B
C
/
2
CD=BC/2
C
D
=
BC
/2
, prove that the triangle
A
B
C
ABC
A
BC
is isosceles.