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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2012 Turkey MO (2nd round)
2012 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
5
1
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Turkish NMO 2012 - P5
Let
P
P
P
be the set of all
2012
2012
2012
tuples
(
x
1
,
x
2
,
…
,
x
2012
)
(x_1, x_2, \dots, x_{2012})
(
x
1
,
x
2
,
…
,
x
2012
)
, where
x
i
∈
{
1
,
2
,
…
20
}
x_i \in \{1,2,\dots 20\}
x
i
∈
{
1
,
2
,
…
20
}
for each
1
≤
i
≤
2012
1\leq i \leq 2012
1
≤
i
≤
2012
. The set
A
⊂
P
A \subset P
A
⊂
P
is said to be decreasing if for each
(
x
1
,
x
2
,
…
,
x
2012
)
∈
A
(x_1,x_2,\dots ,x_{2012} ) \in A
(
x
1
,
x
2
,
…
,
x
2012
)
∈
A
any
(
y
1
,
y
2
,
…
,
y
2012
)
(y_1,y_2,\dots, y_{2012})
(
y
1
,
y
2
,
…
,
y
2012
)
satisfying
y
i
≤
x
i
(
1
≤
i
≤
2012
)
y_i \leq x_i (1\leq i \leq 2012)
y
i
≤
x
i
(
1
≤
i
≤
2012
)
also belongs to
A
A
A
. The set
B
⊂
P
B \subset P
B
⊂
P
is said to be increasing if for each
(
x
1
,
x
2
,
…
,
x
2012
)
∈
B
(x_1,x_2,\dots ,x_{2012} ) \in B
(
x
1
,
x
2
,
…
,
x
2012
)
∈
B
any
(
y
1
,
y
2
,
…
,
y
2012
)
(y_1,y_2,\dots, y_{2012})
(
y
1
,
y
2
,
…
,
y
2012
)
satisfying
y
i
≥
x
i
(
1
≤
i
≤
2012
)
y_i \geq x_i (1\leq i \leq 2012)
y
i
≥
x
i
(
1
≤
i
≤
2012
)
also belongs to
B
B
B
. Find the maximum possible value of
f
(
A
,
B
)
=
∣
A
∩
B
∣
∣
A
∣
⋅
∣
B
∣
f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}
f
(
A
,
B
)
=
∣
A
∣
⋅
∣
B
∣
∣
A
∩
B
∣
, where
A
A
A
and
B
B
B
are nonempty decreasing and increasing sets (
∣
⋅
∣
\mid \cdot \mid
∣
⋅
∣
denotes the number of elements of the set).
4
1
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Turkey NMO 2012 Problem 4
For all positive real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
, show that
x
(
2
x
−
y
)
y
(
2
z
+
x
)
+
y
(
2
y
−
z
)
z
(
2
x
+
y
)
+
z
(
2
z
−
x
)
x
(
2
y
+
z
)
≥
1
\frac{x(2x-y)}{y(2z+x)}+\frac{y(2y-z)}{z(2x+y)}+\frac{z(2z-x)}{x(2y+z)} \geq 1
y
(
2
z
+
x
)
x
(
2
x
−
y
)
+
z
(
2
x
+
y
)
y
(
2
y
−
z
)
+
x
(
2
y
+
z
)
z
(
2
z
−
x
)
≥
1
is true.
6
1
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Turkey NMO 2012 Problem 6
Let
B
B
B
and
D
D
D
be points on segments
[
A
E
]
[AE]
[
A
E
]
and
[
A
F
]
[AF]
[
A
F
]
respectively. Excircles of triangles
A
B
F
ABF
A
BF
and
A
D
E
ADE
A
D
E
touching sides
B
F
BF
BF
and
D
E
DE
D
E
is the same, and its center is
I
I
I
.
B
F
BF
BF
and
D
E
DE
D
E
intersects at
C
C
C
. Let
P
1
,
P
2
,
P
3
,
P
4
,
Q
1
,
Q
2
,
Q
3
,
Q
4
P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4
P
1
,
P
2
,
P
3
,
P
4
,
Q
1
,
Q
2
,
Q
3
,
Q
4
be the circumcenters of triangles
I
A
B
,
I
B
C
,
I
C
D
,
I
D
A
,
I
A
E
,
I
E
C
,
I
C
F
,
I
F
A
IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA
I
A
B
,
I
BC
,
I
C
D
,
I
D
A
,
I
A
E
,
I
EC
,
I
CF
,
I
F
A
respectively.a) Show that points
P
1
,
P
2
,
P
3
,
P
4
P_1, P_2, P_3, P_4
P
1
,
P
2
,
P
3
,
P
4
concylic and points
Q
1
,
Q
2
,
Q
3
,
Q
4
Q_1, Q_2, Q_3, Q_4
Q
1
,
Q
2
,
Q
3
,
Q
4
concylic. b) Denote centers of theese circles as
O
1
O_1
O
1
and
O
2
O_2
O
2
. Prove that
O
1
,
O
2
O_1, O_2
O
1
,
O
2
and
I
I
I
are collinear.
2
1
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Turkey NMO 2012 Problem 2
Let
A
B
C
ABC
A
BC
be a isosceles triangle with
A
B
=
A
C
AB=AC
A
B
=
A
C
an
D
D
D
be the foot of perpendicular of
A
A
A
.
P
P
P
be an interior point of triangle
A
D
C
ADC
A
D
C
such that
m
(
A
P
B
)
>
90
m(APB)>90
m
(
A
PB
)
>
90
and
m
(
P
B
D
)
+
m
(
P
A
D
)
=
m
(
P
C
B
)
m(PBD)+m(PAD)=m(PCB)
m
(
PB
D
)
+
m
(
P
A
D
)
=
m
(
PCB
)
.
C
P
CP
CP
and
A
D
AD
A
D
intersects at
Q
Q
Q
,
B
P
BP
BP
and
A
D
AD
A
D
intersects at
R
R
R
. Let
T
T
T
be a point on
[
A
B
]
[AB]
[
A
B
]
and
S
S
S
be a point on
[
A
P
[AP
[
A
P
and not belongs to
[
A
P
]
[AP]
[
A
P
]
satisfying
m
(
T
R
B
)
=
m
(
D
Q
C
)
m(TRB)=m(DQC)
m
(
TRB
)
=
m
(
D
QC
)
and
m
(
P
S
R
)
=
2
m
(
P
A
R
)
m(PSR)=2m(PAR)
m
(
PSR
)
=
2
m
(
P
A
R
)
. Show that
R
S
=
R
T
RS=RT
RS
=
RT
3
1
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Turkey NMO 2012 Problem 3 functional equation
Find all non-decreasing functions from real numbers to itself such that for all real numbers
x
,
y
x,y
x
,
y
f
(
f
(
x
2
)
+
y
+
f
(
y
)
)
=
x
2
+
2
f
(
y
)
f(f(x^2)+y+f(y))=x^2+2f(y)
f
(
f
(
x
2
)
+
y
+
f
(
y
))
=
x
2
+
2
f
(
y
)
holds.
1
1
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Turkey NMO 2012 Problem 1
Find all polynomials with integer coefficients such that for all positive integers
n
n
n
satisfies
P
(
n
!
)
=
∣
P
(
n
)
∣
!
P(n!)=|P(n)|!
P
(
n
!)
=
∣
P
(
n
)
∣
!