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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2023 Serbia National Math Olympiad
2023 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(5)
5
1
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Unique function
Let
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
be a function which satisfies the following: [*]
f
(
m
)
=
m
f(m)=m
f
(
m
)
=
m
, for all
m
∈
Z
m\in\mathbb{Z}
m
∈
Z
;[*]
f
(
a
+
b
c
+
d
)
=
f
(
a
c
)
+
f
(
b
d
)
2
f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}
f
(
c
+
d
a
+
b
)
=
2
f
(
c
a
)
+
f
(
d
b
)
, for all
a
,
b
,
c
,
d
∈
Z
a, b, c, d\in\mathbb{Z}
a
,
b
,
c
,
d
∈
Z
such that
∣
a
d
−
b
c
∣
=
1
|ad-bc|=1
∣
a
d
−
b
c
∣
=
1
,
c
>
0
c>0
c
>
0
and
d
>
0
d>0
d
>
0
;[*]
f
f
f
is monotonically increasing. (a) Prove that the function
f
f
f
is unique. (b) Find
f
(
5
−
1
2
)
f(\frac{\sqrt{5}-1}{2})
f
(
2
5
−
1
)
.
3
1
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Inequality on a periodic sequence
Given are positive integers
m
,
n
m, n
m
,
n
and a sequence
a
1
,
a
2
,
…
,
a_1, a_2, \ldots,
a
1
,
a
2
,
…
,
such that
a
i
=
a
i
−
n
a_i=a_{i-n}
a
i
=
a
i
−
n
for all
i
>
n
i>n
i
>
n
. For all
1
≤
j
≤
n
1 \leq j \leq n
1
≤
j
≤
n
, let
l
j
l_j
l
j
be the smallest positive integer such that
m
∣
a
j
+
a
j
+
1
+
…
+
a
j
+
l
j
−
1
m \mid a_j+a_{j+1}+\ldots+a_{j+l_j-1}
m
∣
a
j
+
a
j
+
1
+
…
+
a
j
+
l
j
−
1
. Prove that
l
1
+
l
2
+
…
+
l
n
≤
m
n
l_1+l_2+\ldots+l_n \leq mn
l
1
+
l
2
+
…
+
l
n
≤
mn
.
6
1
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2016 ISL G2 vibes?
Given is a triangle
A
B
C
ABC
A
BC
with incenter
I
I
I
and circumcircle
ω
\omega
ω
. The incircle is tangent to
B
C
BC
BC
at
D
D
D
. The perpendicular at
I
I
I
to
A
I
AI
A
I
meets
A
B
,
A
C
AB, AC
A
B
,
A
C
at
E
,
F
E, F
E
,
F
and the circle
(
A
E
F
)
(AEF)
(
A
EF
)
meets
ω
\omega
ω
and
A
I
AI
A
I
at
G
,
H
G, H
G
,
H
. The tangent at
G
G
G
to
ω
\omega
ω
meets
B
C
BC
BC
at
J
J
J
and
A
J
AJ
A
J
meets
ω
\omega
ω
at
K
K
K
. Prove that
(
D
J
K
)
(DJK)
(
D
J
K
)
and
(
G
I
H
)
(GIH)
(
G
I
H
)
are tangent to each other.
4
1
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A number that never can be a power of q
Given a positive integer
n
n
n
and a prime
q
q
q
, prove that the number
n
q
+
(
n
−
1
2
)
2
n^q+(\frac{n-1}{2})^2
n
q
+
(
2
n
−
1
)
2
can't be a power of
q
q
q
.
1
1
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Geo involving centers of (AOH), (BOH), (COH)
Given is a triangle
A
B
C
ABC
A
BC
with circumcenter
O
O
O
and orthocenter
H
H
H
. If
O
a
,
O
b
,
O
c
O_a, O_b, O_c
O
a
,
O
b
,
O
c
denote the circumcenters of
△
A
O
H
\triangle AOH
△
A
O
H
,
△
B
O
H
\triangle BOH
△
BO
H
,
△
C
O
H
\triangle COH
△
CO
H
, then prove that
A
O
a
,
B
O
b
,
C
O
c
AO_a, BO_b, CO_c
A
O
a
,
B
O
b
,
C
O
c
are concurrent.