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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2013 Turkey MO (2nd round)
2013 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(3)
3
2
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Simple connected graph with 100 vertices and 2013 edges
Let
G
G
G
be a simple, undirected, connected graph with
100
100
100
vertices and
2013
2013
2013
edges. It is given that there exist two vertices
A
A
A
and
B
B
B
such that it is not possible to reach
A
A
A
from
B
B
B
using one or two edges. We color all edges using
n
n
n
colors, such that for all pairs of vertices, there exists a way connecting them with a single color. Find the maximum value of
n
n
n
.
n points with integer distances
Let
n
n
n
be a positive integer and
P
1
,
P
2
,
…
,
P
n
P_1, P_2, \ldots, P_n
P
1
,
P
2
,
…
,
P
n
be different points on the plane such that distances between them are all integers. Furthermore, we know that the distances
P
i
P
1
,
P
i
P
2
,
…
,
P
i
P
n
P_iP_1, P_iP_2, \ldots, P_iP_n
P
i
P
1
,
P
i
P
2
,
…
,
P
i
P
n
forms the same sequence for all
i
=
1
,
2
,
…
,
n
i=1,2, \ldots, n
i
=
1
,
2
,
…
,
n
when these numbers are arranged in a non-decreasing order. Find all possible values of
n
n
n
.
2
2
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Perfect cube
Let
m
m
m
be a positive integer. a. Show that there exist infinitely many positive integers
k
k
k
such that
1
+
k
m
3
1+km^3
1
+
k
m
3
is a perfect cube and
1
+
k
n
3
1+kn^3
1
+
k
n
3
is not a perfect cube for all positive integers
n
<
m
n<m
n
<
m
. b. Let
m
=
p
r
m=p^r
m
=
p
r
where
p
≡
2
(
m
o
d
3
)
p \equiv 2 \pmod 3
p
≡
2
(
mod
3
)
is a prime number and
r
r
r
is a positive integer. Find all numbers
k
k
k
satisfying the condition in part a.
Find the maximum value of M
Find the maximum value of
M
M
M
for which for all positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
we have
a
3
+
b
3
+
c
3
−
3
a
b
c
≥
M
(
a
b
2
+
b
c
2
+
c
a
2
−
3
a
b
c
)
a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc)
a
3
+
b
3
+
c
3
−
3
ab
c
≥
M
(
a
b
2
+
b
c
2
+
c
a
2
−
3
ab
c
)
1
2
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Show that four points are concyclic
The circle
ω
1
\omega_1
ω
1
with diameter
[
A
B
]
[AB]
[
A
B
]
and the circle
ω
2
\omega_2
ω
2
with center
A
A
A
intersects at points
C
C
C
and
D
D
D
. Let
E
E
E
be a point on the circle
ω
2
\omega_2
ω
2
, which is outside
ω
1
\omega_1
ω
1
and at the same side as
C
C
C
with respect to the line
A
B
AB
A
B
. Let the second point of intersection of the line
B
E
BE
BE
with
ω
2
\omega_2
ω
2
be
F
F
F
. For a point
K
K
K
on the circle
ω
1
\omega_1
ω
1
which is on the same side as
A
A
A
with respect to the diameter of
ω
1
\omega_1
ω
1
passing through
C
C
C
we have
2
⋅
C
K
⋅
A
C
=
C
E
⋅
A
B
2\cdot CK \cdot AC = CE \cdot AB
2
⋅
C
K
⋅
A
C
=
CE
⋅
A
B
. Let the second point of intersection of the line
K
F
KF
K
F
with
ω
1
\omega_1
ω
1
be
L
L
L
. Show that the symmetric of the point
D
D
D
with respect to the line
B
E
BE
BE
is on the circumcircle of the triangle
L
F
C
LFC
L
FC
.
2^n+n=m!
Find all positive integers
m
m
m
and
n
n
n
satisfying
2
n
+
n
=
m
!
2^n+n=m!
2
n
+
n
=
m
!
.