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2018 China Team Selection Test
1
A Rational Root Problem on Polymonial Sequence
A Rational Root Problem on Polymonial Sequence
Source: 2018 China TST 4 Day 1 Problem 1
March 27, 2018
polynomial
Polynomials
algebra
Rational Root Theorem
Sequence
Problem Statement
Define the polymonial sequence
{
f
n
(
x
)
}
n
≥
1
\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}
{
f
n
(
x
)
}
n
≥
1
with
f
1
(
x
)
=
1
f_1\left ( x \right )=1
f
1
(
x
)
=
1
,
f
2
n
(
x
)
=
x
f
n
(
x
)
,
f
2
n
+
1
(
x
)
=
f
n
(
x
)
+
f
n
+
1
(
x
)
,
n
≥
1.
f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.
f
2
n
(
x
)
=
x
f
n
(
x
)
,
f
2
n
+
1
(
x
)
=
f
n
(
x
)
+
f
n
+
1
(
x
)
,
n
≥
1.
Look for all the rational number
a
a
a
which is a root of certain
f
n
(
x
)
.
f_n\left ( x \right ).
f
n
(
x
)
.
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