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Serbia Contests
Serbia Team Selection Test
2016 Serbia Additional Team Selection Test
1
1
Part of
2016 Serbia Additional Team Selection Test
Problems
(1)
Polynomial induction, Serbia additional TST 2016
Source: Serbia additional TST 2016
4/5/2016
Let
P
0
(
x
)
=
x
3
−
4
x
P_0(x)=x^3-4x
P
0
(
x
)
=
x
3
−
4
x
. Sequence of polynomials is defined as following:\\
P
n
+
1
=
P
n
(
1
+
x
)
P
n
(
1
−
x
)
−
1
P_{n+1}=P_n(1+x)P_n(1-x)-1
P
n
+
1
=
P
n
(
1
+
x
)
P
n
(
1
−
x
)
−
1
.\\ Prove that
x
2016
∣
P
2016
(
x
)
x^{2016}|P_{2016}(x)
x
2016
∣
P
2016
(
x
)
.
algebra
polynomial
Polynomials