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National and Regional Contests
North Macedonia Contests
Memorial "Aleksandar Blazhevski-Cane"
2022 3rd Memorial "Aleksandar Blazhevski-Cane"
2022 3rd Memorial "Aleksandar Blazhevski-Cane"
Part of
Memorial "Aleksandar Blazhevski-Cane"
Subcontests
(6)
P6
1
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Centrally symmetric perfect matchings on a circle
For any integer
n
≥
1
n\geq1
n
≥
1
, we consider a set
P
2
n
P_{2n}
P
2
n
of
2
n
2n
2
n
points placed equidistantly on a circle. A perfect matching on this point set is comprised of
n
n
n
(straight-line) segments whose endpoints constitute
P
2
n
P_{2n}
P
2
n
. Let
M
n
\mathcal{M}_{n}
M
n
denote the set of all non-crossing perfect matchings on
P
2
n
P_{2n}
P
2
n
. A perfect matching
M
∈
M
n
M\in \mathcal{M}_{n}
M
∈
M
n
is said to be centrally symmetric, if it is invariant under point reflection at the circle center. Determine, as a function of
n
n
n
, the number of centrally symmetric perfect matchings within
M
n
\mathcal{M}_{n}
M
n
.Proposed by Mirko Petrusevski
P5
1
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Memorable numbers
We say that a positive integer
n
n
n
is memorable if it has a binary representation with strictly more
1
1
1
's than
0
0
0
's (for example
25
25
25
is memorable because
25
=
(
11001
)
2
25=(11001)_{2}
25
=
(
11001
)
2
has more
1
1
1
's than
0
0
0
's). Are there infinitely many memorable perfect squares?Proposed by Nikola Velov
P4
2
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Incircle diameter lemma concurrence
Let
A
B
C
ABC
A
BC
be an acute triangle with incircle
ω
\omega
ω
, incenter
I
I
I
, and
A
A
A
-excircle
ω
a
\omega_{a}
ω
a
. Let
ω
\omega
ω
and
ω
a
\omega_{a}
ω
a
meet
B
C
BC
BC
at
X
X
X
and
Y
Y
Y
, respectively. Let
Z
Z
Z
be the intersection point of
A
Y
AY
A
Y
and
ω
\omega
ω
which is closer to
A
A
A
. The point
H
H
H
is the foot of the altitude from
A
A
A
. Show that
H
Z
HZ
H
Z
,
I
Y
IY
I
Y
and
A
X
AX
A
X
are concurrent.Proposed by Nikola Velov
Partition into equal-element sets with divisibility condition
Find all positive integers
n
n
n
such that the set
S
=
{
1
,
2
,
3
,
…
2
n
}
S=\{1,2,3, \dots 2n\}
S
=
{
1
,
2
,
3
,
…
2
n
}
can be divided into
2
2
2
disjoint subsets
S
1
S_1
S
1
and
S
2
S_2
S
2
, i.e.
S
1
∩
S
2
=
∅
S_1 \cap S_2 = \emptyset
S
1
∩
S
2
=
∅
and
S
1
∪
S
2
=
S
S_1 \cup S_2 = S
S
1
∪
S
2
=
S
, such that each one of them has
n
n
n
elements, and the sum of the elements of
S
1
S_1
S
1
is divisible by the sum of the elements in
S
2
S_2
S
2
.Proposed by Viktor Simjanoski
P3
1
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An almost symmetric Diophantine Equation
Find all triplets of positive integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
such that
x
2
+
y
2
+
x
+
y
+
z
=
x
y
z
+
1
x^2 + y^2 + x + y + z = xyz + 1
x
2
+
y
2
+
x
+
y
+
z
=
x
yz
+
1
.Proposed by Viktor Simjanoski
P2
1
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Flashback to IMO 2021's inequality
Given an integer
n
≥
2
n\geq2
n
≥
2
, let
x
1
<
x
2
<
⋯
<
x
n
x_1<x_2<\cdots<x_n
x
1
<
x
2
<
⋯
<
x
n
and
y
1
<
y
2
<
⋯
<
y
n
y_1<y_2<\cdots<y_n
y
1
<
y
2
<
⋯
<
y
n
be positive reals. Prove that for every value
C
∈
(
−
2
,
2
)
C\in (-2,2)
C
∈
(
−
2
,
2
)
(by taking
y
n
+
1
=
y
1
y_{n+1}=y_1
y
n
+
1
=
y
1
) it holds that
∑
i
=
1
n
x
i
2
+
C
x
i
y
i
+
y
i
2
<
∑
i
=
1
n
x
i
2
+
C
x
i
y
i
+
1
+
y
i
+
1
2
\hspace{122px}\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_i+y_i^2}<\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_{i+1}+y_{i+1}^2}
∑
i
=
1
n
x
i
2
+
C
x
i
y
i
+
y
i
2
<
∑
i
=
1
n
x
i
2
+
C
x
i
y
i
+
1
+
y
i
+
1
2
.Proposed by Mirko Petrusevski
P1
2
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Chesslike squares with a Connect Four condition
A
6
×
6
6 \times 6
6
×
6
board is given such that each unit square is either red or green. It is known that there are no
4
4
4
adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A
2
×
2
2 \times 2
2
×
2
subsquare of the board is chesslike if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board.Proposed by Nikola Velov
Perpendicular bisector of an altitude in an isosceles triangle
Let
A
B
C
ABC
A
BC
be an acute triangle with altitude
A
D
AD
A
D
(
D
∈
B
C
D \in BC
D
∈
BC
). The line through
C
C
C
parallel to
A
B
AB
A
B
meets the perpendicular bisector of
A
D
AD
A
D
at
G
G
G
. Show that
A
C
=
B
C
AC = BC
A
C
=
BC
if and only if
∠
A
G
C
=
9
0
∘
\angle AGC = 90^{\circ}
∠
A
GC
=
9
0
∘
.