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National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
1998 Turkey Junior National Olympiad
1998 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
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Turkey Junior Olympiad 1998, Part II - P3
We call a positive integer good number, if it is divisible by squares of all its prime factors. Show that there are infinitely many pairs of consequtive numbers both are good.
2
1
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Turkey Junior Olympiad 1998, Part II - P2
The first
9
9
9
positive integers are placed into the squares of a
3
×
3
3\times 3
3
×
3
chessboard. We are taking the smallest number in a column. Let
a
a
a
be the largest of these three smallest number. Similarly, we are taking the largest number in a row. Let
b
b
b
be the smallest of these three largest number. How many ways can we distribute the numbers into the chessboard such that
a
=
b
=
4
a=b=4
a
=
b
=
4
?
1
1
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Turkey Junior Olympiad 1998, Part II - P1
Let
F
F
F
,
D
D
D
, and
E
E
E
be points on the sides
[
A
B
]
[AB]
[
A
B
]
,
[
B
C
]
[BC]
[
BC
]
, and
[
C
A
]
[CA]
[
C
A
]
of
△
A
B
C
\triangle ABC
△
A
BC
, respectively, such that
△
D
E
F
\triangle DEF
△
D
EF
is an isosceles right triangle with hypotenuse
[
E
F
]
[EF]
[
EF
]
. The altitude of
△
A
B
C
\triangle ABC
△
A
BC
passing through
A
A
A
is
10
10
10
cm. If
∣
B
C
∣
=
30
|BC|=30
∣
BC
∣
=
30
cm, and
E
F
∥
B
C
EF \parallel BC
EF
∥
BC
, calculate the perimeter of
△
D
E
F
\triangle DEF
△
D
EF
.