# Problem Set I

In this problem set we present three questions.

(High School Students) Four squares, all of different areas, are cut from a rectangle, leaving a smaller rectangle of dimensions $1\times2$.  If the largest square has area 64, and the other three squares have side lengths that are whole numbers no larger than 7, what are their areas?

(Undergraduate Students) Given $f\in C([0,\infty ))$ such that $f(x)\rightarrow 0$ as $x\rightarrow \infty$ show that for any $\epsilon >0$ there is a polynomial $p$ such that

$\left\vert f(x)-e^{-x}p(x)\right\vert <\epsilon \,\,\, \forall x\in \lbrack 0,\infty ).$

(Masters Students and above) If $K$ is a compact subset of $\mathbb{R}^{n}$ show that the set

$A=\{x\in \mathbb{R}^{n}: d(x,K)=1\}$

has Lebesgue measure $0.$