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Polynomial division

Source: Canadian Mathematical Olympiad - 2010 - Problem 5.

May 6, 2011
algebrapolynomial

Problem Statement

Let P(x)P(x) and Q(x)Q(x) be polynomials with integer coefficients. Let an=n!+na_n = n! +n. Show that if P(an)Q(an)\frac{P(a_n)}{Q(a_n)} is an integer for every nn, then P(n)Q(n)\frac{P(n)}{Q(n)} is an integer for every integer nn such that Q(n)≠0Q(n)\neq 0.
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