Thank you, Henry. That is very counterintuitive. My question was based on my intuition that the most dense packing would have as many spheres kiss as possible.

Fedja, you are correct, but your observation actually proves my point. You really think it is possible to repack the original spheres so that none of them touch at all? How far would the closest two spheres be from one another?

I just wanted to keep things nontrivial.

@PeterTaylor, I want $n$ as close to 2 as possible.

@seva I will think about this more. Thank you again for your help with this.

@seva I ask because I still believe that the inequality in my question is true, even though I haven't proven it.

@Seva, Can you also find a counterexample to $f(x,y)=x^2+y^2-(x+y)^2$ and $g(x,y)=(x+y)^2$.

@Seva it looks like you are correct. Thank you.

@seva, do you have a counter-example?

@seva $|\{f(x,y):x,y \in A\}|+|\{g(x,y):x,y \in A\}| \geq |\{f(x,y)+g(x,y):x,y \in A\}|$. In this problem, $f(x,y)=x^2+y^2-(x+y)^2$ and $g(x,y)=(x+y)^2$.

@seva it is the triangle inequality for set cardinality.