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Problems
Contests
National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2023 JBMO TST - Turkey
2023 JBMO TST - Turkey
Part of
JBMO TST - Turkey
Subcontests
(4)
4
2
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A game with balls and boxes
Initially, Aslı distributes
1000
1000
1000
balls to
30
30
30
boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?
number of divisiors with divisibility
For a prime number
p
p
p
. Can the number of n positive integers that make the expression
n
3
+
n
p
+
1
n
+
p
+
1
\dfrac{n^3+np+1}{n+p+1}
n
+
p
+
1
n
3
+
n
p
+
1
an integer be
777
777
777
?
3
2
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One variable function in Turkey JBMO TST
Find all
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that
f
(
x
+
f
(
x
)
)
=
f
(
−
x
)
f(x+f(x))=f(-x)
f
(
x
+
f
(
x
))
=
f
(
−
x
)
and for all
x
≤
y
x \leq y
x
≤
y
it satisfies
f
(
x
)
≤
f
(
y
)
f(x) \leq f(y)
f
(
x
)
≤
f
(
y
)
Paralell lines and tangent to (ABC)
Let
A
B
C
ABC
A
BC
is triangle and
D
∈
A
B
D \in AB
D
∈
A
B
,
E
∈
A
C
E \in AC
E
∈
A
C
such that
D
E
/
/
B
C
DE//BC
D
E
//
BC
. Let
(
A
B
C
)
(ABC)
(
A
BC
)
meets with
(
B
D
E
)
(BDE)
(
B
D
E
)
and
(
C
D
E
)
(CDE)
(
C
D
E
)
at the second time
K
,
L
K,L
K
,
L
respectively.
B
K
BK
B
K
and
C
L
CL
C
L
intersect at
T
T
T
. Prove that
T
A
TA
T
A
is tangent to the
(
A
B
C
)
(ABC)
(
A
BC
)
2
2
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Concurrent lines with midpoint
Let
A
B
C
ABC
A
BC
is acute angled triangle and
K
,
L
K,L
K
,
L
is points on
A
C
,
B
C
AC,BC
A
C
,
BC
respectively such that
∠
A
K
B
=
∠
A
L
B
\angle{AKB}=\angle{ALB}
∠
A
K
B
=
∠
A
L
B
.
P
P
P
is intersection of
A
L
AL
A
L
and
B
K
BK
B
K
and
Q
Q
Q
is the midpoint of segment
K
L
KL
K
L
. Let
T
,
S
T,S
T
,
S
are the intersection
A
L
,
B
K
AL,BK
A
L
,
B
K
with
(
A
B
C
)
(ABC)
(
A
BC
)
respectively. Prove that
T
K
,
S
L
,
P
Q
TK,SL,PQ
T
K
,
S
L
,
PQ
are concurrent.
marbles placed chessboard
A marble is placed on each
33
33
33
unit square of a
10
∗
10
10*10
10
∗
10
chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?
1
2
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Easy factorization in JBMO TST
Let
n
,
k
n,k
n
,
k
are integers and
p
p
p
is a prime number. Find all
(
n
,
k
,
p
)
(n,k,p)
(
n
,
k
,
p
)
such that
∣
6
n
2
−
17
n
−
39
∣
=
p
k
|6n^2-17n-39|=p^k
∣6
n
2
−
17
n
−
39∣
=
p
k
Inequalities in Turkey JBMO TST
Prove that for all
a
,
b
,
c
a,b,c
a
,
b
,
c
positive real numbers
a
4
+
1
b
3
+
b
2
+
b
+
b
4
+
1
c
3
+
c
2
+
c
+
c
4
+
1
a
3
+
a
2
+
a
≥
2
\dfrac{a^4+1}{b^3+b^2+b}+\dfrac{b^4+1}{c^3+c^2+c}+\dfrac{c^4+1}{a^3+a^2+a} \ge 2
b
3
+
b
2
+
b
a
4
+
1
+
c
3
+
c
2
+
c
b
4
+
1
+
a
3
+
a
2
+
a
c
4
+
1
≥
2