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Moldova 2016 tst B4
Moldova 2016 tst B4
Source: Moldova TST 2016, day2, problem 1
April 25, 2017
number theory
Problem Statement
The sequence of polynomials
(
P
n
(
X
)
)
n
∈
Z
>
0
\left( P_{n}(X)\right)_{n\in Z_{>0}}
(
P
n
(
X
)
)
n
∈
Z
>
0
is defined as follows:
P
1
(
X
)
=
2
X
P_{1}(X)=2X
P
1
(
X
)
=
2
X
P
2
(
X
)
=
2
(
X
2
+
1
)
P_{2}(X)=2(X^2+1)
P
2
(
X
)
=
2
(
X
2
+
1
)
P
n
+
2
(
X
)
=
2
X
⋅
P
n
+
1
(
X
)
−
(
X
2
−
1
)
P
n
(
X
)
P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)
P
n
+
2
(
X
)
=
2
X
⋅
P
n
+
1
(
X
)
−
(
X
2
−
1
)
P
n
(
X
)
, for all positive integers
n
n
n
. Find all
n
n
n
for which
X
2
+
1
∣
P
n
(
X
)
X^2+1\mid P_{n}(X)
X
2
+
1
∣
P
n
(
X
)
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