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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2010 Turkey MO (2nd round)
2010 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(3)
3
2
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Turkey NMO 2010 P3
Prove that for all
n
∈
Z
+
n \in \mathbb{Z^+}
n
∈
Z
+
and for all positive real numbers satisfying
a
1
a
2
.
.
.
a
n
=
1
a_1a_2...a_n=1
a
1
a
2
...
a
n
=
1
∑
i
=
1
n
a
i
a
i
4
+
3
≤
1
2
∑
i
=
1
n
1
a
i
\displaystyle\sum_{i=1}^{n} \frac{a_i}{\sqrt{{a_i}^4+3}} \leq \frac{1}{2}\displaystyle\sum_{i=1}^{n} \frac{1}{a_i}
i
=
1
∑
n
a
i
4
+
3
a
i
≤
2
1
i
=
1
∑
n
a
i
1
Turkey NMO 2010 P6
Let
K
K
K
be the set of all sides and diagonals of a convex
2010
−
g
o
n
2010-gon
2010
−
g
o
n
in the plane. For a subset
A
A
A
of
K
,
K,
K
,
if every pair of line segments belonging to
A
A
A
intersect, then we call
A
A
A
as an intersecting set. Find the maximum possible number of elements of union of two intersecting sets.
2
2
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Turkey NMO 2010 P2
Let
P
P
P
be an interior point of the triangle
A
B
C
ABC
A
BC
which is not on the median belonging to
B
C
BC
BC
and satisfying
∠
C
A
P
=
∠
B
C
P
.
B
P
∩
C
A
=
{
B
′
}
,
C
P
∩
A
B
=
{
C
′
}
\angle CAP = \angle BCP. \: BP \cap CA = \{B'\} \: , \: CP \cap AB = \{C'\}
∠
C
A
P
=
∠
BCP
.
BP
∩
C
A
=
{
B
′
}
,
CP
∩
A
B
=
{
C
′
}
and
Q
Q
Q
is the second point of intersection of
A
P
AP
A
P
and the circumcircle of
A
B
C
.
B
′
Q
ABC. \: B'Q
A
BC
.
B
′
Q
intersects
C
C
′
CC'
C
C
′
at
R
R
R
and
B
′
Q
B'Q
B
′
Q
intersects the line through
P
P
P
parallel to
A
C
AC
A
C
at
S
.
S.
S
.
Let
T
T
T
be the point of intersection of lines
B
′
C
′
B'C'
B
′
C
′
and
Q
B
QB
QB
and
T
T
T
be on the other side of
A
B
AB
A
B
with respect to
C
.
C.
C
.
Prove that
∠
B
A
T
=
∠
B
B
′
Q
⟺
∣
S
Q
∣
=
∣
R
B
′
∣
\angle BAT = \angle BB'Q \: \Longleftrightarrow \: |SQ|=|RB'|
∠
B
A
T
=
∠
B
B
′
Q
⟺
∣
SQ
∣
=
∣
R
B
′
∣
Turkey NMO 2010 P5
For integers
a
a
a
and
b
b
b
with
0
≤
a
,
b
<
2010
18
0 \leq a,b < {2010}^{18}
0
≤
a
,
b
<
2010
18
let
S
S
S
be the set of all polynomials in the form of
P
(
x
)
=
a
x
2
+
b
x
.
P(x)=ax^2+bx.
P
(
x
)
=
a
x
2
+
b
x
.
For a polynomial
P
P
P
in
S
,
S,
S
,
if for all integers n with
0
≤
n
<
2010
18
0 \leq n <{2010}^{18}
0
≤
n
<
2010
18
there exists a polynomial
Q
Q
Q
in
S
S
S
satisfying
Q
(
P
(
n
)
)
≡
n
(
m
o
d
201
0
18
)
,
Q(P(n)) \equiv n \pmod {2010^{18}},
Q
(
P
(
n
))
≡
n
(
mod
201
0
18
)
,
then we call
P
P
P
as a good polynomial. Find the number of good polynomials.
1
2
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Turkey NMO 2010 P1
In a country, there are some two-way roads between the cities. There are
2010
2010
2010
roads connected to the capital city. For all cities different from the capital city, there are less than
2010
2010
2010
roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even.
k
k
k
roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of
k
.
k.
k
.
Turkey NMO 2010 P4
Let
A
A
A
and
B
B
B
be two points on the circle with diameter
[
C
D
]
[CD]
[
C
D
]
and on the different sides of the line
C
D
.
CD.
C
D
.
A circle
Γ
\Gamma
Γ
passing through
C
C
C
and
D
D
D
intersects
[
A
C
]
[AC]
[
A
C
]
different from the endpoints at
E
E
E
and intersects
B
C
BC
BC
at
F
.
F.
F
.
The line tangent to
Γ
\Gamma
Γ
at
E
E
E
intersects
B
C
BC
BC
at
P
P
P
and
Q
Q
Q
is a point on the circumcircle of the triangle
C
E
P
CEP
CEP
different from
E
E
E
and satisfying
∣
Q
P
∣
=
∣
E
P
∣
.
A
B
∩
E
F
=
{
R
}
|QP|=|EP|. \: AB \cap EF =\{R\}
∣
QP
∣
=
∣
EP
∣.
A
B
∩
EF
=
{
R
}
and
S
S
S
is the midpoint of
[
E
Q
]
.
[EQ].
[
EQ
]
.
Prove that
D
R
DR
D
R
is parallel to
P
S
.
PS.
PS
.