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National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2009 Turkey Junior National Olympiad
2009 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
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Turkey Junior Olympiad 2009, Part II - P3
The integer
n
n
n
has exactly six positive divisors, and they are:
1
<
a
<
b
<
c
<
d
<
n
1<a<b<c<d<n
1
<
a
<
b
<
c
<
d
<
n
. Let
k
=
a
−
1
k=a-1
k
=
a
−
1
. If the
k
k
k
-th divisor (according to above ordering) of
n
n
n
is equal to
(
1
+
a
+
b
)
b
(1+a+b)b
(
1
+
a
+
b
)
b
, find the highest possible value of
n
n
n
.
2
1
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Turkey Junior Olympiad 2009, Part II - P2
In the beginnig, each square of a strip formed by
n
n
n
adjacent squares contains
0
0
0
or
1
1
1
. At each step, we are writing
1
1
1
to the squares containing
0
0
0
and to the squares having exactly one neighbour containing
1
1
1
, and we are writing
0
0
0
s into the other squares. Determine all possible values of
n
n
n
such that whatever the initial arrangement of
0
0
0
and
1
1
1
is, after finite number of steps, all squares can turn into
0
0
0
.
1
1
Hide problems
Turkey Junior Olympiad 2009, Part II - P1
Let the tangent line passing through a point
A
A
A
outside the circle with center
O
O
O
touches the circle at
B
B
B
and
C
C
C
. Let
[
B
D
]
[BD]
[
B
D
]
be the diameter of the circle. Let the lines
C
D
CD
C
D
and
A
B
AB
A
B
meet at
E
E
E
. If the lines
A
D
AD
A
D
and
O
E
OE
OE
meet at
F
F
F
, find
∣
A
F
∣
/
∣
F
D
∣
|AF|/|FD|
∣
A
F
∣/∣
F
D
∣
.