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Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2019 Macedonia Junior BMO TST
2019 Macedonia Junior BMO TST
Part of
JBMO TST - Macedonia
Subcontests
(5)
4
1
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2019 JBMO TST- North Macedonia, problem 4
Let the real numbers
a
a
a
,
b
b
b
, and
c
c
c
satisfy the equations
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
=
a
b
c
(a+b)(b+c)(c+a)=abc
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
=
ab
c
and
(
a
9
+
b
9
)
(
b
9
+
c
9
)
(
c
9
+
a
9
)
=
(
a
b
c
)
9
(a^9+b^9)(b^9+c^9)(c^9+a^9)=(abc)^9
(
a
9
+
b
9
)
(
b
9
+
c
9
)
(
c
9
+
a
9
)
=
(
ab
c
)
9
. Prove that at least one of
a
a
a
,
b
b
b
, and
c
c
c
equals
0
0
0
.
5
1
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2019 JBMO TST- North Macedonia, problem 5
Let
p
1
p_{1}
p
1
,
p
2
p_{2}
p
2
, ...,
p
k
p_{k}
p
k
be different prime numbers. Determine the number of positive integers of the form
p
1
α
1
p
2
α
2
.
.
.
p
k
α
k
p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}
p
1
α
1
p
2
α
2
...
p
k
α
k
,
α
i
\alpha_{i}
α
i
∈
\in
∈
N
\mathbb{N}
N
for which
α
1
α
2
.
.
.
α
k
=
p
1
p
2
.
.
.
p
k
\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}
α
1
α
2
...
α
k
=
p
1
p
2
...
p
k
.
3
1
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2019 JBMO TST- North Macedonia, problem 3
Define a colouring in tha plane the following way: - we pick a positive integer
m
m
m
; - let
K
1
K_{1}
K
1
,
K
2
K_{2}
K
2
, ...,
K
m
K_{m}
K
m
be different circles with nonzero radii such that
K
i
⊂
K
j
K_{i}\subset K_{j}
K
i
⊂
K
j
or
K
j
⊂
K
i
K_{j}\subset K_{i}
K
j
⊂
K
i
if
i
≠
j
i \neq j
i
=
j
; - the points in the plane that lie outside an arbitrary circle (one that is amongst the circles we pick) are coloured differently than the points that lie inside the circle. There are
2019
2019
2019
points in the plane such that any
3
3
3
of them are not collinear. Determine the maximum number of colours which we can use to colour the given points.
2
1
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2019 JBMO TST- North Macedonia, problem 2
Circles
ω
1
\omega_{1}
ω
1
and
ω
2
\omega_{2}
ω
2
intersect at points
A
A
A
and
B
B
B
. Let
t
1
t_{1}
t
1
and
t
2
t_{2}
t
2
be the tangents to
ω
1
\omega_{1}
ω
1
and
ω
2
\omega_{2}
ω
2
, respectively, at point
A
A
A
. Let the second intersection of
ω
1
\omega_{1}
ω
1
and
t
2
t_{2}
t
2
be
C
C
C
, and let the second intersection of
ω
2
\omega_{2}
ω
2
and
t
1
t_{1}
t
1
be
D
D
D
. Points
P
P
P
and
E
E
E
lie on the ray
A
B
AB
A
B
, such that
B
B
B
lies between
A
A
A
and
P
P
P
,
P
P
P
lies between
A
A
A
and
E
E
E
, and
A
E
=
2
⋅
A
P
AE = 2 \cdot AP
A
E
=
2
⋅
A
P
. The circumcircle to
△
B
C
E
\bigtriangleup BCE
△
BCE
intersects
t
2
t_{2}
t
2
again at point
Q
Q
Q
, whereas the circumcircle to
△
B
D
E
\bigtriangleup BDE
△
B
D
E
intersects
t
1
t_{1}
t
1
again at point
R
R
R
. Prove that points
P
P
P
,
Q
Q
Q
, and
R
R
R
are collinear.
1
1
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2019 JBMO TST- North Macedonia, problem 1
Determine all prime numbers of the form
1
+
2
p
+
3
p
+
.
.
.
+
p
p
1 + 2^p + 3^p +...+ p^p
1
+
2
p
+
3
p
+
...
+
p
p
where
p
p
p
is a prime number.