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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2018 Turkey MO (2nd Round)
2018 Turkey MO (2nd Round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
6
1
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Boxes and Balls
Initially, there are 2018 distinct boxes on a table. In the first stage, Yazan and Bozan, starting with Yazan, take turns make
2016
2016
2016
moves each, such that, in each move, the person whose turn selects a pair of boxes that is not written on the board, and writes the pair on the board. In the second stage, Bozan enumerates the
4032
4032
4032
pairs with numbers from
1
,
2
,
…
,
4032
1,2,\dots,4032
1
,
2
,
…
,
4032
, in whichever order he wants, and puts
k
k
k
balls in each boxes written contained in the
k
t
h
k^{th}
k
t
h
pair. Is there a strategy for Bozan that guarantees that the number of balls in each box are distinct?
5
1
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Four Numbers with Divisibility Property
Let
a
1
,
a
2
,
a
3
,
a
4
a_1,a_2,a_3,a_4
a
1
,
a
2
,
a
3
,
a
4
be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let
i
,
j
,
k
∈
{
1
,
2
,
3
,
4
}
i,j,k\in\{1,2,3,4\}
i
,
j
,
k
∈
{
1
,
2
,
3
,
4
}
with
i
≠
j
i \neq j
i
=
j
,
j
≠
k
j\neq k
j
=
k
, and
k
≠
i
k\neq i
k
=
i
. Determine the maximum number of triples
(
i
,
j
,
k
)
(i,j,k)
(
i
,
j
,
k
)
for which
(
g
c
d
(
a
i
,
a
j
)
)
2
∣
a
k
.
({\rm gcd}(a_i,a_j))^2|a_k.
(
gcd
(
a
i
,
a
j
)
)
2
∣
a
k
.
4
1
Hide problems
Geometric Inequality with Excircle
In a triangle
A
B
C
ABC
A
BC
, the bisector of the angle
A
A
A
intersects the excircle that is tangential to side
[
B
C
]
[BC]
[
BC
]
at two points
D
D
D
and
E
E
E
such that
D
∈
[
A
E
]
D\in [AE]
D
∈
[
A
E
]
. Prove that,
∣
A
D
∣
∣
A
E
∣
≤
∣
B
C
∣
2
∣
D
E
∣
2
.
\frac{|AD|}{|AE|}\leq \frac{|BC|^2}{|DE|^2}.
∣
A
E
∣
∣
A
D
∣
≤
∣
D
E
∣
2
∣
BC
∣
2
.
3
1
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Integrality of a certain quantity
A sequence
a
1
,
a
2
,
…
a_1,a_2,\dots
a
1
,
a
2
,
…
satisfy
∑
i
=
1
n
a
⌊
n
i
⌋
=
n
10
,
\sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10},
i
=
1
∑
n
a
⌊
i
n
⌋
=
n
10
,
for every
n
∈
N
n\in\mathbb{N}
n
∈
N
. Let
c
c
c
be a positive integer. Prove that, for every positive integer
n
n
n
,
c
a
n
−
c
a
n
−
1
n
\frac{c^{a_n}-c^{a_{n-1}}}{n}
n
c
a
n
−
c
a
n
−
1
is an integer.
2
1
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Triangle
Let
P
P
P
be a point in the interior of the triangle
A
B
C
ABC
A
BC
. The lines
A
P
AP
A
P
,
B
P
BP
BP
, and
C
P
CP
CP
intersect the sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
at
D
,
E
D,E
D
,
E
, and
F
F
F
, respectively. A point
Q
Q
Q
is taken on the ray
[
B
E
[BE
[
BE
such that
E
∈
[
B
Q
]
E\in [BQ]
E
∈
[
BQ
]
and
m
(
E
D
Q
^
)
=
m
(
B
D
F
^
)
m(\widehat{EDQ})=m(\widehat{BDF})
m
(
E
D
Q
)
=
m
(
B
D
F
)
. If
B
E
BE
BE
and
A
D
AD
A
D
are perpendicular, and
∣
D
Q
∣
=
2
∣
B
D
∣
|DQ|=2|BD|
∣
D
Q
∣
=
2∣
B
D
∣
, prove that
m
(
F
D
E
^
)
=
6
0
∘
m(\widehat{FDE})=60^\circ
m
(
F
D
E
)
=
6
0
∘
.
1
1
Hide problems
An equality
Find all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of real numbers that satisfy, \begin{align*} x^2+y^2+x+y &= xy(x+y)-\frac{10}{27}\\ |xy| & \leq \frac{25}{9}. \end{align*}