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Here are some Putnam Exam type problems.  The exam lasts six hours and has 12 
problems on it.  Each answer must be accompanied by a complete proof.
\begin{enumerate}
\item Let $A$ be a positive real number and assume that 
$A = \sum_{n=0}^{\infty}\,x_n$.
What are the 
possible values of $\sum_{n=0}^{\infty} x_n^2$\ ?

\item Prove that there are infinitely many positive integers $n$ such that 
each of the three integers $n$, $n+1$, and $n+2$ is the 
sum of two squares of integers.

\item The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with the 
vertices around the circumference in the given order.  Given that the polygon 
$P_1P_3P_5P_7$ is a square of area 5 and the polygon $P_2P_4P_6P_8$ is a 
rectangle of area 4, find the maximum possible area of the octagon.
\end{enumerate}
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