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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2012 Greece Junior Math Olympiad
2012 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
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Combinatorics
On a plane
Π
\Pi
Π
is given a straight line
ℓ
\ell
ℓ
and on the line
ℓ
\ell
ℓ
are given two different points
A
1
,
A
2
A_1, A_2
A
1
,
A
2
. We consider on the plane
Π
\Pi
Π
, outside the line
ℓ
\ell
ℓ
, two different points
A
3
,
A
4
A_3, A_4
A
3
,
A
4
. Examine if it is possible to put points
A
3
A_3
A
3
and
A
4
A_4
A
4
on such positions such the four points
A
1
,
A
2
,
A
3
,
A
4
A_1, A_2, A_3, A_4
A
1
,
A
2
,
A
3
,
A
4
form the maximal number of possible isosceles triangles, in the following cases: (a) when the points
A
3
,
A
4
A_3, A_4
A
3
,
A
4
belong to dierent semi-planes with respect to
ℓ
\ell
ℓ
; (b) when the points
A
3
,
A
4
A_3, A_4
A
3
,
A
4
belong to the same semi-planes with respect to
ℓ
\ell
ℓ
. Give all possible cases and explain how is possible to construct in each case the points
A
3
A_3
A
3
and
A
4
A_4
A
4
.
3
1
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Number Theory
Given is the equation
(
m
,
n
)
+
[
m
,
n
]
=
m
+
n
(m, n) +[m, n] =m+n
(
m
,
n
)
+
[
m
,
n
]
=
m
+
n
where
m
,
n
m, n
m
,
n
are positive integers and m>n. a) Prove that n divides m. b) If
m
−
n
=
10
m-n=10
m
−
n
=
10
, solve the equation.
2
1
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Easy Algebra equation
For the various values of the parameter
a
∈
R
a \in R
a
∈
R
, solve the equation
∣
∣
x
−
4
∣
−
2
x
+
8
∣
=
a
x
+
4
||x - 4| - 2x + 8| = ax + 4
∣∣
x
−
4∣
−
2
x
+
8∣
=
a
x
+
4
1
1
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prove that AC bisects <DAE, circles related (Greece Junior 2012)
Let
A
B
C
ABC
A
BC
be an acute angled triangle (with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
) inscribed in circle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
(with center
O
O
O
and radius
R
R
R
). Circle
c
1
(
A
,
A
B
)
c_1(A,AB)
c
1
(
A
,
A
B
)
(with center
A
A
A
and radius
A
B
AB
A
B
) intersects side
B
C
BC
BC
at point
D
D
D
and the circumcircle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
at point
E
E
E
. Prove that side
A
C
AC
A
C
bisects angle
∠
D
A
E
\angle DAE
∠
D
A
E
.