TITLE: prove that the greatest number of regions that $n \geq 1$ circles can divide the plane is $n^2-n+2$
QUESTION [7 upvotes]: This is an induction problem, but I have no idea how to do something like this. Any hints?

REPLY [3 votes]: Hint: Assume that you already have $k$ circles, and draw one more.

At how many points will that new circle intersect the earlier ones - at maximum?
How many arcs is that new circle split into (by those points of intersection)?
How many pre-existing regions are split in two (at maximum)?