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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2009 Turkey MO (2nd round)
2009 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(3)
3
2
Hide problems
Turkey NMO 2009 Q3
Alice, who works for the Graph County Electric Works, is commissioned to wire the newly erected utility poles in
k
k
k
days. Each day she either chooses a pole and runs wires from it to as many poles as she wishes, or chooses at most
17
17
17
pairs of poles and runs wires between each pair. Bob, who works for the Graph County Paint Works, claims that, no matter how many poles there are and how Alice connects them, all the poles can be painted using not more than
2009
2009
2009
colors in such a way that no pair of poles connected by a wire is the same color. Determine the greatest value of
k
k
k
for which Bob's claim is valid.
Turkey NMO 2009 Q6
If
1
<
k
1
<
k
2
<
.
.
.
<
k
n
1<k_1<k_2<...<k_n
1
<
k
1
<
k
2
<
...
<
k
n
and
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
are integers such that for every integer
N
,
N,
N
,
k
i
∣
N
−
a
i
k_i \mid N-a_i
k
i
∣
N
−
a
i
for some
1
≤
i
≤
n
,
1 \leq i \leq n,
1
≤
i
≤
n
,
find the smallest possible value of
n
.
n.
n
.
2
2
Hide problems
Turkey NMO 2009 Q2
Let
Γ
\Gamma
Γ
be the circumcircle of a triangle
A
B
C
,
ABC,
A
BC
,
and let
D
D
D
and
E
E
E
be two points different from the vertices on the sides
A
B
AB
A
B
and
A
C
,
AC,
A
C
,
respectively. Let
A
′
A'
A
′
be the second point where
Γ
\Gamma
Γ
intersects the bisector of the angle
B
A
C
,
BAC,
B
A
C
,
and let
P
P
P
and
Q
Q
Q
be the second points where
Γ
\Gamma
Γ
intersects the lines
A
′
D
A'D
A
′
D
and
A
′
E
,
A'E,
A
′
E
,
respectively. Let
R
R
R
and
S
S
S
be the second points of intersection of the lines
A
A
′
AA'
A
A
′
and the circumcircles of the triangles
A
P
D
APD
A
P
D
and
A
Q
E
,
AQE,
A
QE
,
respectively. Show that the lines
D
S
,
E
R
DS, \: ER
D
S
,
ER
and the tangent line to
Γ
\Gamma
Γ
through
A
A
A
are concurrent.
Turkey NMO 2009 Q5
Show that
(
b
+
c
)
(
a
4
−
b
2
c
2
)
a
b
+
2
b
c
+
c
a
+
(
c
+
a
)
(
b
4
−
c
2
a
2
)
b
c
+
2
c
a
+
a
b
+
(
a
+
b
)
(
c
4
−
a
2
b
2
)
c
a
+
2
a
b
+
b
c
≥
0
\frac{(b+c)(a^4-b^2c^2)}{ab+2bc+ca}+\frac{(c+a)(b^4-c^2a^2)}{bc+2ca+ab}+\frac{(a+b)(c^4-a^2b^2)}{ca+2ab+bc} \geq 0
ab
+
2
b
c
+
c
a
(
b
+
c
)
(
a
4
−
b
2
c
2
)
+
b
c
+
2
c
a
+
ab
(
c
+
a
)
(
b
4
−
c
2
a
2
)
+
c
a
+
2
ab
+
b
c
(
a
+
b
)
(
c
4
−
a
2
b
2
)
≥
0
for all positive real numbers
a
,
b
,
c
.
a, \: b , \: c.
a
,
b
,
c
.
1
2
Hide problems
Turkey NMO 2009 Q1
Find all prime numbers
p
p
p
for which
p
3
−
4
p
+
9
p^3-4p+9
p
3
−
4
p
+
9
is a perfect square.
Turkey NMO 2009 Q4
Let
H
H
H
be the orthocenter of an acute triangle
A
B
C
,
ABC,
A
BC
,
and let
A
1
,
B
1
,
C
1
A_1, \: B_1, \: C_1
A
1
,
B
1
,
C
1
be the feet of the altitudes belonging to the vertices
A
,
B
,
C
,
A, \: B, \: C,
A
,
B
,
C
,
respectively. Let
K
K
K
be a point on the smaller
A
B
1
AB_1
A
B
1
arc of the circle with diameter
A
B
AB
A
B
satisfying the condition
∠
H
K
B
=
∠
C
1
K
B
.
\angle HKB = \angle C_1KB.
∠
HK
B
=
∠
C
1
K
B
.
Let
M
M
M
be the point of intersection of the line segment
A
A
1
AA_1
A
A
1
and the circle with center
C
C
C
and radius
C
L
CL
C
L
where
K
B
∩
C
C
1
=
{
L
}
.
KB \cap CC_1=\{L\}.
K
B
∩
C
C
1
=
{
L
}
.
Let
P
P
P
and
Q
Q
Q
be the points of intersection of the line
C
C
1
CC_1
C
C
1
and the circle with center
B
B
B
and radius
B
M
.
BM.
BM
.
Show that
A
,
K
,
P
,
Q
A, \: K, \: P, \: Q
A
,
K
,
P
,
Q
are concyclic.