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National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2005 Turkey Junior National Olympiad
2005 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(3)
3
1
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Turkey Junior Olympiad 2005, Part II - P3
Determine whether or not there exists a sequence of integers
a
1
,
a
2
,
…
,
a
19
,
a
20
a_1,a_2,\dots, a_{19}, a_{20}
a
1
,
a
2
,
…
,
a
19
,
a
20
such that, the sum of all the terms is negative, and the sum of any three of the terms is positive.
2
1
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Turkey Junior Olympiad 2005, Part II - P2
Find all integer pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
x
3
+
y
3
=
(
x
+
y
)
2
x^3+y^3=(x+y)^2
x
3
+
y
3
=
(
x
+
y
)
2
.
1
1
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Turkey Junior Olympiad 2005, Part II - P1
Let
A
B
C
ABC
A
BC
be an acute triangle. Let
H
H
H
and
D
D
D
be points on
[
A
C
]
[AC]
[
A
C
]
and
[
B
C
]
[BC]
[
BC
]
, respectively, such that
B
H
⊥
A
C
BH \perp AC
B
H
⊥
A
C
and
H
D
⊥
B
C
HD \perp BC
HD
⊥
BC
. Let
O
1
O_1
O
1
be the circumcenter of
△
A
B
H
\triangle ABH
△
A
B
H
, and
O
2
O_2
O
2
be the circumcenter of
△
B
H
D
\triangle BHD
△
B
HD
, and
O
3
O_3
O
3
be the circumcenter of
△
H
D
C
\triangle HDC
△
HD
C
. Find the ratio of area of
△
O
1
O
2
O
3
\triangle O_1O_2O_3
△
O
1
O
2
O
3
and
△
A
B
H
\triangle ABH
△
A
B
H
.