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Inequality for a polynomial on Z[x]

Source: Greek national M.O. 1997, Final Round, problem 4

November 20, 2011
inequalitiesalgebrapolynomialnumber theory unsolvednumber theory

Problem Statement

A polynomial PP with integer coefficients has at least 1313 distinct integer roots. Prove that if an integer nn is not a root of PP, then ∣P(n)∣≥7⋅6!2|P(n)| \geq 7 \cdot 6!^2, and give an example for equality.
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