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National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2013 Serbia Additional Team Selection Test
1
1
Part of
2013 Serbia Additional Team Selection Test
Problems
(1)
Serbian additional TST 2013
Source:
5/22/2015
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
number theory