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

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\begin{center}
  {\Large UNIVERSITY OF TORONTO}\\
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  {\large Faculty of Arts and Sciences}\\
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  {\Large APRIL EXAMINATIONS 2015}\\
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  {\Large \href{\myurl/classes/15-475-ProblemSolving/}{Math 475H1 Problem Solving Seminar} --- Final Exam}

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  April 21, 2015\par
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Solve 8 of the following 11 problems. Indicate clearly which problems you wish graded; otherwise an arbitrary subset of the problems you have attempted will be chosen for marking. The problems carry equal weight.

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\noindent{\bf Duration. } You have 3 hours to write this exam.

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\noindent{\bf Allowed Material. } Stationary only.

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\centerline{{\bf Good Luck!}}

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\noindent{\bf Problem 1} (Larson's 1.1.4). Find positive {\red natural} numbers $n$ and $a_1,a_2,\ldots,a_n$ such that the\st{ir} sum ${a_1+\cdots+a_n}$ is 1000 and the\st{ir} product $a_1a_2\cdots a_n$ is as large as possible.

\vfil\noindent{\bf Problem 2} (Larson's 1.2.6). Let $ABC$ be an acute-angled triangle (all angles below $90^\circ$), and let $D$ be on the interior of the segement $AB$. Describe how one can find points $E$ on $AC$ and $F$ on $BC$ such that the triangle $DEF$ will have the minimal possible perimeter.

\vfil\noindent{\bf Problem 3} (Larson's 1.3.1). {\red Let $n$ be a natural number.} Find a general formula for the $n$th derivative of $f(x)=1/(1-x^2)$.

\vfil\noindent{\bf Problem 4} (Larson's 1.5.4). Let $-1 < a_0 < 1$ and define recursively for $n>0$,
\[ a_n=\left(\frac{1+a_{n-1}}{2}\right)^{1/2}. \]
What happens to $4^n(1-a_n)$ as $n\rightarrow\infty$?

\vfil\noindent{\bf Problem 5} (Larson's 1.6.2e). {\red Let $n$ be a natural number.} Of all the $n$-gons which can be inscribed in a given circle, which has the greatest area?

\vfil\noindent{\bf Problem 6} (Larson's 1.7.8). Determine $F(x)$, if for all real $x$ and $y$, $F(x)F(y)-F(xy)=x+y$.

\vfil\noindent{\bf Problem 7} (Larson's 2.6.1). {\red Let $n$ be a natural number.} Given $n+1$ positive integers, none of which exceeds $2n$, show that one of these integers divides another of these integers.

\vfil\noindent{\bf Problem 8} (Larson's 5.4.1). Prove that $e=\sum_{k=0}^\infty\frac{1}{k!}$ is an irrational number.

\vfil\noindent{\bf Problem 9} (Larson's 1.10.8). Let $n$ be odd and $\sigma$ a permutation of $\{1,2,\ldots,n\}$.
Prove that the product
\[ (\sigma_1-1)(\sigma_2-2)\cdots(\sigma_n-n) \]
is even.

\vfil\noindent{\bf Problem 10} (Larson's 3.3.28). Prove that there are infinitely many primes of the form $6n-1$.

\noindent{\bf Hint.} Consider $(p_1p_2\cdots p_k)^2-2$.

\vfil\noindent{\bf Problem 11} (Larson's 1.12.4c, modified) Compute the sum $\displaystyle \sum_{k=0}^n \frac{(-1)^k}{k+1}\binom{n}{k}$.

\vfil \centerline{\bf Good Luck!} \vfil

\noindent{\red In red: post-exam markup.}
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