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National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2006 Turkey Junior National Olympiad
2006 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
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Turkey Junior Olympiad 2006, Part II - P3
In the beginnig, all nine squares of
3
×
3
3\times 3
3
×
3
chessboard contain
0
0
0
. At each step, we choose two squares sharing a common edge, then we add
1
1
1
to them or
−
1
-1
−
1
to them. Show that it is not possible to make all squares
2
2
2
, after a finite number of steps.
2
1
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Turkey Junior Olympiad 2006, Part II - P2
Find all integer triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
such that
x
−
y
z
=
11
x
z
+
y
=
13.
\begin{array}{rcl} x-yz &=& 11 \\ xz+y &=& 13. \end{array}
x
−
yz
x
z
+
y
=
=
11
13.
1
1
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Turkey Junior Olympiad 2006, Part II - P1
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid such that
A
D
∥
B
C
AD\parallel BC
A
D
∥
BC
. The interior angle bisectors of the corners
A
A
A
and
B
B
B
meet on
[
D
C
]
[DC]
[
D
C
]
. If
∣
B
C
∣
=
9
|BC|=9
∣
BC
∣
=
9
and
∣
A
D
∣
=
4
|AD|=4
∣
A
D
∣
=
4
, find
∣
A
B
∣
|AB|
∣
A
B
∣
.