MathDB
Problems
Contests
National and Regional Contests
North Macedonia Contests
Memorial "Aleksandar Blazhevski-Cane"
2023 4th Memorial "Aleksandar Blazhevski-Cane"
2023 4th Memorial "Aleksandar Blazhevski-Cane"
Part of
Memorial "Aleksandar Blazhevski-Cane"
Subcontests
(6)
P6
1
Hide problems
A nice functional divisibility problem
Denote by
N
\mathbb{N}
N
the set of positive integers. Find all functions
f
:
N
→
N
f:\mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
such that: • For all positive integers
a
>
202
3
2023
a> 2023^{2023}
a
>
202
3
2023
it holds that
f
(
a
)
≤
a
f(a) \leq a
f
(
a
)
≤
a
. •
a
2
f
(
b
)
+
b
2
f
(
a
)
f
(
a
)
+
f
(
b
)
\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}
f
(
a
)
+
f
(
b
)
a
2
f
(
b
)
+
b
2
f
(
a
)
is a positive integer for all
a
,
b
∈
N
a,b \in \mathbb{N}
a
,
b
∈
N
.Proposed by Nikola Velov
P5
1
Hide problems
Trivial double counting- graph edition
There are
1000
1000
1000
students in a school. Every student has exactly
4
4
4
friends. A group of three students
{
A
,
B
,
C
}
\left \{A,B,C \right \}
{
A
,
B
,
C
}
is said to be a friendly triplet if any two students in the group are friends. Determine the maximal possible number of friendly triplets.Proposed by Nikola Velov
P4
2
Hide problems
A familiar segment sum condition in a cyclic quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral such that
A
B
=
A
D
+
B
C
AB = AD + BC
A
B
=
A
D
+
BC
and
C
D
<
A
B
CD < AB
C
D
<
A
B
. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
P
P
P
, while the lines
A
D
AD
A
D
and
B
C
BC
BC
intersect at
Q
Q
Q
. The angle bisector of
∠
A
P
B
\angle APB
∠
A
PB
meets
A
B
AB
A
B
at
T
T
T
. Show that the circumcenter of the triangle
C
T
D
CTD
CT
D
lies on the circumcircle of the triangle
C
Q
D
CQD
CQ
D
.Proposed by Nikola Velov
Hidden elliptic curve in a junior level problem
Does the equation
z
(
y
−
x
)
(
x
+
y
)
=
x
3
z(y-x)(x+y)=x^3
z
(
y
−
x
)
(
x
+
y
)
=
x
3
have finitely many solutions in the set of positive integers?Proposed by Nikola Velov
P3
1
Hide problems
A bisector in a complete cyclic quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral inscribed in a circle
ω
\omega
ω
with center
O
O
O
. The lines
A
D
AD
A
D
and
B
C
BC
BC
meet at
E
E
E
, while the lines
A
B
AB
A
B
and
C
D
CD
C
D
meet at
F
F
F
. Let
P
P
P
be a point on the segment
E
F
EF
EF
such that
O
P
⊥
E
F
OP \perp EF
OP
⊥
EF
. The circle
Γ
1
\Gamma_{1}
Γ
1
passes through
A
A
A
and
E
E
E
and is tangent to
ω
\omega
ω
at
A
A
A
, while
Γ
2
\Gamma_{2}
Γ
2
passes through
C
C
C
and
F
F
F
and is tangent to
ω
\omega
ω
at
C
C
C
. If
Γ
1
\Gamma_{1}
Γ
1
and
Γ
2
\Gamma_{2}
Γ
2
meet at
X
X
X
and
Y
Y
Y
, prove that
P
O
PO
PO
is the bisector of
∠
X
P
Y
\angle XPY
∠
XP
Y
.Proposed by Nikola Velov
P2
1
Hide problems
Positive real valued function with positive real inputs
Let
R
+
\mathbb{R}^{+}
R
+
be the set of positive real numbers. Find all functions
f
:
R
+
→
R
+
f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}
f
:
R
+
→
R
+
such that for all
x
,
y
>
0
x,y>0
x
,
y
>
0
we have
f
(
x
y
+
f
(
x
)
)
=
y
f
(
x
)
+
x
.
f(xy+f(x))=yf(x)+x.
f
(
x
y
+
f
(
x
))
=
y
f
(
x
)
+
x
.
Proposed by Nikola Velov
P1
2
Hide problems
A bound on perfect squares as values of a polynomial in m
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be integers. Prove that for any positive integer
n
n
n
, there are at least
⌊
n
4
⌋
\left \lfloor{\frac{n}{4}}\right \rfloor
⌊
4
n
⌋
positive integers
m
≤
n
m \leq n
m
≤
n
such that
m
5
+
d
m
4
+
c
m
3
+
b
m
2
+
2023
m
+
a
m^5 + dm^4 + cm^3 + bm^2 + 2023m + a
m
5
+
d
m
4
+
c
m
3
+
b
m
2
+
2023
m
+
a
is not a perfect square.Proposed by Ilir Snopce
Color the lines!
Let
n
n
n
be a fixed positive integer and fix a point
O
O
O
in the plane. There are
n
n
n
lines drawn passing through the point
O
O
O
. Determine the largest
k
k
k
(depending on
n
n
n
) such that we can always color
k
k
k
of the
n
n
n
lines red in such a way that no two red lines are perpendicular to each other.Proposed by Nikola Velov