Cut circle with straight lines Problem: To cut a circle with $n$ straight lines into the greatest number of possible parts.
What is the greatest number and how can you prove it is optimal?

the number of parts is $\binom{n}{2}+\binom{n}{1}+\binom{n}{0}$
I'll give you a hint first, consider what happens when you add a new line to the circle.  How many lines intersect it? and how many new parts does that create?
Note that this problem generalizes to $n$ planes cutting a sphere: for that, the number of parts would be $\binom{n}{3}+\binom{n}{2}+\binom{n}{1}+\binom{n}{0}$.  Use recursion to prove and by using the formula for the original problem. 
