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0 or 1 is root of p(x)

Source: Iranian National Olympiad (3rd Round) 2004

January 9, 2009
algebrapolynomialnumber theory proposednumber theory

Problem Statement

p(x) p(x) is a polynomial in Z[x] \mathbb{Z}[x] such that for each m,n∈N m,n\in \mathbb{N} there is an integer a a such that n∣p(am) n\mid p(a^m). Prove that 00 or 11 is a root of p(x) p(x).
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