Let $\Omega$ be an $n$-gon inscribed in the unit circle, with vertices $P_1, P_2, ..., P_n$. (a) Show that there exists a point $P$ on the unit circle such that $PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2$. (b) Suppose for each $P$ on the unit circle, the inequality $PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2$ holds. Prove that $\Omega$ is regular.