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.