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Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2019 Serbia Team Selection Test
2019 Serbia Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(6)
P6
1
Hide problems
A very large polyhedron
A figuric is a convex polyhedron with
2
6
5
2019
26^{5^{2019}}
2
6
5
2019
faces. On every face of a figuric we write down a number. When we throw two figurics (who don't necessarily have the same set of numbers on their sides) into the air, the figuric which falls on a side with the greater number wins; if this number is equal for both figurics, we repeat this process until we obtain a winner. Assume that a figuric has an equal probability of falling on any face. We say that one figuric rules over another if when throwing these figurics into the air, it has a strictly greater probability to win than the other figuric (it can be possible that given two figurics, no figuric rules over the other). Milisav and Milojka both have a blank figuric. Milisav writes some (not necessarily distinct) positive integers on the faces of his figuric so that they sum up to
2
7
5
2019
27^{5^{2019}}
2
7
5
2019
. After this, Milojka also writes positive integers on the faces of her figuric so that they sum up to
2
7
5
2019
27^{5^{2019}}
2
7
5
2019
. Is it always possible for Milojka to create a figuric that rules over Milisav's?Proposed by Bojan Basic
P4
1
Hide problems
An old man has a last wish
A trader owns horses of
3
3
3
races, and exacly
b
j
b_j
b
j
of each race (for
j
=
1
,
2
,
3
j=1,2,3
j
=
1
,
2
,
3
). He want to leave these horses heritage to his
3
3
3
sons. He knowns that the boy
i
i
i
for horse
j
j
j
(for
i
,
j
=
1
,
2
,
3
i,j=1,2,3
i
,
j
=
1
,
2
,
3
) would pay
a
i
j
a_{ij}
a
ij
golds, such that for distinct
i
,
j
i,j
i
,
j
holds holds
a
i
i
>
a
i
j
a_{ii}> a_{ij}
a
ii
>
a
ij
and
a
j
j
>
a
i
j
a_{jj} >a_{ij}
a
jj
>
a
ij
.Prove that there exists a natural number
n
n
n
such that whenever it holds
min
{
b
1
,
b
2
,
b
3
}
>
n
\min\{b_1,b_2,b_3\}>n
min
{
b
1
,
b
2
,
b
3
}
>
n
, trader can give the horses to their sons such that after getting the horses each son values his horses more than the other brother is getting, individually.
P3
1
Hide problems
A circle with n points colored blue and red
It is given
n
n
n
a natural number and a circle with circumference
n
n
n
. On the circle, in clockwise direction, numbers
0
,
1
,
2
,
…
n
−
1
0,1,2,\dots n-1
0
,
1
,
2
,
…
n
−
1
are written, in this order and in the same distance to each other. Every number is colored red or blue, and there exists a non-zero number of numbers of each color. It is known that there exists a set
S
⊊
{
0
,
1
,
2
,
…
n
−
1
}
,
∣
S
∣
≥
2
S\subsetneq \{0,1,2,\dots n-1\}, |S|\geq 2
S
⊊
{
0
,
1
,
2
,
…
n
−
1
}
,
∣
S
∣
≥
2
, for wich it holds: if
(
x
,
y
)
,
x
<
y
(x,y), x<y
(
x
,
y
)
,
x
<
y
is a circle sector whose endpoints are of distinct colors, whose distance
y
−
x
y-x
y
−
x
is in
S
S
S
, then
y
y
y
is in
S
S
S
.Prove that there is a divisor
d
d
d
of
n
n
n
different from
1
1
1
and
n
n
n
for wich holds: if
(
x
,
y
)
,
x
<
y
(x,y),x<y
(
x
,
y
)
,
x
<
y
are different points of distinct colors, such that their distance is divisible by
d
d
d
, then both
x
,
y
x,y
x
,
y
are divisible by
d
d
d
.
P1
1
Hide problems
Set of 2019 numbers has no odd prime divisor less that 37
a) Given
2019
2019
2019
different integers wich have no odd prime divisor less than
37
37
37
, prove there exists two of these numbers such that their sum has no odd prime divisor less than
37
37
37
.b)Does the result hold if we change
37
37
37
to
38
38
38
?
P5
1
Hide problems
Diophantine from Serbia TST
Solve the equation in nonnegative integers:\\
2
x
=
5
y
+
3
2^x=5^y+3
2
x
=
5
y
+
3
P2
1
Hide problems
Nice but easy geometry
Given triangle
△
A
B
C
\triangle ABC
△
A
BC
with
A
C
≠
B
C
AC\neq BC
A
C
=
BC
,and let
D
D
D
be a point inside triangle such that
∡
A
D
B
=
9
0
∘
+
1
2
∡
A
C
B
\measuredangle ADB=90^{\circ} + \frac {1}{2}\measuredangle ACB
∡
A
D
B
=
9
0
∘
+
2
1
∡
A
CB
.Tangents from
C
C
C
to the circumcircles of
△
A
B
C
\triangle ABC
△
A
BC
and
△
A
D
C
\triangle ADC
△
A
D
C
intersect
A
B
AB
A
B
and
A
D
AD
A
D
at
P
P
P
and
Q
Q
Q
, respectively.Prove that
P
Q
PQ
PQ
bisects the angle
∡
B
P
C
\measuredangle BPC
∡
BPC
.