MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Olympic Revenge
2023 Turkey Olympic Revenge
2023 Turkey Olympic Revenge
Part of
Turkey Olympic Revenge
Subcontests
(6)
5
1
Hide problems
A Game About Breaking The Most Cups
There are
10
10
10
cups, each having
10
10
10
pebbles in them. Two players
A
A
A
and
B
B
B
play a game, repeating the following in order each move:
∙
\bullet
∙
B
B
B
takes one pebble from each cup and redistributes them as
A
A
A
wishes.
∙
\bullet
∙
After
B
B
B
distributes the pebbles, he tells how many pebbles are in each cup to
A
A
A
. Then
B
B
B
destroys all the cups having no pebbles.
∙
\bullet
∙
B
B
B
switches the places of two cups without telling
A
A
A
.After finitely many moves,
A
A
A
can guarantee that
n
n
n
cups are destroyed. Find the maximum possible value of
n
n
n
. (Note that
A
A
A
doesn't see the cups while playing.)Proposed by Emre Osman
6
1
Hide problems
Isogonal conjugate on a fixed circle
In triangle
A
B
C
ABC
A
BC
,
D
D
D
is a variable point on line
B
C
BC
BC
. Points
E
,
F
E,F
E
,
F
are on segments
A
C
,
A
B
AC, AB
A
C
,
A
B
respectively such that
B
F
=
B
D
BF=BD
BF
=
B
D
and
C
D
=
C
E
CD=CE
C
D
=
CE
. Circles
(
A
E
F
)
(AEF)
(
A
EF
)
and
(
A
B
C
)
(ABC)
(
A
BC
)
meet again at
S
S
S
. Lines
E
F
EF
EF
and
B
C
BC
BC
meet at
P
P
P
and circles
(
P
D
S
)
(PDS)
(
P
D
S
)
and
(
A
E
F
)
(AEF)
(
A
EF
)
meet again at
Q
Q
Q
. Prove that, as
D
D
D
varies, isogonal conjugate of
Q
Q
Q
with respect to triangle
A
B
C
ABC
A
BC
lies on a fixed circle.Proposed by Serdar Bozdag
2
1
Hide problems
Conditional Geometry With Parallel Lines
Let
A
B
C
ABC
A
BC
be a triangle. A point
D
D
D
lies on line
B
C
BC
BC
and points
E
,
F
E,F
E
,
F
are taken on
A
C
,
A
B
AC,AB
A
C
,
A
B
such that
D
E
∥
A
B
DE \parallel AB
D
E
∥
A
B
and
D
F
∥
A
C
DF\parallel AC
D
F
∥
A
C
. Let
G
=
(
A
E
F
)
∩
(
A
B
C
)
≠
A
G = (AEF) \cap (ABC) \neq A
G
=
(
A
EF
)
∩
(
A
BC
)
=
A
and
I
=
(
D
E
F
)
∩
B
C
≠
D
I = (DEF) \cap BC\neq D
I
=
(
D
EF
)
∩
BC
=
D
. Let
H
H
H
and
O
O
O
denote the orthocenter and the circumcenter of triangle
D
E
F
DEF
D
EF
. Prove that
A
,
O
,
I
A,O,I
A
,
O
,
I
are collinear if and only if
G
,
H
,
I
G,H,I
G
,
H
,
I
are collinear.Proposed by Kaan Bilge
1
1
Hide problems
Non-Standard Functional Equation ft. Real Number c
Find all
c
∈
R
c\in \mathbb{R}
c
∈
R
such that there exists a function
f
:
R
→
R
f:\mathbb{R}\to \mathbb{R}
f
:
R
→
R
satisfying
(
f
(
x
)
+
1
)
(
f
(
y
)
+
1
)
=
f
(
x
+
y
)
+
f
(
x
y
+
c
)
(f(x)+1)(f(y)+1)=f(x+y)+f(xy+c)
(
f
(
x
)
+
1
)
(
f
(
y
)
+
1
)
=
f
(
x
+
y
)
+
f
(
x
y
+
c
)
for all
x
,
y
∈
R
x,y\in \mathbb{R}
x
,
y
∈
R
.Proposed by Kaan Bilge
3
1
Hide problems
Polynomial involving sum of digits
Find all polynomials
P
P
P
with integer coefficients such that
s
(
x
)
=
s
(
y
)
⟹
s
(
∣
P
(
x
)
∣
)
=
s
(
∣
P
(
y
)
∣
)
.
s(x)=s(y) \implies s(|P(x)|)=s(|P(y)|).
s
(
x
)
=
s
(
y
)
⟹
s
(
∣
P
(
x
)
∣
)
=
s
(
∣
P
(
y
)
∣
)
.
for all
x
,
y
∈
N
x,y\in \mathbb{N}
x
,
y
∈
N
. Note:
s
(
x
)
s(x)
s
(
x
)
denotes the sum of digits of
x
x
x
.Proposed by Şevket Onur YILMAZ
4
1
Hide problems
f(x)^2+2xf(y)+y^2 is a square
Find all functions
f
:
Z
→
Z
f: \mathbb{Z}\to \mathbb{Z}
f
:
Z
→
Z
such that for all integers
x
x
x
and
y
y
y
, the number
f
(
x
)
2
+
2
x
f
(
y
)
+
y
2
f(x)^2+2xf(y)+y^2
f
(
x
)
2
+
2
x
f
(
y
)
+
y
2
is a perfect square.Proposed by Barış Koyuncu