Suppose $D\subset \mathbb C$ is a smoothly bounded domain and it contains the origin. Let $ds$ denote the arc length measure on $\partial D.$ I am interested in the following two inverse problems (perhaps related to certain moment problem):

**A)** Suppose for some polynomial $p$ there exists $n_0\in\mathbb N$ so that the following identity holds $$\int_{\partial D}\frac{p(z)}{z^n}ds_z=0$$ for all positive integer $n\geq n_0.$
Does it imply that $\partial D$ must be a disk?

**A')** With the same assumption as in (A) we have $$\int_{\partial D}\frac{p(z)}{(z-t)^n}ds_z=0 \,\,\,\,\text{for all}\,\,\, t\in\ D$$

holding for all $n\geq n_0$. Does this force $\partial D$ to be a disk?

In here $ds$ denotes the arc length measure.