MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2021 Serbia National Math Olympiad
2021 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(6)
6
1
Hide problems
Repetition in sequences
A finite sequence of natural numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
is given. A sub-sequence
a
k
+
1
,
a
k
+
2
,
…
,
a
l
a_{k+1}, a_{k+2}, \dots, a_l
a
k
+
1
,
a
k
+
2
,
…
,
a
l
will be called a repetition if there exists a natural number
p
≤
l
−
k
2
p\leq \frac{l-k}2
p
≤
2
l
−
k
such that
a
i
=
a
i
+
p
a_i=a_{i+p}
a
i
=
a
i
+
p
for
k
+
1
≤
i
≤
l
−
p
k+1\leq i\leq l-p
k
+
1
≤
i
≤
l
−
p
, but
a
i
≠
a
i
+
p
a_i\neq a_{i+p}
a
i
=
a
i
+
p
for
i
=
k
i=k
i
=
k
(if
k
>
0
k>0
k
>
0
) and
i
=
l
−
p
+
1
i=l-p+1
i
=
l
−
p
+
1
(if
l
<
n
l<n
l
<
n
).Show that the sequence contains less than
n
n
n
repetitions.
5
1
Hide problems
Functional equation
Find all functions
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
such that for every
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
the following equality holds:
f
(
x
f
(
y
)
+
x
2
+
y
)
=
f
(
x
)
f
(
y
)
+
x
f
(
x
)
+
f
(
y
)
.
f(xf(y)+x^2+y)=f(x)f(y)+xf(x)+f(y).
f
(
x
f
(
y
)
+
x
2
+
y
)
=
f
(
x
)
f
(
y
)
+
x
f
(
x
)
+
f
(
y
)
.
4
1
Hide problems
Rude quadrilateral
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
will be called rude if there exists a convex quadrilateral
P
Q
R
S
PQRS
PQRS
whose points are all in the interior or on the sides of quadrilateral
A
B
C
D
ABCD
A
BC
D
such that the sum of diagonals of
P
Q
R
S
PQRS
PQRS
is larger than the sum of diagonals of
A
B
C
D
ABCD
A
BC
D
. Let
r
>
0
r>0
r
>
0
be a real number. Let us assume that a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is not rude, but every quadrilateral
A
′
B
C
D
A'BCD
A
′
BC
D
such that
A
′
≠
A
A'\neq A
A
′
=
A
and
A
′
A
≤
r
A'A\leq r
A
′
A
≤
r
is rude. Find all possible values of the largest angle of
A
B
C
D
ABCD
A
BC
D
.
3
1
Hide problems
Angle inequality
In a triangle
A
B
C
ABC
A
BC
, let
A
B
AB
A
B
be the shortest side. Points
X
X
X
and
Y
Y
Y
are given on the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
such that
C
X
=
A
X
+
B
X
CX=AX+BX
CX
=
A
X
+
BX
and
C
Y
=
A
Y
+
B
Y
CY=AY+BY
C
Y
=
A
Y
+
B
Y
. Prove that
∡
X
C
Y
<
6
0
o
\measuredangle XCY<60^{o}
∡
XC
Y
<
6
0
o
.
2
1
Hide problems
Traveling through the country
In the country of Graphia there are
100
100
100
towns, each numbered from
1
1
1
to
100
100
100
. Some pairs of towns may be connected by a (direct) road and we call such pairs of towns adjacent. No two roads connect the same pair of towns.Peter, a foreign tourist, plans to visit Graphia
100
100
100
times. For each
i
i
i
,
i
=
1
,
2
,
…
,
100
i=1,2,\dots, 100
i
=
1
,
2
,
…
,
100
, Peter starts his
i
i
i
-th trip by arriving in the town numbered
i
i
i
and then each following day Peter travels from the town he is currently in to an adjacent town with the lowest assigned number, assuming such that a town exists and that he hasn't visited it already on the
i
i
i
-th trip. Otherwise, Peter deems his
i
i
i
-th trip to be complete and returns home. It turns out that after all
100
100
100
trips, Peter has visited each town in Graphia the same number of times. Find the largest possible number of roads in Graphia.
1
1
Hide problems
Interesting number theory
Let
a
>
1
a>1
a
>
1
and
c
c
c
be natural numbers and let
b
≠
0
b\neq 0
b
=
0
be an integer. Prove that there exists a natural number
n
n
n
such that the number
a
n
+
b
a^n+b
a
n
+
b
has a divisor of the form
c
x
+
1
cx+1
c
x
+
1
,
x
∈
N
x\in\mathbb{N}
x
∈
N
.