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Name (Last, First): $\underline{\hspace{3in}}$ \hfill Student ID: $\underline{\hspace{2in}}$

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{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1415}{2014-15}:
  \href{\myurl/classes/15-475-ProblemSolving}{MAT 475 Problem Solving Seminar}:
  \hfill\url{http://drorbn.net/15-475}
}

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{\large\bf Quiz 4} ``Choose an Effective Notation'', February 3, 2015. You have 30 minutes to solve as much as you can of the following problems. Please write on both sides of the page. \hfill {\bf Good Luck!}

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\noindent{\bf Problem 1} (Larson's 1.5.1). A couple days ago in the morning it started snowing at a heavy and constant rate. A snowplow started out at 8:00AM. At 9:00AM, it had gone 2km. By 10:00AM, it had gone 3km. Assuming that the snowplow removes a constant volume of snow per hour, determine the time at which it started snowing.

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\noindent{\bf Problem 2} (Larson's 1.5.10).  A well known theorem asserts that a
prime $p>2$ can be written as a sum of two perfect squares ($p=m^2+n^2$
with $m$ and $n$ integers) iff $p$ is 1 mod 4. Assuming this, prove:
\begin{enumerate}
\item Every prime which is 1 mod 8 can be written as $x^2+16y^2$, with $x$
and $y$ integers.
\item Every prime which is 5 mod 8 can be written as $(2x+y)^2+4y^2$, with
$x$ and $y$ integers.
\end{enumerate}

\noindent{\bf Problem 3} (no credit, yet the best solutions will be advertised). What is your favourite ``Modify the Problem'' or ``Choose an Effective Notation'' problem?

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