October 6, 2016

We started by working on the following problem from the 2014 Virginia Tech Regional Math Contest:

Suppose we are given a 19×19 chessboard (a table with \(19^2\) squares) and remove the central square. Is it possible to tile the remaining \(19^2 -1 = 360\) squares with 4 × 1 and 1 × 4 rectangles? (So each of the 360 squares is covered by exactly one rectangle.) Justify your answer.

We proved that the answer is no, by adapting the proof of a classical problem: it is not possible to tile an 8×8 chessboard with 2 opposite corners removed using 2×1 and 1×2 tiles.

Then we worked on solutions for three problems from the 2010 Putnam:

  • B1: We used the suggestion from last week’s post, and also saw how to do a proof using the Cauchy-Schwarz inequality.
  • A2: Using ideas similar to the ones we used in problem 8 of the proof strategies handout, we could prove that the derivative of f is periodic with period 1. Then we took the second derivative of f, which turned out to be zero and thus f must be of the form ax + b for some constants a and b.
  • A3: Here we first thought about a similar but simpler problem: \(h(x) = ah'(x) \). Using standard ODE techniques,\(h(x) = Ce^{-x/a}\), so the boundedness condition forces the constant C to be 0 and thus h is zero as well. For the two variable case, fix x and y and take derivatives of \(h(x+at,y+bt) \) with respect to t.

Today’s nuggets of wisdom:

  • Once again, invariants made an appearance. Without them, it can be quite difficult to prove that something does not exist!
  • Simplification is often a good idea to get you started.

 

Virginia Tech Regional Math Contest 2016

This year, OU will be participating in the Virginia Tech Regional Mathematics Contest for the first time!

This competition was conceived as a regional preliminary contest for the Putnam Competition: it is shorter, and you could think of it as a mini-Putnam. The flavor of the problems is slightly different, but it can be a very good way of experiencing trying to solve problems against the clock. More information, including exams from previous years, is available at

https://www.math.vt.edu/people/plinnell/Vtregional/

The competition will take place on Saturday October 22, from 9:00 to 11:30am, in the Physical Sciences Center (exact room to be determined depending on the number of participants). If you are interested in participating, please fill out the registration form linked below:

https://docs.google.com/forms/d/e/1FAIpQLSfnfJ05xk71YTat9Ko29DwYbeadbN4waYeDMby8i0r2D2Am5w/viewform

 

September 29, 2016

Today we worked on the problems from the 2010 Putnam Exam (handed out a couple of weeks ago). We had quite a good time, working in small groups.

Different people worked on different problems, and every problem was tried by at least one person. The only one we presented at the front was problem A1.

A few comments we made about other problems:

  • A4: It is a good idea to try to find a nontrivial divisor when \(n=1\), we may be able to get inspiration from that.
  • A5: What does it mean for \(G\). to be a group with operation \(\ast\)?
  • A6: In general, to show that the integral of a positive function \(h(x)\) diverges it is useful to compare it a function \(g(x)\) such that \(h(x) \ge g(x) \ge 0\) and the integral of \(g(x)\) diverges.
  • B1: First we need to make a reasonable guess regarding whether it is possible or not to find such a sequence. Notice that if \(a>1\) then \(a^m\) increases as \(m\) increases, but the opposite happens when \(0<a<1\).

Today’s nuggets of wisdom:

  • In a problem where many partitions/configurations/etc. are possible, look for an invariant: a quantity that never changes. For example, in problem A1 the total sum of the numbers never changes.
  • We need to be careful when showing that \(k\) is the maximum value for which something is possible: it does not suffice to be convinced that we have found the maximum, we need to prove that it is in fact the maximum.

September 22, 2016

We continued working on the problems from last week’s proof strategies handout.

  • We solved problem 4 using classical induction on the number of vertices of the polygon. When reducing the case of \(n+1\) vertices to the case of \(n\), we had to be careful. In particular, making sure that we still had three colors was a bit of a fine point.
  • We observed that problem 4 might also be done using strong induction. We did not finish the details of the argument, since it did not appear to be easier.
  • The theme of the day was becoming induction, so we worked on problem 8. We observed that having the equality for \(m\) allowed us to prove it for \(2m\), and thus from the hypothesis we get it for all powers of 2 by induction. To finish, we noticed that we could prove the equality for \(n=3\) assuming the case \(n=4\), and more generally we could prove the case of \(n-1\) from the case for \(n\).
  • There was some discussion about what functions \(f : \mathbb{R} \to \mathbb{R}\) can satisfy the equality in problem 8. We noticed that linear functions (\(f(x) = ax\) for a real constant \(a\)) and affine functions  (\(f(x)= ax+b\) for real constants \(a,b\)) do satisfy it, and we looked at the geometrical interpretation of the equality. For those who have taken linear algebra, we also pointed out that any \(\mathbb{Q}\)-linear function \(\mathbb{R} \to \mathbb{R}\) satisfies the equality, and there are such \(\mathbb{Q}\)-linear functions which are not \(\mathbb{R}\)-linear (though we cannot write them explicitly, we can only prove their existence using arguments that depend on the Axiom of Choice).

Today’s nuggets of wisdom:

Arguments by induction come in several different flavors, including:

  • Classical: We assume the statement holds for \(n\) , and prove it for \(n+1\).
  • Strong: We assume the statement holds for \(1,2,\dotsc,n\) , and prove it for \(n+1\).
  • Back and forth: This one has two parts:
    • Using classical induction, we prove the statement for an increasing sequence of natural numbers (say, powers of 2).
    • We assume the statement holds for \(n\) , and prove it for \(n-1\).

 

September 15, 2016

Today was our first session for the Fall 2016 semester!

We worked on problems from the following two handouts:

  • Examples of several proof strategies:
    Including contradiction, induction, the pigeonhole principle and extremality. We solved problems 1, 2 and 3, and started thinking about problem 4. Problem 2 was especially interesting, because we had to work hard and use a mix of strategies. Though we did find a solution to Problem 3, we are still trying to find a different one using induction.
  • The 2010 Putnam Exam:
    A few people started working on these, but we have not yet presented any ideas in front of everybody.

Today’s nuggets of wisdom:

  • If we have a triangle, and we move one of its vertices along the line passing through it and parallel to the opposite side, the area of the triangle does not change.
  • In order to use extremality, we need to have a number associated to each object or configuration. Examples:
    • Length, area, volume, etc.
    • Number of: sides, vertices, edges, regions, etc.
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Welcome

The Putnam Mathematical Competition is the most prestigious math contest for undergraduate students in the U.S. and Canada. The exam is designed to test creative thinking in addition to technical competence, and is held every year on the first Saturday of December. The problems range across the undergraduate mathematics curriculum, but most do not require specialized knowledge of mathematics beyond calculus.

Unlike standard textbook problems, where many of them are quite similar, Putnam problems are challenging (and fun!) precisely because it is generally not immediately obvious how to approach them. There are some standard techniques that often can be useful to start making progress towards a solution, but a good amount of practice is needed to develop an intuition of when to apply each one. For this reason, we hold the Putnam Problem-Solving Seminar at OU each fall: we meet weekly to solve Putnam-style problems and learn about some problem-solving strategies.

Anyone interested in problem-solving is welcome at the seminar, whether they’re thinking of participating in the competition or not. Working on difficult, but interesting, mathematical problems is a good way to improve your analytical abilities and have fun at the same time. You’ll not only be better at mathematics proper, you’ll be able to work better in any field that requires analytical reasoning!

Below is information about the 2016-2020 seminars, and at this link you can find the archives for 2015 and before.

For the fall 2020 semester, the Putnam Problem-Solving Seminar is being run by Alejandro Chávez-Domínguez (jachavezd[at]ou[dot]edu) and Roi Docampo (roi[at]ou[dot]edu).

We are meeting virtually on Thursdays 5:00-6:30pm using Zoom, starting on September 3rd. Please do join us even if you need to arrive late or leave early, this is not a class and the format easily allows for people to come and go as they need.

Meeting ID: 987 0807 8441
Passcode: Putnam2020

Fall 2016: first meeting

Our first meeting of the semester will be held:

  • Thursday, September 15, 5:00-6:30pm
  • PHSC 1025

Please do come even if you need to show up a bit later or leave early, this is not a class and the format easily allows for people to come and go as they need.

 

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