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National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2003 Turkey Junior National Olympiad
2003 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
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Turkey Junior Olympiad 2003, Part II - P3
How many subsets of
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
}
\{1,2,3,4,5,6,7,8,9,10,11\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
}
contain no two consequtive numbers?
2
1
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Turkey Junior Olympiad 2003, Part II - P2
From the positive integers,
m
,
m
+
1
,
…
,
m
+
n
m,m+1,\dots,m+n
m
,
m
+
1
,
…
,
m
+
n
, only the sum of digits of
m
m
m
and the sum of digits of
m
+
n
m+n
m
+
n
are divisible by
8
8
8
. Find the maximum value of
n
n
n
.
1
1
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Turkey Junior Olympiad 2003, Part II - P1
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral, and
E
E
E
be the intersection of its diagonals. If
m
(
A
D
B
^
)
=
22.
5
∘
m(\widehat{ADB}) = 22.5^\circ
m
(
A
D
B
)
=
22.
5
∘
,
∣
B
D
∣
=
6
|BD|=6
∣
B
D
∣
=
6
, and
∣
A
D
∣
⋅
∣
C
E
∣
=
∣
D
C
∣
⋅
∣
A
E
∣
|AD|\cdot|CE|=|DC|\cdot|AE|
∣
A
D
∣
⋅
∣
CE
∣
=
∣
D
C
∣
⋅
∣
A
E
∣
, find the area of the quadrilateral
A
B
C
D
ABCD
A
BC
D
.