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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2010 Greece Junior Math Olympiad
2010 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
3
1
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Inequality
If
a
,
b
a, b
a
,
b
are positive real numbers with sum
3
3
3
and the positive real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
have product
1
1
1
, prove that:
(
a
x
+
b
)
(
a
y
+
b
)
(
a
z
+
b
)
≥
27
(ax + b)(ay + b)(az + b) \ge 27
(
a
x
+
b
)
(
a
y
+
b
)
(
a
z
+
b
)
≥
27
. When equality holds?
4
1
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Combinatorics
Three parallel lines
ℓ
1
,
ℓ
2
\ell_1, \ell_2
ℓ
1
,
ℓ
2
and
ℓ
3
\ell_3
ℓ
3
of a plane are given such that the line
ℓ
2
\ell_2
ℓ
2
has the same distance
a
a
a
from
ℓ
1
\ell_1
ℓ
1
and
ℓ
3
\ell_3
ℓ
3
. We put
5
5
5
points
M
1
,
M
2
,
M
3
,
M
4
M_1, M_2, M_3,M_4
M
1
,
M
2
,
M
3
,
M
4
and
M
5
M_5
M
5
on the lines
ℓ
1
,
ℓ
2
\ell_1, \ell_2
ℓ
1
,
ℓ
2
and
ℓ
3
\ell_3
ℓ
3
in such a way that each line contains at least one point. Detennine the maximal number of isosceles triangles that are possible to be formed with vertices three of the points
M
1
,
M
2
,
M
3
,
M
4
M_1, M_2, M_3, M_4
M
1
,
M
2
,
M
3
,
M
4
and
M
5
M_5
M
5
in the following cases: (i)
M
1
,
M
2
,
M
3
∈
ℓ
2
,
M
4
∈
ℓ
1
M_1,M_2,M_3 \in \ell_2, M_4 \in \ell_1
M
1
,
M
2
,
M
3
∈
ℓ
2
,
M
4
∈
ℓ
1
and
M
5
∈
ℓ
3
M_5 \in \ell_3
M
5
∈
ℓ
3
. (ii)
M
1
,
M
2
∈
ℓ
1
,
M
3
,
M
4
∈
ℓ
3
M_1,M_2 \in \ell_1, M_3,M_4 \in \ell_3
M
1
,
M
2
∈
ℓ
1
,
M
3
,
M
4
∈
ℓ
3
and
M
5
∈
ℓ
2
M_5 \in \ell_2
M
5
∈
ℓ
2
.
1
1
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Number Theory
Determine the number of all positive integers which cannot be written in the form
80
k
+
3
m
80k + 3m
80
k
+
3
m
, where
k
,
m
∈
N
=
{
0
,
1
,
2
,
.
.
.
,
}
k,m \in N = \{0,1,2,...,\}
k
,
m
∈
N
=
{
0
,
1
,
2
,
...
,
}
2
1
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rectangle, equilateral, equal segments in extensions (Greece Junior 2010)
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with sides
A
B
=
a
AB=a
A
B
=
a
and
B
C
=
b
BC=b
BC
=
b
. Let
O
O
O
be the intersection point of it's diagonals. Extent side
B
A
BA
B
A
towards
A
A
A
at a segment
A
E
=
A
O
AE=AO
A
E
=
A
O
, and diagonal
D
B
DB
D
B
towards
B
B
B
at a segment
B
Z
=
B
O
BZ=BO
BZ
=
BO
. If the triangle
E
Z
C
EZC
EZC
is an equilateral, then prove that: i)
b
=
a
3
b=a\sqrt3
b
=
a
3
ii)
A
Z
=
E
O
AZ=EO
A
Z
=
EO
iii)
E
O
⊥
Z
D
EO \perp ZD
EO
⊥
Z
D