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\headers{Math378}{Homework 8}{Due Fri. Mar 13, 2026}

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\noindent
\textbf{Directions:} Solve the following problems.  All written work must be your own.  See the course syllabus for detailed rules.

\begin{enumerate}

\item {} Determine the minimum integer $n$ such that every $2$-coloring of $\{1,\ldots,n\}$ contains a monochromatic solution to $x+2y=z$.  (Hint: consider the case that $1$ and $2$ have the same color and the case that $1$ and $2$ have distinct colors.)

\item {} [3.1.6] How many odd five-digit integers start with an even digit?

\item {} [3.1.8] Count the functions $f\st \{1,\ldots,7\} \to \{1,2,3,4\}$.

\item {} [3.1.14] How many ways are there to color the vertices of a pentagon with three colors such that no two adjacent vertices receive the same color?


\end{enumerate}
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