IMO Shortlist 1997 problem 11
Dodao/la:
arhiva2. travnja 2012. Let
![P(x)](/media/m/c/d/7/cd7664875343d44cd5f96a566b582b0e.png)
be a polynomial with real coefficients such that
![P(x) > 0](/media/m/0/f/f/0fff1696658baa5adc83ad85e6ea87c0.png)
for all
![x \geq 0.](/media/m/f/6/4/f64fc57c0675e3c9456b074ffab4d069.png)
Prove that there exists a positive integer n such that
![(1 + x)^n \cdot P(x)](/media/m/6/f/0/6f00586a65376acdbd192b338f1c837a.png)
is a polynomial with nonnegative coefficients.
%V0
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 $(1 + x)^n \cdot P(x)$ is a polynomial with nonnegative coefficients.
Izvor: Međunarodna matematička olimpijada, shortlist 1997