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(1 + x)^n*P(x) is polynomial with nonnegative coefficients

Source: IMO Shortlist 1997, Q11

August 10, 2008

algebrapolynomialcoefficientsIMO Shortlist

Problem Statement

Let

P(x)

be a polynomial with real coefficients such that

P(x) > 0

for all

x \geq 0.

Prove that there exists a positive integer n such that (1 \plus{} x)^n \cdot P(x) is a polynomial with nonnegative coefficients.