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Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2018 Turkey Team Selection Test
2018 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(9)
8
1
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2018 TST P8
For integers
m
≥
3
m\geq 3
m
≥
3
,
n
n
n
and
x
1
,
x
2
,
…
,
x
m
x_1,x_2, \ldots , x_m
x
1
,
x
2
,
…
,
x
m
if
x
i
+
1
−
x
i
≡
x
i
−
x
i
−
1
(
m
o
d
n
)
x_{i+1}-x_i \equiv x_i-x_{i-1} (mod n)
x
i
+
1
−
x
i
≡
x
i
−
x
i
−
1
(
m
o
d
n
)
for every
2
≤
i
≤
m
−
1
2\leq i \leq m-1
2
≤
i
≤
m
−
1
, we say that the
m
m
m
-tuple
(
x
1
,
x
2
,
…
,
x
m
)
(x_1,x_2,\ldots , x_m)
(
x
1
,
x
2
,
…
,
x
m
)
is an arithmetic sequence in
(
m
o
d
n
)
(mod n)
(
m
o
d
n
)
. Let
p
≥
5
p\geq 5
p
≥
5
be a prime number and
1
<
a
<
p
−
1
1<a<p-1
1
<
a
<
p
−
1
be an integer. Let
a
1
,
a
2
,
…
,
a
k
{a_1,a_2,\ldots , a_k}
a
1
,
a
2
,
…
,
a
k
be the set of all possible remainders when positive powers of
a
a
a
are divided by
p
p
p
. Show that if a permutation of
a
1
,
a
2
,
…
,
a
k
{a_1,a_2,\ldots , a_k}
a
1
,
a
2
,
…
,
a
k
is an arithmetic sequence in
(
m
o
d
p
)
(mod p)
(
m
o
d
p
)
, then
k
=
p
−
1
k=p-1
k
=
p
−
1
.
3
1
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Operations on a permutation
A Retired Linguist (R.L.) writes in the first move a word consisting of
n
n
n
letters, which are all different. In each move, he determines the maximum
i
i
i
, such that the word obtained by reversing the first
i
i
i
letters of the last word hasn't been written before, and writes this new word. Prove that R.L. can make
n
!
n!
n
!
moves.
1
1
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Divisors of a quadratic polynomial
Prove that, for all integers
a
,
b
a, b
a
,
b
, there exists a positive integer
n
n
n
, such that the number
n
2
+
a
n
+
b
n^2+an+b
n
2
+
an
+
b
has at least
2018
2018
2018
different prime divisors.
6
1
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Sequence with a choice
a
0
,
a
1
,
…
,
a
100
a_0, a_1, \ldots, a_{100}
a
0
,
a
1
,
…
,
a
100
and
b
1
,
b
2
,
…
,
b
100
b_1, b_2,\ldots, b_{100}
b
1
,
b
2
,
…
,
b
100
are sequences of real numbers, for which the property holds: for all
n
=
0
,
1
,
…
,
99
n=0, 1, \ldots, 99
n
=
0
,
1
,
…
,
99
, either a_{n+1}=\frac{a_n}{2} \text{and} b_{n+1}=\frac{1}{2}-a_n, or a_{n+1}=2a_n^2 \text{and} b_{n+1}=a_n. Given
a
100
≤
a
0
a_{100}\leq a_0
a
100
≤
a
0
, what is the maximal value of
b
1
+
b
2
+
⋯
+
b
100
b_1+b_2+\cdots+b_{100}
b
1
+
b
2
+
⋯
+
b
100
?
4
1
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Point on median
In a non-isosceles acute triangle
A
B
C
ABC
A
BC
,
D
D
D
is the midpoint of the edge
[
B
C
]
[BC]
[
BC
]
. The points
E
E
E
and
F
F
F
lie on
[
A
C
]
[AC]
[
A
C
]
and
[
A
B
]
[AB]
[
A
B
]
, respectively, and the circumcircles of
C
D
E
CDE
C
D
E
and
A
E
F
AEF
A
EF
intersect in
P
P
P
on
[
A
D
]
[AD]
[
A
D
]
. The angle bisector from
P
P
P
in triangle
E
F
P
EFP
EFP
intersects
E
F
EF
EF
in
Q
Q
Q
. Prove that the tangent line to the circumcirle of
A
Q
P
AQP
A
QP
at
A
A
A
is perpendicular to
B
C
BC
BC
.
7
1
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Graph consisting of lattice points
For integers
a
,
b
a, b
a
,
b
, call the lattice point with coordinates
(
a
,
b
)
(a,b)
(
a
,
b
)
basic if
g
c
d
(
a
,
b
)
=
1
gcd(a,b)=1
g
c
d
(
a
,
b
)
=
1
. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between
(
a
1
,
b
1
)
(a_1,b_1)
(
a
1
,
b
1
)
and
(
a
2
,
b
2
)
(a_2,b_2)
(
a
2
,
b
2
)
if and only if
2
a
1
=
2
a
2
∈
{
b
1
−
b
2
,
b
2
−
b
1
}
2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}
2
a
1
=
2
a
2
∈
{
b
1
−
b
2
,
b
2
−
b
1
}
or
2
b
1
=
2
b
2
∈
{
a
1
−
a
2
,
a
2
−
a
1
}
2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}
2
b
1
=
2
b
2
∈
{
a
1
−
a
2
,
a
2
−
a
1
}
. Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?
9
1
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Set of Simson Lines
For a triangle
T
T
T
and a line
d
d
d
, if the feet of perpendicular lines from a point in the plane to the edges of
T
T
T
all lie on
d
d
d
, say
d
d
d
focuses
T
T
T
. If the set of lines focusing
T
1
T_1
T
1
and the set of lines focusing
T
2
T_2
T
2
are the same, say
T
1
T_1
T
1
and
T
2
T_2
T
2
are equivalent. Prove that, for any triangle in the plane, there exists exactly one equilateral triangle which is equivalent to it.
5
1
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P5 Combinatorics 2018 Turkey TST
We say that a group of
25
25
25
students is a team if any two students in this group are friends. It is known that in the school any student belongs to at least one team but if any two students end their friendships at least one student does not belong to any team. We say that a team is special if at least one student of the team has no friend outside of this team. Show that any two friends belong to some special team.
2
1
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Functional Equation
Find all
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
surjective functions such that
f
(
x
f
(
y
)
+
y
2
)
=
f
(
(
x
+
y
)
2
)
−
x
f
(
x
)
f(xf(y)+y^2)=f((x+y)^2)-xf(x)
f
(
x
f
(
y
)
+
y
2
)
=
f
((
x
+
y
)
2
)
−
x
f
(
x
)
for all real numbers
x
,
y
x,y
x
,
y
.