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Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2019 Turkey Junior National Olympiad
2019 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
4
1
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Combinatorics about 54 boxes
There are
27
27
27
cardboard and
27
27
27
plastic boxes. There are balls of certain colors inside the boxes. It is known that any two boxes of the same kind do not have a ball with the same color. Boxes of different kind have at least one ball of the same color. At each step we select two boxes that have a ball of same color and switch this common color into any other color we wish. Find the smallest number
n
n
n
of moves required.
3
1
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Geometry about incircle
In
A
B
C
ABC
A
BC
triangle
I
I
I
is incenter and incircle of
A
B
C
ABC
A
BC
tangents to
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
at
D
,
E
,
F
D,E,F
D
,
E
,
F
, respectively. If
A
I
AI
A
I
intersects
D
E
DE
D
E
and
D
F
DF
D
F
at
P
P
P
and
Q
Q
Q
, prove that the circumcenter of
D
P
Q
DPQ
D
PQ
triangle is the midpoint of
B
C
BC
BC
.
2
1
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Inequality about x^5+y^5+z^5
x
,
y
,
z
∈
R
+
x,y,z \in \mathbb{R}^+
x
,
y
,
z
∈
R
+
and
x
5
+
y
5
+
z
5
=
x
y
+
y
z
+
z
x
x^5+y^5+z^5=xy+yz+zx
x
5
+
y
5
+
z
5
=
x
y
+
yz
+
z
x
. Prove that
3
≥
x
2
y
+
y
2
z
+
z
2
x
3 \ge x^2y+y^2z+z^2x
3
≥
x
2
y
+
y
2
z
+
z
2
x
1
1
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Easiest number theory question ever
Solve
2
a
2
+
3
a
−
44
=
3
p
n
2a^2+3a-44=3p^n
2
a
2
+
3
a
−
44
=
3
p
n
in positive integers where
p
p
p
is a prime.