MathDB
Problems
Contests
National and Regional Contests
North Macedonia Contests
Macedonia National Olympiad
2023 Macedonian Mathematical Olympiad
2023 Macedonian Mathematical Olympiad
Part of
Macedonia National Olympiad
Subcontests
(5)
Problem 5
1
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Entropy of children on a round table
There are
n
n
n
boys and
n
n
n
girls sitting around a circular table, where
n
>
3
n>3
n
>
3
. In every move, we are allowed to swap the places of
2
2
2
adjacent children. The entropy of a configuration is the minimal number of moves such that at the end of them each child has at least one neighbor of the same gender.Find the maximal possible entropy over the set of all configurations.Proposed by Viktor Simjanoski
Problem 4
1
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Circle TaTbTc passes through the orthocenter
Let
A
B
C
ABC
A
BC
be a scalene acute triangle with orthocenter
H
H
H
. The circle with center
A
A
A
and radius
A
H
AH
A
H
meets the circumcircle of
B
H
C
BHC
B
H
C
at
T
a
≠
H
T_{a} \neq H
T
a
=
H
. Define
T
b
T_{b}
T
b
and
T
c
T_{c}
T
c
similarly. Show that
H
H
H
lies on the circumcircle of
T
a
T
b
T
c
T_{a}T_{b}T_{c}
T
a
T
b
T
c
.Proposed by Nikola Velov
Problem 3
1
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Friendship in the City of Gnomes
In a city of gnomes there are
1000
1000
1000
identical towers, each of which has
1000
1000
1000
stories, with exactly one gnome living on each story. Every gnome in the city wears a hat colored in one of
1000
1000
1000
possible colors and any two gnomes in the same tower have different hats. A pair of gnomes are friends if they wear hats of the same color, one of them lives in the
k
k
k
-th story of his tower and the other one in the
(
k
+
1
)
(k+1)
(
k
+
1
)
-st story of his tower. Determine the maximal possible number of pairs of gnomes which are friends.Proposed by Nikola Velov
Problem 2
1
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A little bit of fun with orders
Let
p
p
p
and
q
q
q
be odd prime numbers and
a
a
a
a positive integer so that
p
∣
a
q
+
1
p|a^q+1
p
∣
a
q
+
1
and
q
∣
a
p
+
1
q|a^p+1
q
∣
a
p
+
1
. Show that
p
∣
a
+
1
p|a+1
p
∣
a
+
1
or
q
∣
a
+
1
q|a+1
q
∣
a
+
1
.Proposed by Nikola Velov
Problem 1
1
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Functional equation with sum of squares
Determine all functions
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
we have:
x
f
(
x
+
y
)
+
y
f
(
y
−
x
)
=
f
(
x
2
+
y
2
)
.
xf(x+y)+yf(y-x) = f(x^2+y^2)\,.
x
f
(
x
+
y
)
+
y
f
(
y
−
x
)
=
f
(
x
2
+
y
2
)
.
Proposed by Nikola Velov