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Junior Balkan MO
2019 Junior Balkan MO
2019 Junior Balkan MO
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Junior Balkan MO
Subcontests
(4)
4
1
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2019 Junior Balkan MO, Problem 4
A
5
×
100
5 \times 100
5
×
100
table is divided into
500
500
500
unit square cells, where
n
n
n
of them are coloured black and the rest are coloured white. Two unit square cells are called adjacent if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of
n
n
n
.
3
1
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2019 Junior Balkan MO, Problem 3
Triangle
A
B
C
ABC
A
BC
is such that
A
B
<
A
C
AB < AC
A
B
<
A
C
. The perpendicular bisector of side
B
C
BC
BC
intersects lines
A
B
AB
A
B
and
A
C
AC
A
C
at points
P
P
P
and
Q
Q
Q
, respectively. Let
H
H
H
be the orthocentre of triangle
A
B
C
ABC
A
BC
, and let
M
M
M
and
N
N
N
be the midpoints of segments
B
C
BC
BC
and
P
Q
PQ
PQ
, respectively. Prove that lines
H
M
HM
H
M
and
A
N
AN
A
N
meet on the circumcircle of
A
B
C
ABC
A
BC
.
2
1
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2019 Junior Balkan MO, Problem 2
Let
a
a
a
,
b
b
b
be two distinct real numbers and let
c
c
c
be a positive real numbers such that
a
4
−
2019
a
=
b
4
−
2019
b
=
c
a^4 - 2019a = b^4 - 2019b = c
a
4
−
2019
a
=
b
4
−
2019
b
=
c
.Prove that
−
c
<
a
b
<
0
- \sqrt{c} < ab < 0
−
c
<
ab
<
0
.
1
1
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2019 Junior Balkan MO, Problem 1
Find all prime numbers
p
p
p
for which there exist positive integers
x
x
x
,
y
y
y
, and
z
z
z
such that the number
x
p
+
y
p
+
z
p
−
x
−
y
−
z
x^p + y^p + z^p - x - y - z
x
p
+
y
p
+
z
p
−
x
−
y
−
z
is a product of exactly three distinct prime numbers.