\documentclass[12pt]{article}
\begin{document}
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}
\end{document}