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

1. Determine the minimum integer $n$ such that every $2$-coloring of $\{1,\dots ,n\}$ 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.)
2. [3.1.6] How many odd five-digit integers start with an even digit?
3. [3.1.8] Count the functions $f:\{1,\dots ,7\}\to \{1,2,3,4\}$.
4. [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?