This looks very familiar. I think something similar appeared on the forum recently. First of all, notice that $m=1$ doesn't work: We may assume WLOG that none of the polygons contain $O$. Consider a line $AB$ through $O$ with $A,B$ on opposite sides of $O$, and rays $(OC,(OD$ on the same side of $AB$, with $(OC$ between $(OB$ and $(OD$. We can now take our "polygons" to be $BC,CD,DA$ (they can, of course, be "thickened" into proper polygons and not just segments). Next, we can show that $m=2$ works: For each polygon in the collection consider the minimal region contained between two lines through $O$ which contains the polygon. Now pick a polygon $\mathcal P$ in the collection whose region is minimal wrt inclusion, in the sense that there's no other polygon whose region is properly included in the $\mathcal P$'s region. The two lines through $O$ which bound the region of $\mathcal P$ have the required properties, as can easily be seen. I hope it's correct.