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Modulo Polynomial

Source: IMO Shortlist 1992, Problem 19

August 13, 2008
algebrapolynomialnumber theorycoefficientsDivisibilityIMO Shortlist

Problem Statement

Let f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1. Let p>3 p > 3 be a prime and suppose there exists an integer z z such that p p divides f(z). f(z). Prove that there exist integers z1,z2,…,z8 z_1, z_2, \ldots, z_8 such that if g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8), then all coefficients of f(x) \minus{} g(x) are divisible by p. p.
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