MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2011 German National Olympiad
6
6
Part of
2011 German National Olympiad
Problems
(1)
Sequence of polynomials divisible by p
Source: Germany 2011 - Problem 6
12/6/2022
Let
p
>
2
p>2
p
>
2
be a prime. Define a sequence
(
Q
n
(
x
)
)
(Q_{n}(x))
(
Q
n
(
x
))
of polynomials such that
Q
0
(
x
)
=
1
,
Q
1
(
x
)
=
x
Q_{0}(x)=1, Q_{1}(x)=x
Q
0
(
x
)
=
1
,
Q
1
(
x
)
=
x
and
Q
n
+
1
(
x
)
=
x
Q
n
(
x
)
+
n
Q
n
−
1
(
x
)
Q_{n+1}(x) =xQ_{n}(x) + nQ_{n-1}(x)
Q
n
+
1
(
x
)
=
x
Q
n
(
x
)
+
n
Q
n
−
1
(
x
)
for
n
≥
1.
n\geq 1.
n
≥
1.
Prove that
Q
p
(
x
)
−
x
p
Q_{p}(x)-x^p
Q
p
(
x
)
−
x
p
is divisible by
p
p
p
for all integers
x
.
x.
x
.
algebra
polynomial
number theory
Sequence
prime