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National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2008 Turkey Junior National Olympiad
2008 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
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Turkey Junior Olympiad 2008, Part II - P3
There are
24
24
24
cups on a table. In the beginning, only three of them placed upside-down. At each step, we are turning four cups. Can we turn all the cups right-side up in at most
100
100
100
steps?
2
1
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Turkey Junior Olympiad 2008, Part II - P2
Find all solutions of the equation
4
x
+
3
y
=
z
2
4^x+3^y=z^2
4
x
+
3
y
=
z
2
in positive integers.
1
1
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Turkey Junior Olympiad 2008, Part II - P1
Let
A
B
C
ABC
A
BC
be a right triangle with
m
(
C
^
)
=
9
0
∘
m(\widehat {C}) = 90^\circ
m
(
C
)
=
9
0
∘
, and
D
D
D
be its incenter. Let
N
N
N
be the intersection of the line
A
D
AD
A
D
and the side
C
B
CB
CB
. If
∣
C
A
∣
+
∣
A
D
∣
=
∣
C
B
∣
|CA|+|AD|=|CB|
∣
C
A
∣
+
∣
A
D
∣
=
∣
CB
∣
, and
∣
C
N
∣
=
2
|CN|=2
∣
CN
∣
=
2
, then what is
∣
N
B
∣
|NB|
∣
NB
∣
?