Solutions to Selected Problems in 308HI Pratice Midterm

Problem 2. Determine whether the following matrix is invertible:

Many of you tried mod 2 and mod 3 and the result is that the determinant is indeed 0 mod 2 and mod 3. One natural choice is then to try mod 4 and it worked! Indeed, we have:

The “Mod Trick” (2)

Here is another problem in which the “mod trick” applies:

Let be a by matrix with even integer entries. Prove that if is an odd integer, then is not an eigenvalue of .

Here is a solution:

The invariance principle (1)

Example 1 Initially an urn contains 100 black marbles and 100 white marbles. Repeatedly,
three marbles are removed from the urn and replaced from a pile outside the
urn as follows:
MARBLES REMOVED REPLACED WITH
3 black                                                                           1 black
2 black, 1 white                                                    1 black, 1 white
1 black, 2 white                                                           2 white
3 white                                                                     1 black, 1 white.

Which of the following sets of marbles could be the contents of the urn after
repeated applications of this procedure?

(A) 2 black marbles       (B) 2 white marbles          (C) 1 black marble
(D) 1 black and 1 white marble              (E) 1 white marble

Since this is a multiple choice problem, it is best if we can simply eliminate

A good question from one of my students

During a recent visit of my high school, Island School, one of my students asked the following questions:

Given an 8 by 8 chessboard with the opposite corners removed. How many rectangles are there?

First of all, I believe the motivation comes from the following two problems:

1) Given an 8 by 8 chessboard, how many rectangles are there?

2) Given an 8 by 8 chessboard with the opposite corners removed, can you tile it using 31 dominos?

The student simply asked the problem in the other setting. While the other problem formed this way “can you tile an 8 by 8 chessboard with dominos” is stupid, asking the number of ways to tile such a board using dominos is very difficult.  Before diving into this, let’s answer the question raised by the student.

In fact, we will answer the following: given an n by n chessboard with opposite corners removed, how many rectangles are there.

Proof: First label the opposite corners as

Personal(1): Teaching Statement

Students’ experience with mathematics influences their lives in many different ways, from making financial decisions to interpreting statistics in the news to understanding medical advice. It is my ambition to ensure that my students acquire the analytical skills required to help them in such daily experiences. For college teaching experience, I have taught both undergraduate and graduate level classes, and I have served as teaching assistant. I have also served as coaches for Putnam competition and the International Mathematical Olympiad as well as advising student projects for competition similar to Intel’s Science Talent Search. For my teaching philosophy, I think the most important point is

An Interesting Game in IMO Training Session

I once played the following game at an IMO training session for the advanced candidates. I simply asked the students one by one, to state theorems related to number theory. This turn out to be an excellent game to get the students involved and they are really actively engaged. According to my memory, here are the theorems stated:

1) Fermat’s little theorem

2) Euler’s theorem

3) Wilson’s theorem

4) Fermat’s Last Theorem

5) Fibonacci power theorem

6) q-th power lemma

7) Fermat’s two-square theorem

Lagrange’s four-square theorem

9) The prime number theorem

10) Hasse-Weil bound

11) Formula for primitive pythagorean triples

12) Solution to Pell’s equation

13) Quadratic reciprocty

14) Dirichlet’s theorem on infinitude of primes in arithmetic progression

15) Iwaniec’s theorem on the largest prime of n^2 + 1

16) Chen’s theorem

17) Iwaniec’s theorem on the primes and almost primes of n^2 + 1

18) Catalan’s theorem

I think there are a few more, but I do not remember them. Please add to the list if you have interesting theorems to share.

Sum of Squares (2)

Recall the question, let be a prime, in how many ways can we write as a sum of two squares?

The answer is , i.e. if , then the pair works. However, the following pairs also work:

The reason is simple if one thinks in terms of Gaussian integers Recall that the Gaussian primes are precisely the following:

1) .

2) Rational primes of the form

3) such that , a rational prime.

Since has unique factorization, if , we must have , therefore must be a Gaussian prime, and hence the prime divisors of are precisely and its unit multiples, namely: Likewise for the conjugate , we also have . Therefore, this gives 8 different ways to write as a sum of two squares.

For the general question: For a given integer , how many ways can we write it as a sum of two squares. For this, let to denote the number of ways to write as a sum of squares. We have:

For the proof, one may try to use Gaussian integers. There is a more general method that deals with finding , which uses the theory of modular forms. I will discuss that later.

Coming up: IMO 2008 Problem 3.

Hello world!

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