Let $f$ have only poles on the complex plane. Suppose that for every polynomial $p(z)$ and every closed curve $C$ not passing through the poles of $f$ we have: $$\int_C p(z)f(z)\,\mathrm{d}z = 0.$$ Prove that $f$ is entire.
I thought about the "Closed Curve Theorem" for entire functions so that $p(z)f(z)$ has to be entire. But I couldn't relate the functions poles being on the complex plane. If you can show me how to approach, I will appreciate.