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 . 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 such that as show that for any there is a polynomial such that

**(Masters Students and above)** If is a compact subset of show that the set

has Lebesgue measure