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Problems
Contests
National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2022 Junior Macedonian Mathematical Olympiad
2022 Junior Macedonian Mathematical Olympiad
Part of
JBMO TST - Macedonia
Subcontests
(5)
P4
1
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Covering triangular board of side length 2022 with figures
An equilateral triangle
T
T
T
with side length
2022
2022
2022
is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures shown on the image below, which consist of
4
4
4
equilateral unit triangles and can be rotated by any angle
k
⋅
6
0
∘
k \cdot 60^{\circ}
k
⋅
6
0
∘
for
k
∈
{
1
,
2
,
3
,
4
,
5
}
k \in \left \{1,2,3,4,5 \right \}
k
∈
{
1
,
2
,
3
,
4
,
5
}
. The covering satisfies the following conditions:
1
)
1)
1
)
It is possible not to use figures of some type and it is possible to use several figures of the same type. The unit triangles in the figures correspond to the unit triangles in the grid.
2
)
2)
2
)
Every unit triangle in the grid is covered, no two figures overlap and every figure is fully contained in
T
T
T
.Determine the smallest possible number of figures of type
1
1
1
that can be used in such a covering.Proposed by Ilija Jovcheski
P5
1
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Two polynomials cannot be perfect cubes at the same time
Let
n
n
n
be a positive integer such that
n
5
+
n
3
+
2
n
2
+
2
n
+
2
n^5+n^3+2n^2+2n+2
n
5
+
n
3
+
2
n
2
+
2
n
+
2
is a perfect cube. Prove that
2
n
2
+
n
+
2
2n^2+n+2
2
n
2
+
n
+
2
is not a perfect cube.Proposed by Anastasija Trajanova
P3
1
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Circle centered at orthocenter meets side of triangle
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle with orthocenter
H
H
H
. The circle
Γ
\Gamma
Γ
with center
H
H
H
and radius
A
H
AH
A
H
meets the lines
A
B
AB
A
B
and
A
C
AC
A
C
at the points
E
E
E
and
F
F
F
respectively. Let
E
′
E'
E
′
,
F
′
F'
F
′
and
H
′
H'
H
′
be the reflections of the points
E
E
E
,
F
F
F
and
H
H
H
with respect to the line
B
C
BC
BC
, respectively. Prove that the points
A
A
A
,
E
′
E'
E
′
,
F
′
F'
F
′
and
H
′
H'
H
′
lie on a circle.Proposed by Jasna Ilieva
P2
1
Hide problems
Inequality with one variable rational functions
Let
a
a
a
,
b
b
b
and
c
c
c
be positive real numbers such that
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
. Prove the inequality
a
3
a
2
+
1
+
b
3
b
2
+
1
+
c
3
c
2
+
1
≥
3
2
.
\frac{a^3}{a^2+1}+\frac{b^3}{b^2+1}+\frac{c^3}{c^2+1} \geq \frac{3}{2}.
a
2
+
1
a
3
+
b
2
+
1
b
3
+
c
2
+
1
c
3
≥
2
3
.
Proposed by Anastasija Trajanova
P1
1
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Diophantine equation with factorial
Determine all positive integers
a
a
a
,
b
b
b
and
c
c
c
which satisfy the equation
a
2
+
b
2
+
1
=
c
!
.
a^2+b^2+1=c!.
a
2
+
b
2
+
1
=
c
!
.
Proposed by Nikola Velov