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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2021 Serbia Team Selection Test
2021 Serbia Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(5)
P5
1
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Operations on positive integers
The following operation is allowed on the positive integers: if a number is even, we can divide it by
2
2
2
, otherwise we can multiply it by a power of
3
3
3
(different from
3
0
3^0
3
0
) and add
1
1
1
. Prove that we can reach
1
1
1
from any starting positive integer
n
n
n
.
P6
1
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NT Functional divisibility
Let
S
=
{
1
,
2
,
…
,
1
0
10
}
S=\{1,2, \ldots ,10^{10}\}
S
=
{
1
,
2
,
…
,
1
0
10
}
. Find all functions
f
:
S
→
S
f:S \rightarrow S
f
:
S
→
S
, such that
f
(
x
+
1
)
=
f
(
f
(
x
)
)
+
1
(
m
o
d
1
0
10
)
f(x+1)=f(f(x))+1 \pmod {10^{10}}
f
(
x
+
1
)
=
f
(
f
(
x
))
+
1
(
mod
1
0
10
)
for each
x
∈
S
x \in S
x
∈
S
(assume
f
(
1
0
10
+
1
)
=
f
(
1
)
f(10^{10}+1)=f(1)
f
(
1
0
10
+
1
)
=
f
(
1
)
).
P4
1
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Sequences with sum 2021
Given that
a
1
,
a
2
,
…
,
a
2020
a_1, a_2, \ldots,a_{2020}
a
1
,
a
2
,
…
,
a
2020
are integers, find the maximal number of subsequences
a
i
,
a
i
+
1
,
.
.
.
,
a
j
a_i,a_{i+1}, ..., a_j
a
i
,
a
i
+
1
,
...
,
a
j
(
0
<
i
≤
j
<
2021
0<i\leq j<2021
0
<
i
≤
j
<
2021
) with with sum
2021
2021
2021
P3
1
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Number theoretical equations with a prime
Given is a prime number
p
p
p
. Find the number of positive integer solutions
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
of the system of equations
a
c
+
b
d
=
p
(
a
+
c
)
ac+bd = p(a+c)
a
c
+
b
d
=
p
(
a
+
c
)
and
b
c
−
a
d
=
p
(
b
−
d
)
bc-ad = p(b-d)
b
c
−
a
d
=
p
(
b
−
d
)
.
P2
1
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Geometry with fixed point from Serbia TST
Let
D
D
D
be an arbitrary point on the side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
. Points
E
E
E
and
F
F
F
are on
C
A
CA
C
A
and
B
A
BA
B
A
are such that
C
D
=
C
E
CD=CE
C
D
=
CE
and
B
D
=
B
F
BD=BF
B
D
=
BF
. Lines
B
E
BE
BE
and
C
F
CF
CF
intersect at point
P
P
P
. Prove that when point
D
D
D
varies along the line
B
C
BC
BC
,
P
D
PD
P
D
passes through a fixed point.