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.