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IMC 2017 Problem 7

Source:

August 3, 2017
college contestsIMCimc 2017

Problem Statement

Let p(x)p(x) be a nonconstant polynomial with real coefficients. For every positive integer~nn, let qn(x)=(x+1)np(x)+xnp(x+1).q_n(x) = (x+1)^np(x)+x^n p(x+1) .
Prove that there are only finitely many numbers nn such that all roots of qn(x)q_n(x) are real.
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