MathDB
Polynomial

Source: Nordic MO 2015

January 22, 2016
number theoryalgebrapolynomial

Problem Statement

Let n>1n > 1 and p(x)=xn+an−1xn−1+...+a0p(x)=x^n+a_{n-1}x^{n-1} +...+a_0 be a polynomial with nn real roots (counted with multiplicity). Let the polynomial qq be defined by q(x)=∏j=12015p(x+j)q(x) = \prod_{j=1}^{2015} p(x + j). We know that p(2015)=2015p(2015) = 2015. Prove that qq has at least 19701970 different roots r1,...,r1970r_1, ..., r_{1970} such that ∣rj∣<2015|r_j| < 2015 for all j=1,...,1970 j = 1, ..., 1970.
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