For a), let $x=y \cdot e^{i\theta}$, easy to observe that $|e^{i\theta}-1| \le 1$. thus $$|z-x|+|z-y| \ge |x-y|=|e^{i\theta}-1| \ge 1 \ge |e^{-i\theta}| \ge |z-e^{-i\theta}|-|z|=|zx-y|-|z|$$For b), from the condition we know that $\Delta OXY$ has three angles $\le \frac{2\pi}{3}$, thus its First Fermat Point lies in the triangle(or the boundary). Construct an equaleteral triangle $\Delta TXY$ with $T,O$ lie in the different side of $XY$, thus $t-x=e^{i\frac{\pi}{3}}\cdot(x-y)$, then $$LHS \ge |t|=|t \cdot e^{i\frac{\pi}{6}}|=RHS$$

Proposed by Benyamin Ghaseminia.