MathDB
Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2020 Junior Macedonian National Olympiad
2020 Junior Macedonian National Olympiad
Part of
JBMO TST - Macedonia
Subcontests
(5)
5
1
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2020 Macedonian Junior BMO TST - P5
Let
T
T
T
be a triangle whose vertices have integer coordinates, such that each side of
T
T
T
contains exactly
m
m
m
points with integer coordinates. If the area of
T
T
T
is less than
2020
2020
2020
, determine the largest possible value of
m
m
m
.
4
1
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2020 Macedonian Junior BMO TST- P4
Let
A
B
C
ABC
A
BC
be an isosceles triangle with base
A
C
AC
A
C
. Points
D
D
D
and
E
E
E
are chosen on the sides
A
C
AC
A
C
and
B
C
BC
BC
, respectively, such that
C
D
=
D
E
CD = DE
C
D
=
D
E
. Let
H
,
J
,
H, J,
H
,
J
,
and
K
K
K
be the midpoints of
D
E
,
A
E
,
DE, AE,
D
E
,
A
E
,
and
B
D
BD
B
D
, respectively. The circumcircle of triangle
D
H
K
DHK
DHK
intersects
A
D
AD
A
D
at point
F
F
F
, whereas the circumcircle of triangle
H
E
J
HEJ
H
E
J
intersects
B
E
BE
BE
at
G
G
G
. The line through
K
K
K
parallel to
A
C
AC
A
C
intersects
A
B
AB
A
B
at
I
I
I
. Let
I
H
∩
G
F
=
IH \cap GF =
I
H
∩
GF
=
{
M
M
M
}. Prove that
J
,
M
,
J, M,
J
,
M
,
and
K
K
K
are collinear points.
3
1
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2020 Macedonian Junior BMO TST- P3
Solve the following equation in the set of integers
x
5
+
2
=
3
⋅
10
1
y
x^5 + 2 = 3 \cdot 101^y
x
5
+
2
=
3
⋅
10
1
y
.
2
1
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2020 Macedonian Junior BMO TST- P2
Let
x
,
y
,
x, y,
x
,
y
,
and
z
z
z
be positive real numbers such that
x
y
+
y
z
+
z
x
=
27
xy + yz + zx = 27
x
y
+
yz
+
z
x
=
27
. Prove that
x
+
y
+
z
≥
3
x
y
z
x + y + z \ge \sqrt{3xyz}
x
+
y
+
z
≥
3
x
yz
.When does equality hold?
1
1
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2020 Macedonian Junior BMO TST- P1
Let
S
S
S
be the set of all positive integers
n
n
n
such that each of the numbers
n
+
1
n + 1
n
+
1
,
n
+
3
n + 3
n
+
3
,
n
+
4
n + 4
n
+
4
,
n
+
5
n + 5
n
+
5
,
n
+
6
n + 6
n
+
6
, and
n
+
8
n + 8
n
+
8
is composite. Determine the largest integer
k
k
k
with the following property: For each
n
∈
S
n \in S
n
∈
S
there exist at least
k
k
k
consecutive composite integers in the set {
n
,
n
+
1
,
n
+
2
,
n
+
3
,
n
+
4
,
n
+
5
,
n
+
6
,
n
+
7
,
n
+
8
,
n
+
9
n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9
n
,
n
+
1
,
n
+
2
,
n
+
3
,
n
+
4
,
n
+
5
,
n
+
6
,
n
+
7
,
n
+
8
,
n
+
9
}.