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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2024 Turkey Team Selection Test
2024 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(9)
6
1
Hide problems
n-variable inequality from Turkey TST
For a positive integer
n
n
n
and real numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots ,a_n
a
1
,
a
2
,
…
,
a
n
we'll define
b
1
,
b
2
,
…
,
b
n
+
1
b_1, b_2, \dots ,b_{n+1}
b
1
,
b
2
,
…
,
b
n
+
1
such that
b
k
=
a
k
+
max
(
a
k
+
1
,
a
k
+
2
)
b_k=a_k+\max({a_{k+1},a_{k+2}})
b
k
=
a
k
+
max
(
a
k
+
1
,
a
k
+
2
)
for all
1
≤
k
≤
n
1\leq k \leq n
1
≤
k
≤
n
and
b
n
+
1
=
b
1
b_{n+1}=b_1
b
n
+
1
=
b
1
. (Also
a
n
+
1
=
a
1
a_{n+1}=a_1
a
n
+
1
=
a
1
and
a
n
+
2
=
a
2
a_{n+2}=a_2
a
n
+
2
=
a
2
) Find the least possible value of
λ
\lambda
λ
such that for all
n
,
a
1
,
…
,
a
n
n, a_1, \dots, a_n
n
,
a
1
,
…
,
a
n
the inequality
λ
[
∑
i
=
1
n
(
a
i
−
a
i
+
1
)
2024
]
≥
∑
i
=
1
n
(
b
i
−
b
i
+
1
)
2024
\lambda \Biggl[ \sum_{i=1}^n(a_i-a_{i+1})^{2024} \Biggr] \geq \sum_{i=1}^n(b_i-b_{i+1})^{2024}
λ
[
i
=
1
∑
n
(
a
i
−
a
i
+
1
)
2024
]
≥
i
=
1
∑
n
(
b
i
−
b
i
+
1
)
2024
holds.
3
1
Hide problems
Turkey TST Combinatorics... well kinda
If
S
S
S
is a set which consists of
12
12
12
elements, what is the maximum number of pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
such that
a
,
b
∈
S
a, b\in S
a
,
b
∈
S
and
b
a
\frac{b}{a}
a
b
is a prime number?
1
1
Hide problems
Favorite line IO
In triangle
A
B
C
ABC
A
BC
, the incenter is
I
I
I
and the circumcenter is
O
O
O
. Let
A
I
AI
A
I
intersects
(
A
B
C
)
(ABC)
(
A
BC
)
second time at
P
P
P
. The line passes through
I
I
I
and perpendicular to
A
I
AI
A
I
intersects
B
C
BC
BC
at
X
X
X
. The feet of the perpendicular from
X
X
X
to
I
O
IO
I
O
is
Y
Y
Y
. Prove that
A
,
P
,
X
,
Y
A,P,X,Y
A
,
P
,
X
,
Y
cyclic.
8
1
Hide problems
A sequence related to the sigma function
For an integer
n
n
n
,
σ
(
n
)
\sigma(n)
σ
(
n
)
denotes the sum of postitive divisors of
n
n
n
. A sequence of positive integers
(
a
i
)
i
=
0
∞
(a_i)_{i=0}^{\infty}
(
a
i
)
i
=
0
∞
with
a
0
=
1
a_0 =1
a
0
=
1
is defined as follows: For each
n
>
1
n>1
n
>
1
,
a
n
a_n
a
n
is the smallest integer greater than
1
1
1
that satisfies
σ
(
a
0
a
1
…
a
n
−
1
)
∣
σ
(
a
0
a
1
…
a
n
)
.
\sigma{(a_0a_1\dots a_{n-1})} \vert \sigma{(a_0a_1\dots a_{n})}.
σ
(
a
0
a
1
…
a
n
−
1
)
∣
σ
(
a
0
a
1
…
a
n
)
.
Determine the number of divisors of
202
4
2024
2024^{2024}
202
4
2024
amongst the sequence.
2
1
Hide problems
Easy function in turkey TST
Find all
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
functions such that
f
(
x
+
y
)
3
=
(
x
+
2
y
)
f
(
x
2
)
+
f
(
f
(
y
)
)
(
x
2
+
3
x
y
+
y
2
)
f(x+y)^3=(x+2y)f(x^2)+f(f(y))(x^2+3xy+y^2)
f
(
x
+
y
)
3
=
(
x
+
2
y
)
f
(
x
2
)
+
f
(
f
(
y
))
(
x
2
+
3
x
y
+
y
2
)
for all real numbers
x
,
y
x,y
x
,
y
5
1
Hide problems
Complex bash geo
In a scalene triangle
A
B
C
ABC
A
BC
,
H
H
H
is the orthocenter, and
G
G
G
is the centroid. Let
A
b
A_b
A
b
and
A
c
A_c
A
c
be points on
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
B
B
B
,
C
C
C
,
A
b
A_b
A
b
,
A
c
A_c
A
c
are cyclic, and the points
A
b
A_b
A
b
,
A
c
A_c
A
c
,
H
H
H
are collinear.
O
a
O_a
O
a
is the circumcenter of the triangle
A
A
b
A
c
AA_bA_c
A
A
b
A
c
.
O
b
O_b
O
b
and
O
c
O_c
O
c
are defined similarly. Prove that the centroid of the triangle
O
a
O
b
O
c
O_aO_bO_c
O
a
O
b
O
c
lies on the line
H
G
HG
H
G
.
7
1
Hide problems
Coloring numbers somewhat equally
Let
r
≥
2
r\geq 2
r
≥
2
be a positive integer, and let each positive integer be painted in one of
r
r
r
different colors. For every positive integer
n
n
n
and every pair of colors
a
a
a
and
b
b
b
, if the difference between the number of divisors of
n
n
n
that are painted in color
a
a
a
and the number of divisors of
n
n
n
that are painted in color
b
b
b
is at most
1
1
1
, find all possible values of
r
r
r
.
9
1
Hide problems
Hard geo finale with the cursed line
In a scalene triangle
A
B
C
,
ABC,
A
BC
,
I
I
I
is the incenter and
O
O
O
is the circumcenter. The line
I
O
IO
I
O
intersects the lines
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at points
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. Let
A
1
A_1
A
1
be the intersection of
B
E
BE
BE
and
C
F
CF
CF
. The points
B
1
B_1
B
1
and
C
1
C_1
C
1
are defined similarly. The incircle of
A
B
C
ABC
A
BC
is tangent to sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at points
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
respectively. Let the lines
X
A
1
,
Y
B
1
XA_1, YB_1
X
A
1
,
Y
B
1
and
Z
C
1
ZC_1
Z
C
1
intersect
I
O
IO
I
O
at points
A
2
,
B
2
,
C
2
A_2,B_2,C_2
A
2
,
B
2
,
C
2
respectively. Prove that the circles with diameters
A
A
2
,
B
B
2
AA_2,BB_2
A
A
2
,
B
B
2
and
C
C
2
CC_2
C
C
2
have a common point.
4
1
Hide problems
A regular NT problem in Turkey TST
Find all positive integer pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
such that,
1
0
a
!
−
3
b
+
1
2
a
\frac{10^{a!} - 3^b +1}{2^a}
2
a
1
0
a
!
−
3
b
+
1
is a perfect square.