1)Let $\displaystyle A,B\in \mathbb{R}^+$. Define $\displaystyle a_0=A$, $\displaystyle a_1=B$ and $\displaystyle a_n = a_{n-1}+a_{n-2} \mbox{ for }n\geq 2$. Find the radius of convergence of $\displaystyle \sum_{n=0}^{\infty}a_nx^n$.

The next two problems are for the younger kids, so give them a chance.

2)Let $\displaystyle S$ be a set (non-empty) of finite terms. A "partition" is breaking the set into two sets (non-empty) so that together they have the all the elements of $\displaystyle S$ but none of eachother. For example, let $\displaystyle S=\{1,2,3,4,5\}$ then $\displaystyle \{1,2,3\} \mbox{ and }\{4,5\}$ are partitions. But $\displaystyle \{1,2,3,4\} \mbox{ and }\{4,5\}$ are not. Say that $\displaystyle S$ has $\displaystyle n$ elements. How many different paritions are there in terms of $\displaystyle n$?

3)Given an $\displaystyle 8\times 8$ checkerboard what is the maximum number of checkers which can be placed so that no two are adjacent.Proveyour answer. "Adjacent" means either horizontally or veritcally next to eachothernotdiagnolly.