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Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2007 Turkey Junior National Olympiad
2007 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
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Turkey Junior Olympiad 2007, Part II - P3
Find all odd postive integers less than
2007
2007
2007
such that the sum of all of its positive divisors is odd.
2
1
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Turkey Junior Olympiad 2007, Part II - P2
In a qualification group with
15
15
15
volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceeding
N
N
N
. If there are at least
7
7
7
teams qualified, find the possible least value of
N
N
N
.
1
1
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Turkey Junior Olympiad 2007, 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
and
∣
A
B
∣
=
∣
B
C
∣
|AB|=|BC|
∣
A
B
∣
=
∣
BC
∣
. Let
E
E
E
and
F
F
F
be the midpoints of
[
B
C
]
[BC]
[
BC
]
and
[
A
D
]
[AD]
[
A
D
]
, respectively. If the internal angle bisector of
△
A
B
C
\triangle ABC
△
A
BC
passes through
F
F
F
, find
∣
B
D
∣
/
∣
E
F
∣
|BD|/|EF|
∣
B
D
∣/∣
EF
∣
.