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\begin{document}

$\;$\hfill Do not need to submit.

\begin{center}
{\LARGE\bf Homework 12}
\end{center}

NOTE: Hints or solutions will be posted by the end of the week. You are welcome to
submit your writing, but it will not be counted towards your score for this course.

\begin{hw} Let $e$ be an edge in $K_n$. What is the number of spanning trees
that contain $e$ as an edge?
\end{hw}

\begin{hw} Let $A$ be the adjacency matrix of the complete bipartite graph $K_{3,3}$,
compute the eigenvalues of $A$.
\end{hw}

\begin{hw} Let $a_1 < ... < a_n$  and $b_1 < ... < b_n$ be two sequence of positive integers.
Let $M$ be the matrix where the $(i,j)$ entry is $\binom{a_i}{b_j}$. Then $\det M \geq 0$,
and it is 0 iff $a_i < b_i$ for some $i$.
\end{hw}

Consider the following a bonus problem.
Since we did not have time to cover the relevant thoughts in class,
it has nothing to do with the final exam.

\begin{hw}[I.M.O. 2007] Let $n$ be a positive integer. Consider
\[ S = \{ (x, y, z) : x, y, z \in [n] \cup \{0\}, x + y + z > 0 \}\]
as a set of $(n+1)^3 - 1$ points in $\br^3$. Determine the smallest number
of planes, the union of which contains $S$ but does not inlcude $(0, 0, 0)$.
\end{hw}

\end{document}

