Let A₁, A₂, A₃, . . . , A₈ be the vertices of a regular octagon that lie on a circle of radius 2. Let P be a point on the circle, and let P A_k denote the distance between the points P and A_k, for k = 1, 2, . . . , 8. If P varies over the circle, then the maximum value of the product P A₁ · P A₂ · . . . · P A₈ is:

The correct answer is 512.

Step 1: Understanding the Geometry in the Complex Plane
Let us place the circle of radius $R=2$ in the complex plane, centered at the origin $0$.
The vertices of the regular octagon $A_1,A_2,\dots ,A_8$ lie on this circle. Therefore, their complex coordinates are given by the 8th roots of $R^8$, or specifically:

$z_k=2e^{i(\theta +\frac{2k\pi }{8})}$

Without loss of generality, we can orient the octagon such that its vertices correspond to the complex numbers $z$ satisfying the equation:

$z^8-2^8=0$

Thus, the 8 vertices $A_k$ are the roots of the polynomial $P\left(z\right)=z^8-2^8$.

Step 2: Expressing the Product of Distances
Let $P$ be a point on the circle of radius $2$. Its complex representation is $z=2e^{i\varphi }$ for some real angle $\varphi$.
The distance $P{A}_k$ between the point $P$ and vertex $A_k$ is given by the modulus of their difference in the complex plane:

$P{A}_k=|z-z_k|$

Therefore, the product of the distances from $P$ to all 8 vertices is:

$\prod _{k=1}^{8}P{A}_k=\prod _{k=1}^8|z-z_k|=\left|\prod _{k=1}^8\right(z-z_k\left)\right|$

Since $z_k$ are the roots of $z^8-2^8$, we have the factorization:

$\prod _{k=1}^8(z-z_k)=z^8-2^8$

Thus, the product of distances becomes:

Step 3: Finding the Maximum Value
Substitute $z=2e^{i\varphi }$ into the expression:

$z^8={\left(2e^{i\varphi }\right)}^8=2^8{e}^{i8\varphi }$

So the product is:

$|2^8{e}^{i8\varphi }-2^8|=2^8|e^{i8\varphi }-1|$

Using the triangle inequality or by maximizing the distance from $e^{i8\varphi }$ to $1$ on the unit circle:

The maximum value of $|e^{i8\varphi }-1|$ occurs when $e^{i8\varphi }=-1$, giving a maximum value of $2$.
Therefore, the maximum value of the product of the distances is:

$\text{Maximum Value}=2^8\times 2=2^9=512$