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Contests
National and Regional Contests
North Macedonia Contests
Memorial "Aleksandar Blazhevski-Cane"
2021 2nd Memorial "Aleksandar Blazhevski-Cane"
2021 2nd Memorial "Aleksandar Blazhevski-Cane"
Part of
Memorial "Aleksandar Blazhevski-Cane"
Subcontests
(6)
3
1
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Combinatorial inequality for n-tuples with sum 1
Given a positive integer
n
≥
3
n \geq 3
n
≥
3
, let
C
n
C_{n}
C
n
be the collection of all
n
n
n
-tuples
a
=
(
a
1
,
a
2
,
.
.
.
,
a
n
)
a=(a_{1},a_{2},...,a_{n})
a
=
(
a
1
,
a
2
,
...
,
a
n
)
of nonnegative reals
a
i
a_i
a
i
,
i
=
1
,
.
.
.
,
n
i=1,...,n
i
=
1
,
...
,
n
, such that
a
1
+
a
2
+
.
.
.
+
a
n
=
1
a_{1}+a_{2}+...+a_{n}=1
a
1
+
a
2
+
...
+
a
n
=
1
. For
k
∈
{
1
,
.
.
.
,
n
−
1
}
k \in \left \{ 1,...,n-1 \right \}
k
∈
{
1
,
...
,
n
−
1
}
and
a
∈
C
n
a \in C_{n}
a
∈
C
n
, consider the sum set
σ
k
(
a
)
=
{
a
1
+
.
.
.
+
a
k
,
a
2
+
.
.
.
+
a
k
+
1
,
.
.
.
,
a
n
−
k
+
1
+
.
.
.
+
a
n
}
\sigma_{k}(a) = \left \{a_{1}+...+a_{k},a_{2}+...+a_{k+1},...,a_{n-k+1}+...+a_{n} \right \}
σ
k
(
a
)
=
{
a
1
+
...
+
a
k
,
a
2
+
...
+
a
k
+
1
,
...
,
a
n
−
k
+
1
+
...
+
a
n
}
. Show the following. (a) There exist
m
k
=
max
{
min
σ
k
(
a
)
:
a
∈
C
n
}
m_k=\max\{\min\sigma_k(a):a\in\mathcal{C}_n\}
m
k
=
max
{
min
σ
k
(
a
)
:
a
∈
C
n
}
and
M
k
=
min
{
max
σ
k
(
a
)
:
a
∈
C
n
}
M_k=\min\{\max\sigma_k(a):a\in\mathcal{C}_n\}
M
k
=
min
{
max
σ
k
(
a
)
:
a
∈
C
n
}
. (b) It holds that
1
≤
∑
k
=
1
n
−
1
(
1
M
k
−
1
m
k
)
≤
n
−
2
\displaystyle{1\leq\sum_{k=1}^{n-1}(\frac{1}{M_k}-\frac{1}{m_k})\leq n-2}
1
≤
k
=
1
∑
n
−
1
(
M
k
1
−
m
k
1
)
≤
n
−
2
. Moreover, on the left side, equality is attained only for finitely many values of
n
n
n
, whereas on the right side, equality holds for infinitely values of
n
n
n
.
2
1
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Set of residues mod p closed with respect to the operation ab+1 (mod p)
Let
p
p
p
be a prime number and
F
=
{
0
,
1
,
2
,
.
.
.
,
p
−
1
}
F=\left \{0,1,2,...,p-1 \right \}
F
=
{
0
,
1
,
2
,
...
,
p
−
1
}
. Let
A
A
A
be a proper subset of
F
F
F
that satisfies the following property: if
a
,
b
∈
A
a,b \in A
a
,
b
∈
A
, then
a
b
+
1
ab+1
ab
+
1
(mod
p
p
p
)
∈
A
\in A
∈
A
. How many elements can
A
A
A
have? (Justify your answer.)
1
1
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Cyclic quadrilateral with equal sides and unusual length condition
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral such that
A
B
=
A
D
AB=AD
A
B
=
A
D
. Let
E
E
E
and
F
F
F
be points on the sides
B
C
BC
BC
and
C
D
CD
C
D
, respectively, such that
B
E
+
D
F
=
E
F
BE+DF=EF
BE
+
D
F
=
EF
. Prove that
∠
B
A
D
=
2
∠
E
A
F
\angle BAD = 2 \angle EAF
∠
B
A
D
=
2∠
E
A
F
.
6
1
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Almost symmetric functional equation in positive reals
Let
R
+
\mathbb{R}^{+}
R
+
be the set of all positive real numbers. Find all the 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
+
,
f
(
x
)
f
(
y
)
=
f
(
y
)
f
(
x
f
(
y
)
)
+
1
x
y
.
f(x)f(y) = f(y)f(xf(y)) + \frac{1}{xy}.
f
(
x
)
f
(
y
)
=
f
(
y
)
f
(
x
f
(
y
))
+
x
y
1
.
4
1
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Positive integers with specified number of divisors function
Find all positive integers
n
n
n
that have precisely
n
+
1
\sqrt{n+1}
n
+
1
natural divisors.
5
1
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Interesting geometry. Collinearity
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle with circumcenter
O
O
O
. The perpendicular bisectors of the segments
O
A
,
O
B
OA,OB
O
A
,
OB
and
O
C
OC
OC
intersect the lines
B
C
,
C
A
BC,CA
BC
,
C
A
and
A
B
AB
A
B
at
D
,
E
D,E
D
,
E
and
F
F
F
, respectively. Prove that
D
,
E
,
F
D,E,F
D
,
E
,
F
are collinear.