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Problems
Contests
National and Regional Contests
North Macedonia Contests
Memorial "Aleksandar Blazhevski-Cane"
2024 5th Memorial "Aleksandar Blazhevski-Cane"
2024 5th Memorial "Aleksandar Blazhevski-Cane"
Part of
Memorial "Aleksandar Blazhevski-Cane"
Subcontests
(6)
P6
1
Hide problems
Your level of friendship gets assessed
In a group of
2
n
2n
2
n
students, each student has exactly
3
3
3
friends within the group. The friendships are mutual and for each two students
A
A
A
and
B
B
B
which are not friends, there is a sequence
C
1
,
C
2
,
.
.
.
,
C
r
C_1, C_2, ..., C_r
C
1
,
C
2
,
...
,
C
r
of students such that
A
A
A
is a friend of
C
1
C_1
C
1
,
C
1
C_1
C
1
is a friend of
C
2
C_2
C
2
, et cetera, and
C
r
C_r
C
r
is a friend of
B
B
B
. Every student was asked to assess each of his three friendships with: "acquaintance", "friend" and "BFF". It turned out that each student either gave the same assessment to all of his friends or gave every assessment exactly once. We say that a pair of students is in conflict if they gave each other different assessments. Let
D
D
D
be the set of all possible values of the total number of conflicts. Prove that
∣
D
∣
≥
3
n
|D| \geq 3n
∣
D
∣
≥
3
n
with equality if and only if the group can be partitioned into two subsets such that each student is separated from all of his friends.
P4
1
Hide problems
Angle condition, circles and concurrent lines
Let
D
D
D
be a point inside
△
A
B
C
\triangle ABC
△
A
BC
such that
∠
C
D
A
+
∠
C
B
A
=
18
0
∘
.
\angle CDA + \angle CBA = 180^{\circ}.
∠
C
D
A
+
∠
CB
A
=
18
0
∘
.
The line
C
D
CD
C
D
meets the circle
⊙
A
B
C
\odot ABC
⊙
A
BC
at the point
E
E
E
for the second time. Let
G
G
G
be the common point of the circle centered at
C
C
C
with radius
C
D
CD
C
D
and the arc
A
C
⌢
\overset{\LARGE \frown}{AC}
A
C
⌢
of
⊙
A
B
C
\odot ABC
⊙
A
BC
which does not contain the point
B
B
B
. The circle centered at
A
A
A
with radius
A
D
AD
A
D
meets
⊙
B
C
D
\odot BCD
⊙
BC
D
for the second time at
F
F
F
. Prove that the lines
G
E
,
F
D
,
C
B
GE, FD, CB
GE
,
F
D
,
CB
are concurrent or parallel.
P5
1
Hide problems
Orders running around in circles
For a given integer
k
≥
1
k \geq 1
k
≥
1
, find all
k
k
k
-tuples of positive integers
(
n
1
,
n
2
,
.
.
.
,
n
k
)
(n_1,n_2,...,n_k)
(
n
1
,
n
2
,
...
,
n
k
)
with
GCD
(
n
1
,
n
2
,
.
.
.
,
n
k
)
=
1
\text{GCD}(n_1,n_2,...,n_k) = 1
GCD
(
n
1
,
n
2
,
...
,
n
k
)
=
1
and
n
2
∣
(
n
1
+
1
)
n
1
−
1
n_2|(n_1+1)^{n_1}-1
n
2
∣
(
n
1
+
1
)
n
1
−
1
,
n
3
∣
(
n
2
+
1
)
n
2
−
1
n_3|(n_2+1)^{n_2}-1
n
3
∣
(
n
2
+
1
)
n
2
−
1
, ... ,
n
1
∣
(
n
k
+
1
)
n
k
−
1
n_1|(n_k+1)^{n_k}-1
n
1
∣
(
n
k
+
1
)
n
k
−
1
.Proposed by Pavel Dimovski
P1
1
Hide problems
Graph theory in the contest halls
This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant
X
X
X
, let
t
(
X
)
t(X)
t
(
X
)
be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true:
1
)
1)
1
)
For any two friends
X
′
X'
X
′
and
X
′
′
X''
X
′′
, we have
t
(
X
′
)
≠
t
(
X
′
′
)
,
t(X') \neq t(X''),
t
(
X
′
)
=
t
(
X
′′
)
,
2
)
2)
2
)
For every contestant
X
X
X
, the set
{
t
(
Y
)
:
Y
is a friend of
X
}
\{ t(Y) : Y \text{ is a friend of } X \}
{
t
(
Y
)
:
Y
is a friend of
X
}
consists of consecutive integers. The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?
P2
1
Hide problems
Asymetric inequality with determinant-like condition
Let
x
,
y
x,y
x
,
y
and
z
z
z
be positive real numbers such that
x
y
+
z
2
=
8
xy+z^2=8
x
y
+
z
2
=
8
. Determine the smallest possible value of the expression
x
+
y
z
+
y
+
z
x
2
+
z
+
x
y
2
.
\frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}.
z
x
+
y
+
x
2
y
+
z
+
y
2
z
+
x
.
P3
1
Hide problems
Bounded integer function with prime quasiperiodicity
Find all functions
f
:
N
→
Z
f: \mathbb{N} \rightarrow \mathbb{Z}
f
:
N
→
Z
such that
∣
f
(
k
)
∣
≤
k
|f(k)| \leq k
∣
f
(
k
)
∣
≤
k
for all positive integers
k
k
k
and there is a prime number
p
>
2024
p>2024
p
>
2024
which satisfies both of the following conditions:
1
)
1)
1
)
For all
a
∈
N
a \in \mathbb{N}
a
∈
N
we have
a
f
(
a
+
p
)
=
a
f
(
a
)
+
p
f
(
a
)
,
af(a+p) = af(a)+pf(a),
a
f
(
a
+
p
)
=
a
f
(
a
)
+
p
f
(
a
)
,
2
)
2)
2
)
For all
a
∈
N
a \in \mathbb{N}
a
∈
N
we have
p
∣
a
p
+
1
2
−
f
(
a
)
.
p|a^{\frac{p+1}{2}}-f(a).
p
∣
a
2
p
+
1
−
f
(
a
)
.
Proposed by Nikola Velov