Ratio of product from one point and minimum distance Given points $A_0,A_1,\ldots,A_n$ in the plane, let $m$ denote the minimum distance among any two points. What is the minimum value of $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$

Here $|AB|$ denotes the Euclidean distance between points $A$ and $B$.

[Source: Based on Chinese competition problem]
Lets consider

$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}$

as

$\dfrac{|A_0A_1|}{m}\cdot\dfrac{|A_0A_2|}{m}$...$\dfrac{|A_0A_n|}{m}$

We know that for any k:

$\dfrac{|A_0A_k|}{m}\geq1$

Thus the minimum value for 

$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}$

would be $1$, but we must show that such a configuration can indeed exist on the plane. For this we will just pick three points $A_0,A_1,A_2$ such that they form an equilateral triangle and we get that our expression evaluates to $1$, so we are finished.

In the case where $m=0$ and $2$ points coincied our expression is undefined so it is not at a minimum there either.
Lets consider

$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}$

as

$\dfrac{|A_0A_1|}{m}\cdot\dfrac{|A_0A_2|}{m}$...$\dfrac{|A_0A_n|}{m}$

We know that for any k:

$\dfrac{|A_0A_k|}{m}\geq1$

Thus the minimum value for 

$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}$

would be $1$, but we must show that such a configuration can indeed exist on the plane. For this we will just pick three points $A_0,A_1,A_2$ such that they form an equilateral triangle and we get that our expression evaluates to $1$, so we are finished.

In the case where $m=0$ and $2$ points coincied our expression is undefined so it is not at a minimum there either.