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National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2004 Turkey Junior National Olympiad
2004 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
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Turkey Junior Olympiad 2004, Part II - P3
On the evening, more than
1
3
\frac 13
3
1
of the students of a school are going to the cinema. On the same evening, More than
3
10
\frac {3}{10}
10
3
are going to the theatre, and more than
4
11
\frac {4}{11}
11
4
are going to the concert. At least how many students are there in this school?
2
1
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Turkey Junior Olympiad 2004, Part II - P2
The positive integer
n
n
n
is the sum of two positive integers that divide
n
+
6
n+6
n
+
6
. Find all possible values of
n
n
n
1
1
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Turkey Junior Olympiad 2004, Part II - P1
Let
[
A
D
]
[AD]
[
A
D
]
and
[
C
E
]
[CE]
[
CE
]
be internal angle bisectors of
△
A
B
C
\triangle ABC
△
A
BC
such that
D
D
D
is on
[
B
C
]
[BC]
[
BC
]
and
E
E
E
is on
[
A
B
]
[AB]
[
A
B
]
. Let
K
K
K
and
M
M
M
be the feet of perpendiculars from
B
B
B
to the lines
A
D
AD
A
D
and
C
E
CE
CE
, respectively. If
∣
B
K
∣
=
∣
B
M
∣
|BK|=|BM|
∣
B
K
∣
=
∣
BM
∣
, show that
△
A
B
C
\triangle ABC
△
A
BC
is isosceles.