Inductive proof problem

Good Anonymouses, this anonymous is having a little trouble with starting this induction proof:

Two non-overlaping circles of radius 1 intersect in exactly two points. Prove by induction that given n >= 2 non-overlapping circles in the plane that the total number of intersections is never more than n(n-1).

I know this is true and I can see why, but I don't know what to do after I've done the base case and the inductive hypothesis.

"Two non-overlaping circles of radius 1 intersect in exactly two points."

Uh... what about one with center (-1,0) and one with center (1,0)?

That part is a little ambigous but it says that when there are 2 or more circles, then the total number of intersections is never MORE then n(n-1), so it can be less. In the example you've given it is less.

>>1
Adding a circle to n circles will at most add 2n intersections, and n(n-1)+2n == (n+1)((n+1)-1).

>>3
He's referring to the 'exactly', which should probably be 'no more than'. I don't even see why that entire sentence is there, the problem works fine without it.

>>4
How would one add that to the proof? Wouldn't I need to prove that adding a circle to n circles will at most add 2n intersections first before I can use it?

This is actually the problem I'm having, its the word problems. If it was a simple "2^n < n!" or something I can do it but when it comes to word problems I don't know how to formula a sort of equation that would let me prove something.

>>6
If you have n circles and you add another one, and that adds more than 2n intersections then the added circle must intersect at least one of the n circles in more than 2 points.

Wouldn't I need to prove that adding a circle to n circles will at most add 2n intersections first before I can use it?
It follows directly from the base case (which you said you already proved, and which is actually what the first sentence of the problem states as a given with the change from >>5). Observe:

You have n circles. You add one. The intersections between the n circles won't change, but the added circle can make at most 2 more intersections with each of the n circles, hence at most 2n intersections are added.

>>7
>>8

Ah yes I see. I guess since it wasn't in the form of a plain and easy to see equation that I thought I needed to prove it first. I better start getting used to these word problems...