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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2018 Turkey Junior National Olympiad
2018 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
3
1
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Geometry problem about ABC and its circumcircle
In an acute
A
B
C
ABC
A
BC
triangle which has a circumcircle center called
O
O
O
, there is a line that perpendiculars to
A
O
AO
A
O
line cuts
[
A
B
]
[AB]
[
A
B
]
and
[
A
C
]
[AC]
[
A
C
]
respectively on
D
D
D
and
E
E
E
points. There is a point called
K
K
K
that is different from
A
O
AO
A
O
and
B
C
BC
BC
's junction point on
[
B
C
]
[BC]
[
BC
]
.
A
K
AK
A
K
line cuts the circumcircle of
A
D
E
ADE
A
D
E
on
L
L
L
that is different from
A
A
A
.
M
M
M
is the symmetry point of
A
A
A
according to
D
E
DE
D
E
line. Prove that
K
K
K
,
L
L
L
,
M
M
M
,
O
O
O
are circular.
2
1
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A combinatorics question about the chess table
We are placing rooks on a
n
⋅
n
n \cdot n
n
⋅
n
chess table that providing this condition: Every two rooks will threaten an empty square at least. What is the most number of rooks?
4
1
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İnequality for all x,y,z positive real numbers
For all
x
,
y
,
z
x,y,z
x
,
y
,
z
positive real numbers, find the all
c
c
c
positive real numbers that providing
x
3
y
+
y
3
z
+
z
3
x
x
+
y
+
z
+
4
c
x
y
z
≥
2
c
+
2
\frac{x^3y+y^3z+z^3x}{x+y+z}+\frac{4c}{xyz}\ge2c+2
x
+
y
+
z
x
3
y
+
y
3
z
+
z
3
x
+
x
yz
4
c
≥
2
c
+
2
1
1
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Common divisors of a,b anda 2a+3b
Let
s
(
n
)
s(n)
s
(
n
)
be the number of positive integer divisors of
n
n
n
. Find the all positive values of
k
k
k
that is providing
k
=
s
(
a
)
=
s
(
b
)
=
s
(
2
a
+
3
b
)
k=s(a)=s(b)=s(2a+3b)
k
=
s
(
a
)
=
s
(
b
)
=
s
(
2
a
+
3
b
)
.