Question:

**Step 1: Use complex numbers** Let the circle be in the complex plane, centered at the origin with radius $R = 2$. The vertices $A_1, \dots, A_8$ can be represented by the complex numbers $z_k = R e^{i 2\pi k / 8}$ for $k = 1, \dots, 8$. These are the roots of the equation $z^8 - R^8 = 0$. **Step 2: Express the product in terms of polynomials** The point $P$ lies on the same circle, so it can be represented by a complex number $z$ with $|z| = R = 2$. The distance $PA_k$ is given by $|z - z_k|$.\nThe product of the distances is $\prod_{k=1}^8 |z - z_k| = \left| \prod_{k=1}^8 (z - z_k) \right|$.\nSince $z_k$ are the roots of $w^8 - R^8 = 0$, we have $\prod_{k=1}^8 (z - z_k) = z^8 - R^8$. **Step 3: Maximize the product** The product is $M = |z^8 - R^8|$. By the triangle inequality, $|z^8 - R^8| \le |z^8| + |-R^8| = |z|^8 + R^8$.\nSince $P$ lies on the circle, $|z| = R = 2$. So the maximum value is $2^8 + 2^8 = 256 + 256 = 512$.\nThis maximum is achieved when $z^8 = -R^8$, i.e., $z = R e^{i \pi (2k+1) / 8}$, which geometrically means $P$ is exactly halfway between any two adjacent vertices.