MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Olympic Revenge
2024 Turkey Olympic Revenge
2024 Turkey Olympic Revenge
Part of
Turkey Olympic Revenge
Subcontests
(6)
6
1
Hide problems
Stop Azer :)
Let
n
n
n
be a positive integer. On a number line, Azer is at point
0
0
0
in his car which have fuel capacity of
2
n
2^n
2
n
units and is initially full. At each positive integer
m
m
m
, there is a gas station. Azer only moves to the right with constant speed and doesn't stop anywhere except the gas stations. Each time his car moves to the right by some amount, its fuel decreases by the same amount. Azer may choose to stop at a gas station or pass it.There are thieves at some gas stations. (A station may have multiple thieves) If Azer stops at a station which have
k
≥
0
k\ge 0
k
≥
0
thieves while its car have fuel capacity
d
d
d
, his cars new fuel capacity becomes
d
2
k
\frac{d}{2^k}
2
k
d
. After that, Azer fulls his cars tank and leaves the station. Find the minimum number of thieves needed to guarantee that Azer will eventually run out of fuel.Proposed by Mehmet Can Baştemir and Deniz Can Karaçelebi
5
1
Hide problems
2020 N6 vibes...
Let
a
a
a
be a positive real number. Prove that a) There exists
n
∈
N
n\in \mathbb{N}
n
∈
N
with
σ
(
φ
(
n
)
)
φ
(
σ
(
n
)
)
>
a
\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} > a
φ
(
σ
(
n
))
σ
(
φ
(
n
))
>
a
.b) There exists
n
∈
N
n\in \mathbb{N}
n
∈
N
with
σ
(
φ
(
n
)
)
φ
(
σ
(
n
)
)
<
a
\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} < a
φ
(
σ
(
n
))
σ
(
φ
(
n
))
<
a
.(As usual,
σ
(
n
)
=
∑
d
∣
n
d
\sigma(n) = \sum_{d\mid n} d
σ
(
n
)
=
∑
d
∣
n
d
and
φ
(
n
)
\varphi(n)
φ
(
n
)
is the number of integers
1
≤
m
≤
n
1\le m\le n
1
≤
m
≤
n
which are coprime with
n
n
n
.)Proposed by Deniz Can Karaçelebi
4
1
Hide problems
Geo with unnecessary condition
Let the circumcircle of a triangle
A
B
C
ABC
A
BC
be
Γ
\Gamma
Γ
. The tangents to
Γ
\Gamma
Γ
at
B
,
C
B,C
B
,
C
meet at point
E
E
E
. For a point
F
F
F
on line
B
C
BC
BC
which is not on the segment
B
C
BC
BC
, let the midpoint of
E
F
EF
EF
be
G
G
G
. Lines
G
B
,
G
C
GB,GC
GB
,
GC
meet
Γ
\Gamma
Γ
again at points
I
,
H
I,H
I
,
H
respectively. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Prove that the points
F
,
I
,
H
,
M
F,I,H,M
F
,
I
,
H
,
M
lie on a circle.Proposed by Mehmet Can Baştemir
3
1
Hide problems
Operations on complete graph
In a simple graph
G
G
G
, an operation is defined as taking two neighbor vertices
u
,
v
u,v
u
,
v
which have a common neighbor, deleting the edge between
u
,
v
u,v
u
,
v
and adding a new vertex
w
w
w
whose neighbors are exactly the common neighbors of
u
u
u
and
v
v
v
. Starting with the complete graph
G
=
K
n
G=K_n
G
=
K
n
where
n
≥
3
n\ge 3
n
≥
3
is a positive integer, find the maximum number of operations that can be applied.Proposed by Deniz Can Karaçelebi
2
1
Hide problems
Computational geo
In the plane, three distinct non-collinear points
A
,
B
,
C
A,B,C
A
,
B
,
C
are marked. In each step, Ege can do one of the following: [*] For marked points
X
,
Y
X,Y
X
,
Y
, mark the reflection of
X
X
X
across
Y
Y
Y
.[*]For distinct marked points
X
,
Y
,
Z
,
T
X,Y,Z,T
X
,
Y
,
Z
,
T
which do not form a parallelogram, mark the center of spiral similarity which takes segment
X
Y
XY
X
Y
to
Z
T
ZT
ZT
.[*] For distinct marked points
X
,
Y
,
Z
,
T
X,Y,Z,T
X
,
Y
,
Z
,
T
, mark the intersection of lines
X
Y
XY
X
Y
and
Z
T
ZT
ZT
. No matter how the points
A
,
B
,
C
A,B,C
A
,
B
,
C
are marked in the beginning, can Ege always mark, after finitely many moves,a) The circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
.b) The incenter of
△
A
B
C
\triangle ABC
△
A
BC
.Proposed by Deniz Can Karaçelebi
1
1
Hide problems
Colorful board with stones
Let
m
,
n
m,n
m
,
n
be positive integers. An
n
×
n
n\times n
n
×
n
board has rows and columns numbered
1
,
2
,
…
,
n
1,2,\dots,n
1
,
2
,
…
,
n
from left to right and top to bottom, respectively. This board is colored with colors
r
1
,
r
2
,
…
,
r
m
r_1,r_2,\dots,r_m
r
1
,
r
2
,
…
,
r
m
such that the cell at the intersection of
i
i
i
th row and
j
j
j
th column is colored with
r
i
+
j
−
1
r_{i+j-1}
r
i
+
j
−
1
where indices are taken modulo
m
m
m
. After the board is colored, Ahmet wants to put
n
n
n
stones to the board so that each row and column has exactly one stone, also he wants to put the same amount of stones to each color. Find all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
for which he can accomplish his goal. Proposed by Sena Başaran