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1986 IMO Longlists
34
Polynomial identity
Polynomial identity
Source:
August 29, 2010
algebra
polynomial
algebra proposed
Problem Statement
For each non-negative integer
n
n
n
,
F
n
(
x
)
F_n(x)
F
n
(
x
)
is a polynomial in
x
x
x
of degree
n
n
n
. Prove that if the identity
F
n
(
2
x
)
=
∑
r
=
0
n
(
−
1
)
n
−
r
(
n
r
)
2
r
F
r
(
x
)
F_n(2x)=\sum_{r=0}^{n} (-1)^{n-r} \binom nr 2^r F_r(x)
F
n
(
2
x
)
=
r
=
0
∑
n
(
−
1
)
n
−
r
(
r
n
)
2
r
F
r
(
x
)
holds for each n, then
F
n
(
t
x
)
=
∑
r
=
0
n
(
n
r
)
t
r
(
1
−
t
)
n
−
r
F
r
(
x
)
F_n(tx)=\sum_{r=0}^{n} \binom nr t^r (1-t)^{n-r} F_r(x)
F
n
(
t
x
)
=
r
=
0
∑
n
(
r
n
)
t
r
(
1
−
t
)
n
−
r
F
r
(
x
)
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