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Ireland National Math Olympiad
2009 Irish Math Olympiad
1
Polynomials
Polynomials
Source:
February 2, 2012
algebra
polynomial
Problem Statement
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with rational coefficients. Prove that there exists a positive integer
n
n
n
such that the polynomial
Q
(
x
)
Q(x)
Q
(
x
)
defined by
Q
(
x
)
=
P
(
x
+
n
)
ā
P
(
x
)
Q(x)= P(x+n)-P(x)
Q
(
x
)
=
P
(
x
+
n
)
ā
P
(
x
)
has integer coefficients.
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