IMO Shortlist 1976 problem 8
Dodao/la:
arhiva2. travnja 2012. Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
be a polynomial with real coefficients such that
![P(x) > 0](/media/m/0/f/f/0fff1696658baa5adc83ad85e6ea87c0.png)
if
![x > 0](/media/m/1/9/4/1945adeeed2d2765b5d8b1595074b738.png)
. Prove that there exist polynomials
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
and
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
with nonnegative coefficients such that
![\displaystyle P(x) = \frac{Q(x)}{R(x)}](/media/m/d/b/b/dbb1adff98dd39bc944fac8c441b87c9.png)
if
%V0
Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $\displaystyle P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$
Izvor: Međunarodna matematička olimpijada, shortlist 1976