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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2022 Serbia National Math Olympiad
2022 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(6)
P6
1
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Algebra with primes
Let
p
p
p
and
q
q
q
be different primes, and
α
∈
(
0
,
3
)
\alpha\in (0, 3)
α
∈
(
0
,
3
)
a real number. Prove that in sequence
[
α
]
,
[
2
α
]
,
[
3
α
]
…
\left[ \alpha \right] , \left[ 2\alpha \right] , \left[ 3\alpha \right] \dots
[
α
]
,
[
2
α
]
,
[
3
α
]
…
exists number less than
2
p
q
2pq
2
pq
, divisible by
p
p
p
or
q
q
q
.
P3
1
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Combinatorics
The table of dimensions
n
×
n
n\times n
n
×
n
,
n
∈
N
n\in\mathbb{N}
n
∈
N
, is filled with numbers from
1
1
1
to
n
2
n^2
n
2
, but the difference any two numbers on adjacent fields is at most
n
n
n
, and that for every
k
=
1
,
2
,
…
,
n
2
k = 1, 2,\dots , n^2
k
=
1
,
2
,
…
,
n
2
set of fields whose numbers are
1
,
2
,
…
,
k
1, 2,\dots , k
1
,
2
,
…
,
k
is connected, as well as the set of fields whose numbers are
k
,
k
+
1
,
…
,
n
2
k, k + 1,\dots , n^2
k
,
k
+
1
,
…
,
n
2
. Neighboring fields are fields with a common side, while a set of fields is considered connected if from each field to every other field of that set can be reached going only to the neighboring fields within that set. We call a pair of adjacent numbers, ie. numbers on adjacent fields, good, if their absolute difference is exactly
n
n
n
(one number can be found in several good pairs). Prove that the table has at least
2
(
n
−
1
)
2 (n - 1)
2
(
n
−
1
)
good pairs.
P5
1
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Game with numbers
On the board are written
n
n
n
natural numbers,
n
∈
N
n\in \mathbb{N}
n
∈
N
. In one move it is possible to choose two equal written numbers and increase one by
1
1
1
and decrease the other by
1
1
1
. Prove that in this the game cannot be played more than
n
3
6
\frac{n^3}{6}
6
n
3
moves.
P4
1
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Number theory
Let
f
(
n
)
f(n)
f
(
n
)
be number of numbers
x
∈
{
1
,
2
,
⋯
,
n
}
x \in \{1,2,\cdots ,n\}
x
∈
{
1
,
2
,
⋯
,
n
}
,
n
∈
N
n\in\mathbb{N}
n
∈
N
, such that
g
c
d
(
x
,
n
)
gcd(x, n)
g
c
d
(
x
,
n
)
is either
1
1
1
or prime. Prove
∑
d
∣
n
f
(
d
)
+
φ
(
n
)
≥
2
n
\sum_{d|n} f(d) + \varphi(n) \geq 2n
d
∣
n
∑
f
(
d
)
+
φ
(
n
)
≥
2
n
For which
n
n
n
does equality hold?
P1
1
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Geometry
Let
k
k
k
be incircle of acute triangle
A
B
C
ABC
A
BC
,
A
C
≠
B
C
AC\neq BC
A
C
=
BC
, and
l
l
l
be excircle that touches
A
B
AB
A
B
. Line
p
p
p
through the
C
C
C
is orthogonal to
A
B
AB
A
B
,
p
∩
k
=
{
X
,
Y
}
p\cap k = \{X, Y\}
p
∩
k
=
{
X
,
Y
}
,
p
∩
l
=
{
Z
,
T
}
p\cap l = \{Z, T\}
p
∩
l
=
{
Z
,
T
}
and the point arrangement is
X
−
Y
−
Z
−
T
X-Y-Z-T
X
−
Y
−
Z
−
T
. Circle
m
m
m
through
X
X
X
and
Z
Z
Z
intersects
A
B
AB
A
B
at
D
D
D
and
E
E
E
. Prove that points
D
,
Y
,
E
,
T
D,Y,E,T
D
,
Y
,
E
,
T
are concyclic.
P2
1
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Inequality
Let
a
a
a
,
b
b
b
and
c
c
c
be positive real numbers and
a
3
+
b
3
+
c
3
=
3
a^3+b^3+c^3=3
a
3
+
b
3
+
c
3
=
3
. Prove
1
3
−
2
a
+
1
3
−
2
b
+
1
3
−
2
c
≥
3
\frac{1}{3-2a}+\frac{1}{3-2b}+\frac{1}{3-2c}\geq 3
3
−
2
a
1
+
3
−
2
b
1
+
3
−
2
c
1
≥
3