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1978 IMO Longlists
47
Proving an identity and P is a polynomial.
Proving an identity and P is a polynomial.
Source:
October 31, 2010
algebra
polynomial
induction
algebra unsolved
Problem Statement
Given the expression
P
n
(
x
)
=
1
2
n
[
(
x
+
x
2
−
1
)
n
+
(
x
−
x
2
−
1
)
n
]
,
P_n(x) =\frac{1}{2^n}\left[(x +\sqrt{x^2 - 1})^n+(x-\sqrt{x^2 - 1})^n\right],
P
n
(
x
)
=
2
n
1
[
(
x
+
x
2
−
1
)
n
+
(
x
−
x
2
−
1
)
n
]
,
prove:
(
a
)
P
n
(
x
)
(a) P_n(x)
(
a
)
P
n
(
x
)
satisfies the identity
P
n
(
x
)
−
x
P
n
−
1
(
x
)
+
1
4
P
n
−
2
(
x
)
≡
0.
P_n(x) - xP_{n-1}(x) + \frac{1}{4}P_{n-2}(x) \equiv 0.
P
n
(
x
)
−
x
P
n
−
1
(
x
)
+
4
1
P
n
−
2
(
x
)
≡
0.
(
b
)
P
n
(
x
)
(b) P_n(x)
(
b
)
P
n
(
x
)
is a polynomial in
x
x
x
of degree
n
.
n.
n
.
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