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ThePerfectHacker · #1 September 16th, 2007, 08:41 PM

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

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

2)Let $S$ 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 $S$ but none of eachother. For example, let $S=\{1,2,3,4,5\}$ then $\{1,2,3\} \mbox{ and }\{4,5\}$ are partitions. But $\{1,2,3,4\} \mbox{ and }\{4,5\}$ are not. Say that $S$ has $n$ elements. How many different paritions are there in terms of $n$ ?

3)Given an $8\times 8$ checkerboard what is the maximum number of checkers which can be placed so that no two are adjacent. Prove your answer. "Adjacent" means either horizontally or veritcally next to eachother not diagnolly.