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Find all polynomials such that f(a) divides f(2a^k)

Source: Romania TST 6 2009, Problem 2

May 5, 2012

algebrapolynomialalgebra proposed

Problem Statement

Let

n

and

k

be positive integers. Find all monic polynomials

f\in \mathbb{Z}[X]

, of degree

n

, such that

f(a)

divides

f(2a^k)

for

a\in \mathbb{Z}

with

f(a)\neq 0