Let $n$ be a positive integer, and $n$ lines drawn in a ring such that each one of them intersects with all of them, but no more than two intersect at one point. prove that the lines cut the disk $n^2+n+2\over 2$.
I can't figure out how to approach to question. Can you give me some hints ?
Suppose you already have $(n-1)$ chords in your circle, and you add the $n$th. It will intersect the other $(n-1)$ chords in $(n-1)$ distinct points, so it will be cut into $(n-1)+1=n$ segments. Each such segment splits one previous region into two new ones. So you added $n$ regions. Now show that this inductive step fits your formula, and also check the base case of $n=1$ or even $n=0$.
