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Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2018 Macedonia JBMO TST
2018 Macedonia JBMO TST
Part of
JBMO TST - Macedonia
Subcontests
(5)
5
1
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2018 JBMO TST- Macedonia, problem 5
A regular
2018
2018
2018
-gon is inscribed in a circle. The numbers
1
,
2
,
.
.
.
,
2018
1, 2, ..., 2018
1
,
2
,
...
,
2018
are arranged on the vertices of the
2018
2018
2018
-gon, with each vertex having one number on it, such that the sum of any
2
2
2
neighboring numbers (
2
2
2
numbers are neighboring if the vertices they are on lie on a side of the polygon) equals the sum of the
2
2
2
numbers that are on the antipodes of those
2
2
2
vertices (with respect to the given circle). Determine the number of different arrangements of the numbers. (Two arrangements are identical if you can get from one of them to the other by rotating around the center of the circle).
4
1
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2018 JBMO TST- Macedonia, problem 4
Determine all pairs
(
p
,
q
)
(p, q)
(
p
,
q
)
,
p
,
q
∈
N
p, q \in \mathbb {N}
p
,
q
∈
N
, such that
(
p
+
1
)
p
−
1
+
(
p
−
1
)
p
+
1
=
q
q
(p + 1)^{p - 1} + (p - 1)^{p + 1} = q^q
(
p
+
1
)
p
−
1
+
(
p
−
1
)
p
+
1
=
q
q
.
3
1
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2018 JBMO TST- Macedonia, problem 3
Let
x
x
x
,
y
y
y
, and
z
z
z
be positive real numbers such that
x
+
y
+
z
=
1
x + y + z = 1
x
+
y
+
z
=
1
. Prove that
(
x
+
y
)
3
z
+
(
y
+
z
)
3
x
+
(
z
+
x
)
3
y
+
9
x
y
z
≥
9
(
x
y
+
y
z
+
z
x
)
\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx)
z
(
x
+
y
)
3
+
x
(
y
+
z
)
3
+
y
(
z
+
x
)
3
+
9
x
yz
≥
9
(
x
y
+
yz
+
z
x
)
.When does equality hold?
2
1
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2018 JBMO TST- Macedonia, problem 2
We are given a semicircle
k
k
k
with center
O
O
O
and diameter
A
B
AB
A
B
. Let
C
C
C
be a point on
k
k
k
such that
C
O
⊥
A
B
CO \bot AB
CO
⊥
A
B
. The bisector of
∠
A
B
C
\angle ABC
∠
A
BC
intersects
k
k
k
at point
D
D
D
. Let
E
E
E
be a point on
A
B
AB
A
B
such that
D
E
⊥
A
B
DE \bot AB
D
E
⊥
A
B
and let
F
F
F
be the midpoint of
C
B
CB
CB
. Prove that the quadrilateral
E
F
C
D
EFCD
EFC
D
is cyclic.
1
1
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2018 JBMO TST- Macedonia, problem 1
Determine all positive integers
n
>
2
n>2
n
>
2
, such that
n
=
a
3
+
b
3
n = a^3 + b^3
n
=
a
3
+
b
3
, where
a
a
a
is the smallest positive divisor of
n
n
n
greater than
1
1
1
and
b
b
b
is an arbitrary positive divisor of
n
n
n
.