MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2016 JBMO TST - Turkey
2016 JBMO TST - Turkey
Part of
JBMO TST - Turkey
Subcontests
(8)
8
1
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Connected graph with 2016 vertices
Let
G
G
G
be a simple connected graph with
2016
2016
2016
vertices and
k
k
k
edges. We want to choose a set of vertices where there is no edge between them and delete all these chosen vertices (we delete both the vertices and all edges of these vertices) such that the remaining graph becomes unconnected. If we can do this task no matter how these
k
k
k
edges are arranged (by making the graph connected), find the maximal value of
k
k
k
.
7
1
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An equation involving two prime numbers
Find all pairs
(
p
,
q
)
(p, q)
(
p
,
q
)
of prime numbers satisfying
p
3
+
7
q
=
q
9
+
5
p
2
+
18
p
.
p^3+7q=q^9+5p^2+18p.
p
3
+
7
q
=
q
9
+
5
p
2
+
18
p
.
6
1
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An inequality with a condition
Prove that
(
x
4
+
y
)
(
y
4
+
z
)
(
z
4
+
x
)
≥
(
x
+
y
2
)
(
y
+
z
2
)
(
z
+
x
2
)
(x^4+y)(y^4+z)(z^4+x) \geq (x+y^2)(y+z^2)(z+x^2)
(
x
4
+
y
)
(
y
4
+
z
)
(
z
4
+
x
)
≥
(
x
+
y
2
)
(
y
+
z
2
)
(
z
+
x
2
)
for all positive real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
satisfying
x
y
z
≥
1
xyz \geq 1
x
yz
≥
1
.
5
1
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Show that two lines are perpendicular
In an acute triangle
A
B
C
ABC
A
BC
, the feet of the perpendiculars from
A
A
A
and
C
C
C
to the opposite sides are
D
D
D
and
E
E
E
, respectively. The line passing through
E
E
E
and parallel to
B
C
BC
BC
intersects
A
C
AC
A
C
at
F
F
F
, the line passing through
D
D
D
and parallel to
A
B
AB
A
B
intersects
A
C
AC
A
C
at
G
G
G
. The feet of the perpendiculars from
F
F
F
to
D
G
DG
D
G
and
G
E
GE
GE
are
K
K
K
and
L
L
L
, respectively.
K
L
KL
K
L
intersects
E
D
ED
E
D
at
M
M
M
. Prove that
F
M
⊥
E
D
FM \perp ED
FM
⊥
E
D
.
4
1
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Collinearity
In a trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
B
<
C
D
AB<CD
A
B
<
C
D
and
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
, the diagonals intersect each other at
E
E
E
. Let
F
F
F
be the midpoint of the arc
B
C
BC
BC
(not containing the point
E
E
E
) of the circumcircle of the triangle
E
B
C
EBC
EBC
. The lines
E
F
EF
EF
and
B
C
BC
BC
intersect at
G
G
G
. The circumcircle of the triangle
B
F
D
BFD
BF
D
intersects the ray
[
D
A
[DA
[
D
A
at
H
H
H
such that
A
∈
[
H
D
]
A \in [HD]
A
∈
[
HD
]
. The circumcircle of the triangle
A
H
B
AHB
A
H
B
intersects the lines
A
C
AC
A
C
and
B
D
BD
B
D
at
M
M
M
and
N
N
N
, respectively.
B
M
BM
BM
intersects
G
H
GH
G
H
at
P
P
P
,
G
N
GN
GN
intersects
A
C
AC
A
C
at
Q
Q
Q
. Prove that the points
P
,
Q
,
D
P, Q, D
P
,
Q
,
D
are collinear.
3
1
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Primes and divisibility
Let
n
n
n
be a positive integer,
p
p
p
and
q
q
q
be prime numbers such that pq \mid n^p+2 \text{and} n+2 \mid n^p+q^p. Prove that there exists a positive integer
m
m
m
satisfying
q
∣
4
m
⋅
n
+
2
q \mid 4^m \cdot n +2
q
∣
4
m
⋅
n
+
2
.
2
1
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A game on a pyramid
A and B plays a game on a pyramid whose base is a
2016
2016
2016
-gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the
k
k
k
colors such that none of the sides with a common vertex have the same color. If A starts the game, find the minimal value of
k
k
k
for which
B
B
B
can guarantee that all sides are colored.
1
1
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Equation System
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of real numbers satisfying the equations \begin{align*} x^2+y&=xy^2 \\ 2x^2y+y^2&=x+y+3xy. \end{align*}