MathDB
Problems
Contests
National and Regional Contests
PEN Problems
PEN Q Problems
9
Q 9
Q 9
Source:
May 25, 2007
function
algebra
polynomial
calculus
integration
rational function
Polynomials
Problem Statement
For non-negative integers
n
n
n
and
k
k
k
, let
P
n
,
k
(
x
)
P_{n, k}(x)
P
n
,
k
(
x
)
denote the rational function
(
x
n
−
1
)
(
x
n
−
x
)
⋯
(
x
n
−
x
k
−
1
)
(
x
k
−
1
)
(
x
k
−
x
)
⋯
(
x
k
−
x
k
−
1
)
.
\frac{(x^{n}-1)(x^{n}-x) \cdots (x^{n}-x^{k-1})}{(x^{k}-1)(x^{k}-x) \cdots (x^{k}-x^{k-1})}.
(
x
k
−
1
)
(
x
k
−
x
)
⋯
(
x
k
−
x
k
−
1
)
(
x
n
−
1
)
(
x
n
−
x
)
⋯
(
x
n
−
x
k
−
1
)
.
Show that
P
n
,
k
(
x
)
P_{n, k}(x)
P
n
,
k
(
x
)
is actually a polynomial for all
n
,
k
∈
N
n, k \in \mathbb{N}
n
,
k
∈
N
.
Back to Problems
View on AoPS