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Canada National Olympiad
2010 Canada National Olympiad
5
5
Part of
2010 Canada National Olympiad
Problems
(1)
Polynomial division
Source: Canadian Mathematical Olympiad - 2010 - Problem 5.
5/6/2011
Let
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
be polynomials with integer coefficients. Let
a
n
=
n
!
+
n
a_n = n! +n
a
n
=
n
!
+
n
. Show that if
P
(
a
n
)
Q
(
a
n
)
\frac{P(a_n)}{Q(a_n)}
Q
(
a
n
)
P
(
a
n
)
is an integer for every
n
n
n
, then
P
(
n
)
Q
(
n
)
\frac{P(n)}{Q(n)}
Q
(
n
)
P
(
n
)
is an integer for every integer
n
n
n
such that
Q
(
n
)
≠
0
Q(n)\neq 0
Q
(
n
)
=
0
.
algebra
polynomial