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Problems
Contests
National and Regional Contests
North Macedonia Contests
Macedonian Team Selection Test
2021 Macedonian Team Selection Test
2021 Macedonian Team Selection Test
Part of
Macedonian Team Selection Test
Subcontests
(6)
Problem 6
1
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Three tangents lemma plus a tangent circle
Let
A
B
C
ABC
A
BC
be an acute triangle such that
A
B
<
A
C
AB<AC
A
B
<
A
C
with orthocenter
H
H
H
. The altitudes
B
H
BH
B
H
and
C
H
CH
C
H
intersect
A
C
AC
A
C
and
A
B
AB
A
B
at
B
1
B_{1}
B
1
and
C
1
C_{1}
C
1
. Denote by
M
M
M
the midpoint of
B
C
BC
BC
. Let
l
l
l
be the line parallel to
B
C
BC
BC
passing through
A
A
A
. The circle around
C
M
C
1
CMC_{1}
CM
C
1
meets the line
l
l
l
at points
X
X
X
and
Y
Y
Y
, such that
X
X
X
is on the same side of the line
A
H
AH
A
H
as
B
B
B
and
Y
Y
Y
is on the same side of
A
H
AH
A
H
as
C
C
C
. The lines
M
X
MX
MX
and
M
Y
MY
M
Y
intersect
C
C
1
CC_{1}
C
C
1
at
U
U
U
and
V
V
V
respectively. Show that the circumcircles of
M
U
V
MUV
M
U
V
and
B
1
C
1
H
B_{1}C_{1}H
B
1
C
1
H
are tangent.Proposed by Nikola Velov
Problem 5
1
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Functional Inequality with Divisibility Condition
Determine all functions
f
:
N
→
N
f:\mathbb{N}\to \mathbb{N}
f
:
N
→
N
such that for all
a
,
b
∈
N
a, b \in \mathbb{N}
a
,
b
∈
N
the following conditions hold:
(
i
)
(i)
(
i
)
f
(
f
(
a
)
+
b
)
∣
b
a
−
1
f(f(a)+b) \mid b^a-1
f
(
f
(
a
)
+
b
)
∣
b
a
−
1
;
(
i
i
)
(ii)
(
ii
)
f
(
f
(
a
)
)
≥
f
(
a
)
−
1
f(f(a))\geq f(a)-1
f
(
f
(
a
))
≥
f
(
a
)
−
1
.
Problem 4
1
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Cycle conditions in group of permutations determine f(2021)
Let
S
=
{
1
,
2
,
3
,
…
2021
}
S=\{1, 2, 3, \dots 2021\}
S
=
{
1
,
2
,
3
,
…
2021
}
and
f
:
S
→
S
f:S \to S
f
:
S
→
S
be a function such that
f
(
n
)
(
n
)
=
n
f^{(n)}(n)=n
f
(
n
)
(
n
)
=
n
for each
n
∈
S
n \in S
n
∈
S
. Find all possible values for
f
(
2021
)
f(2021)
f
(
2021
)
. (Here,
f
(
n
)
(
n
)
=
f
(
f
(
f
…
f
(
⏟
n
times
n
)
)
)
…
)
)
f^{(n)}(n) = \underbrace{f(f(f\dots f(}_{n \text{ times} }n)))\dots))
f
(
n
)
(
n
)
=
n
times
f
(
f
(
f
…
f
(
n
)))
…
))
.)Proposed by Viktor Simjanoski
Problem 3
1
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Partitioning graphs in even subgraphs
A group of people is said to be good if every member has an even number (zero included) of acquaintances in it. Prove that any group of people can be partitioned into two (possibly empty) parts such that each part is good.
Problem 2
1
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Orthogonal lines meeting on a circle formed by midpoints
Let
A
B
C
ABC
A
BC
be an acute triangle such that
A
B
<
A
C
AB<AC
A
B
<
A
C
. Denote by
A
′
A'
A
′
the reflection of
A
A
A
with respect to
B
C
BC
BC
. The circumcircle of
A
′
B
C
A'BC
A
′
BC
meets the rays
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
respectively, such that
B
B
B
is between
A
A
A
and
D
D
D
, and
E
E
E
is between
A
A
A
and
C
C
C
. Denote by
P
P
P
and
Q
Q
Q
the midpoints of the segments
C
D
CD
C
D
and
B
E
BE
BE
, and let
S
S
S
be the midpoint of
B
C
BC
BC
. Show that the lines
B
C
BC
BC
and
A
A
′
AA'
A
A
′
meet on the circumcircle of
P
Q
S
PQS
PQS
.Proposed by Nikola Velov
Problem 1
1
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Analysis flavored inequality for monotone sequence
Let
k
≥
2
k\geq 2
k
≥
2
be a natural number. Suppose that
a
1
,
a
2
,
…
a
2021
a_1, a_2, \dots a_{2021}
a
1
,
a
2
,
…
a
2021
is a monotone decreasing sequence of non-negative numbers such that
∑
i
=
n
2021
a
i
≤
k
a
n
\sum_{i=n}^{2021}a_i\leq ka_n
i
=
n
∑
2021
a
i
≤
k
a
n
for all
n
=
1
,
2
,
…
2021
n=1,2,\dots 2021
n
=
1
,
2
,
…
2021
. Prove that
a
2021
≤
4
(
1
−
1
k
)
2021
a
1
a_{2021}\leq 4(1-\frac{1}{k})^{2021}a_1
a
2021
≤
4
(
1
−
k
1
)
2021
a
1
.