In our first blog post Problem set 1 we presented three questions. We received many emails regarding the blog and were happy to see the interest it generated. There were also (different) solutions posted on our blog site. In this blog we present our set of solutions, we look forward to feedback on the same. Our next set of questions will be posted in middle of May.
(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?
Solution : The only possible areas in the answer are among the perfect squares 1, 4, 9, 16, 25, 36, 49. You can experiment with different arrangements. Graph paper would be helpful, and you might enjoy cutting the possible squares out of paper and shifting them around physically.
(Undergraduate Students) Given such that as show that for any there is a polynomial such that
Solution : Initial hunch is to hammer this problem to with the Weierstrass Approximation Theorem (see Terrence Tao’s blog for a more general version of the same). However, one needs to be careful in getting uniformity in approximation. Any simplification of the below proof is welcome.
Define a function by
Let be given. By Weierstrass Approximation Theorem ( is continuous ?) we can find a polynomial such that
for Suppose then from the above it is easy to note that (why ?). Translating the inequality in (1) for we have
for all Note that is now fixed. To finish the proof, using (2), we need to show that for and any $\eta >0$ there is a polynomial such that
for We prove the above by the method of mathematical induction on Let $\eta >0$ be given.
- Verification of base case: For we take
- Inductive step assumption: For assume that there exists a polynomial such that
- Proof for , : Let Using (4) with replaced by we have
We will now to find a polynomial such that approximates Let Then
We estimate the right hand side by observing:-
(b) for any
where we have used the inequality
and the fact that attains its maximum at
(c) Stirling’s formula, implies that there exists such that if then
Therefore if and
If we define the polynomial by for , from (5), (6), and (7) we have
As was arbitrary we have proved the case ..
This completes the proof.
(Masters Students and above) If is a compact subset of show that the set
has Lebesgue measure
Suppose Let Let with Then
Now note that and Hence
Similarly if then
Therefore we have shown that then
for any .
Using polar coordinates (See Real and Complex Analysis by Walter Rudin, 3rd Edition
Problem 6, Chapter 8) we conclude from that has measure
By translation the same conclusion holds if is contained in some
open ball of radius The general case is handled by noting
that is the union of a finite number of compact subsets of diameter not
exceeding and any point in has distance from one of
This completes the proof.