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Contests
National and Regional Contests
North Macedonia Contests
Macedonian Team Selection Test
2023 Macedonian Team Selection Test
2023 Macedonian Team Selection Test
Part of
Macedonian Team Selection Test
Subcontests
(6)
Problem 6
1
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Lucky and Jinx color segments of a polygon
Lucky and Jinx were given a paper with
2023
2023
2023
points arranged as the vertices of a regular polygon. They were then tasked to color all the segments connecting these points such that no triangle formed with these points has all edges in the same color, nor in three different colors and no quadrilateral (not necessarily convex) has all edges in the same color. After the coloring it was determined that Jinx used at least two more colors than Lucky. How many colors did each of them use?Proposed by Ilija Jovcheski
Problem 5
1
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Maximal value of Q(2023) under class of polynomials
Let
Q
(
x
)
=
a
2023
x
2023
+
a
2022
x
2022
+
⋯
+
a
1
x
+
a
0
∈
Z
[
x
]
Q(x) = a_{2023}x^{2023}+a_{2022}x^{2022}+\dots+a_{1}x+a_{0} \in \mathbb{Z}[x]
Q
(
x
)
=
a
2023
x
2023
+
a
2022
x
2022
+
⋯
+
a
1
x
+
a
0
∈
Z
[
x
]
be a polynomial with integer coefficients. For an odd prime number
p
p
p
we define the polynomial
Q
p
(
x
)
=
a
2023
p
−
2
x
2023
+
a
2022
p
−
2
x
2022
+
⋯
+
a
1
p
−
2
x
+
a
0
p
−
2
.
Q_{p}(x) = a_{2023}^{p-2}x^{2023}+a_{2022}^{p-2}x^{2022}+\dots+a_{1}^{p-2}x+a_{0}^{p-2}.
Q
p
(
x
)
=
a
2023
p
−
2
x
2023
+
a
2022
p
−
2
x
2022
+
⋯
+
a
1
p
−
2
x
+
a
0
p
−
2
.
Assume that there exist infinitely primes
p
p
p
such that
Q
p
(
x
)
−
Q
(
x
)
p
\frac{Q_{p}(x)-Q(x)}{p}
p
Q
p
(
x
)
−
Q
(
x
)
is an integer for all
x
∈
Z
x \in \mathbb{Z}
x
∈
Z
. Determine the largest possible value of
Q
(
2023
)
Q(2023)
Q
(
2023
)
over all such polynomials
Q
Q
Q
.Proposed by Nikola Velov
Problem 4
1
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Similar to BMO SL 2022 A6
Let
f
:
R
2
→
R
f: \mathbb{R}^2 \to \mathbb{R}
f
:
R
2
→
R
be a function satisfying the following property: If
A
,
B
,
C
∈
R
2
A, B, C \in \mathbb{R}^2
A
,
B
,
C
∈
R
2
are the vertices of an equilateral triangle with sides of length
1
1
1
, then
f
(
A
)
+
f
(
B
)
+
f
(
C
)
=
0.
f(A) + f(B) + f(C) = 0.
f
(
A
)
+
f
(
B
)
+
f
(
C
)
=
0.
Show that
f
(
x
)
=
0
f(x) = 0
f
(
x
)
=
0
for all
x
∈
R
2
x \in \mathbb{R}^2
x
∈
R
2
.Proposed by Ilir Snopce
Problem 3
1
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Maximal value of 2023 under monotonic function
Let
f
:
N
→
N
f:\mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
be a monotonically increasing function over the natural numbers, such that
f
(
f
(
n
)
)
=
n
2
f(f(n))=n^{2}
f
(
f
(
n
))
=
n
2
. What is the smallest, and what is the largest value that
f
(
2023
)
f(2023)
f
(
2023
)
can take?Proposed by Ilija Jovcheski
Problem 2
1
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Geometric inequality with reflections and tangents
Let
A
B
C
ABC
A
BC
be an acute triangle such that
A
B
<
A
C
AB<AC
A
B
<
A
C
and
A
B
<
B
C
AB<BC
A
B
<
BC
. Let
P
P
P
be a point on the segment
B
C
BC
BC
such that
∠
A
P
B
=
∠
B
A
C
\angle APB = \angle BAC
∠
A
PB
=
∠
B
A
C
. The tangent to the circumcircle of triangle
A
B
C
ABC
A
BC
at
A
A
A
meets the circumcircle of triangle
A
P
B
APB
A
PB
at
Q
≠
A
Q \neq A
Q
=
A
. Let
Q
′
Q'
Q
′
be the reflection of
Q
Q
Q
with respect to the midpoint of
A
B
AB
A
B
. The line
P
Q
PQ
PQ
meets the segment
A
Q
′
AQ'
A
Q
′
at
S
S
S
. Prove that
1
A
B
+
1
A
C
>
1
C
S
.
\frac{1}{AB}+\frac{1}{AC} > \frac{1}{CS}.
A
B
1
+
A
C
1
>
CS
1
.
Proposed by Nikola Velov
Problem 1
1
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Constructing numbers with large prime divisors
Let
s
(
n
)
s(n)
s
(
n
)
denote the smallest prime divisor and
d
(
n
)
d(n)
d
(
n
)
denote the number of positive divisors of a positive integer
n
>
1
n>1
n
>
1
. Is it possible to choose
2023
2023
2023
positive integers
a
1
,
a
2
,
.
.
.
,
a
2023
a_{1},a_{2},...,a_{2023}
a
1
,
a
2
,
...
,
a
2023
with
a
1
<
a
2
−
1
<
.
.
.
<
a
2023
−
2022
a_{1}<a_{2}-1<...<a_{2023}-2022
a
1
<
a
2
−
1
<
...
<
a
2023
−
2022
such that for all
k
=
1
,
.
.
.
,
2022
k=1,...,2022
k
=
1
,
...
,
2022
we have
d
(
a
k
+
1
−
a
k
−
1
)
>
202
3
k
d(a_{k+1}-a_{k}-1)>2023^{k}
d
(
a
k
+
1
−
a
k
−
1
)
>
202
3
k
and
s
(
a
k
+
1
−
a
k
)
>
202
3
k
s(a_{k+1}-a_{k}) > 2023^{k}
s
(
a
k
+
1
−
a
k
)
>
202
3
k
?Proposed by Nikola Velov