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n integer roots

Source: French TST 2005 pb 6.

May 27, 2005
inequalitiestriangle inequalityalgebra unsolvedalgebra

Problem Statement

Let PP be a polynom of degree n≥5n \geq 5 with integer coefficients given by P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0   with ai∈Za_i \in \mathbb{Z}, an≠0a_n \neq 0. Suppose that PP has nn different integer roots (elements of Z\mathbb{Z}) : 0,α2,…,αn0,\alpha_2,\ldots,\alpha_n. Find all integers k∈Zk \in \mathbb{Z} such that P(P(k))=0P(P(k))=0.
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