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National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2013 Serbia Additional Team Selection Test
2013 Serbia Additional Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
3
1
Hide problems
Serbia additional TST 2013
Let
p
>
3
p > 3
p
>
3
be a given prime number. For a set
S
⊆
Z
S \subseteq \mathbb{Z}
S
⊆
Z
and
a
∈
N
a \in \mathbb{N}
a
∈
N
, define
S
a
=
{
x
∈
{
0
,
1
,
2
,
.
.
.
,
p
−
1
}
S_a = \{ x \in \{ 0,1, 2,...,p-1 \}
S
a
=
{
x
∈
{
0
,
1
,
2
,
...
,
p
−
1
}
|
(
∃
s
∈
S
)
x
≡
p
a
⋅
s
}
(\exists_s \in S) x \equiv_p a \cdot s \}
(
∃
s
∈
S
)
x
≡
p
a
⋅
s
}
.
(
a
)
(a)
(
a
)
How many sets
S
⊆
{
1
,
2
,
.
.
.
,
p
−
1
}
S \subseteq \{ 1, 2,...,p-1 \}
S
⊆
{
1
,
2
,
...
,
p
−
1
}
are there for which the sequence
S
1
,
S
2
,
.
.
.
,
S
p
−
1
S_1 , S_2 , ..., S_{p-1}
S
1
,
S
2
,
...
,
S
p
−
1
contains exactly two distinct terms?
(
b
)
(b)
(
b
)
Determine all numbers
k
∈
N
k \in \mathbb{N}
k
∈
N
for which there is a set
S
⊆
{
1
,
2
,
.
.
.
,
p
−
1
}
S \subseteq \{ 1, 2,...,p-1 \}
S
⊆
{
1
,
2
,
...
,
p
−
1
}
such that the sequence
S
1
,
S
2
,
.
.
.
,
S
p
−
1
S_1 , S_2 , ..., S_{p-1}
S
1
,
S
2
,
...
,
S
p
−
1
contains exactly
k
k
k
distinct terms.Proposed by Milan Basic and Milos Milosavljevic
2
1
Hide problems
Serbia additional TST 2013
In an acute
△
A
B
C
\triangle ABC
△
A
BC
(
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
) with angle
α
\alpha
α
at the vertex
A
A
A
, point
E
E
E
is the nine-point center, and
P
P
P
a point on the segment
A
E
AE
A
E
. If
∠
A
B
P
=
∠
A
C
P
=
x
\angle ABP = \angle ACP = x
∠
A
BP
=
∠
A
CP
=
x
, prove that
x
=
90
x = 90
x
=
90
°
−
2
α
-2 \alpha
−
2
α
.Proposed by Dusan Djukic
1
1
Hide problems
Serbian additional TST 2013
We call polynomials
A
(
x
)
=
a
n
x
n
+
.
.
.
+
a
1
x
+
a
0
A(x) = a_n x^n +. . .+a_1 x+a_0
A
(
x
)
=
a
n
x
n
+
...
+
a
1
x
+
a
0
and
B
(
x
)
=
b
m
x
m
+
.
.
.
+
b
1
x
+
b
0
B(x) = b_m x^m +. . .+b_1 x+b_0
B
(
x
)
=
b
m
x
m
+
...
+
b
1
x
+
b
0
(
a
n
b
m
≠
0
a_n b_m \neq 0
a
n
b
m
=
0
) similar if the following conditions hold:
(
i
)
(i)
(
i
)
n
=
m
n = m
n
=
m
;
(
i
i
)
(ii)
(
ii
)
There is a permutation
π
\pi
π
of the set
{
0
,
1
,
.
.
.
,
n
}
\{ 0, 1, . . . , n\}
{
0
,
1
,
...
,
n
}
such that
b
i
=
a
π
(
i
)
b_i = a_{\pi (i)}
b
i
=
a
π
(
i
)
for each
i
∈
0
,
1
,
.
.
.
,
n
i \in {0, 1, . . . , n}
i
∈
0
,
1
,
...
,
n
. Let
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
be similar polynomials with integer coefficients. Given that
P
(
16
)
=
3
2012
P(16) = 3^{2012}
P
(
16
)
=
3
2012
, find the smallest possible value of
∣
Q
(
3
2012
)
∣
|Q(3^{2012})|
∣
Q
(
3
2012
)
∣
.Proposed by Milos Milosavljevic