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January 30 – Razvan Gelca – Check the Extremes!

January 30, 2010 – Dr. Razvan Gelca – “Check the Extremes”

Dr. Gelca will show how to solve mathematical problems by looking at the largest or the smallest element.

Art and Craft of Mathematical Problem Solving

Famous problem solver and lecturer, Paul Zeitz of the University of San Francisco, has just published a lecture series through the Teaching Company titled:  Art and Craft of Mathematical Problem Solving

This should be a great resource for any of our students and their families.  Here is a description from the Teaching Company site:

In 24 mind-enriching lectures, The Art and Craft of Mathematical Problem Solving conducts you through scores of problems—at all levels of difficulty—under the inspiring guidance of award-winning Professor Paul Zeitz of the University of San Francisco, a former champion “mathlete” in national and international math competitions and a firm believer that mathematical problem solving is an important skill that can be nurtured in practically everyone.

These are not mathematical exercises, which Professor Zeitz defines as questions that you know how to answer by applying a specific procedure. Instead, problems are questions that you initially have no idea how to answer. A problem by its very nature requires exploration, resourcefulness, and adventure—and a rigorous proof is less important than no-holds-barred investigation.

Think More Lucidly, Logically, Creatively

Not only is solving such problems fun, but the techniques you learn come in handy whenever you are presented with an unfamiliar problem in mathematics, giving you the confidence to try different approaches until you make a breakthrough. Also, by learning a range of different problem-solving approaches in algebra, geometry, combinatorics, number theory, and other fields, you see how all of mathematics is tied together, and how techniques in one area can be used to solve problems in another.

Furthermore, entertaining math problems sharpen the mind, stimulating you to think more lucidly, logically, and creatively and allowing you to tackle intellectual challenges you might never have imagined.

And for those in high school or college, this course serves as an enriching mathematical experience, equal to anything available in the top schools. Professor Zeitz is a masterful coach of math teams at every level of competition, from beginners through international champions, and he knows how to inspire, encourage, and instruct.

Strategies, Tactics, and Tools of Math Masters

The Art and Craft of Mathematical Problem Solving is more than a bag of math tricks. Instead, Professor Zeitz has designed a series of lessons that take you through increasingly more challenging problems, illustrating a variety of strategies, tactics, and tools that you can use to overcome difficult math obstacles. His goal is to give you the persistence and creativity to turn over a problem in your mind for however long it takes to reach a solution.

The first step is to come up with a strategy—an overall plan of attack. Among the many strategies that Professor Zeitz discusses are these:

• Get your hands dirty: Dive in! Plug in numbers and see what happens. This is a superb starting strategy because it almost always shows a way to keep on investigating. You’ll be surprised at how often a pattern emerges that takes you to the next step.
• Think outside the box: Break the bounds of conventional thinking. Professor Zeitz shows you the original think-outside-the-box problem, in which the key idea is to disregard the boundaries of an implied box. He also explains why he prefers to call this strategy “chainsaw the giraffe.”
• Wishful thinking: Turn a hard problem into an easy one by removing the hard part. For example, substitute small numbers for big ones. This is a confidence-builder that often gives you a partial solution that shows you how to solve the original problem.
• Change your point of view: Every problem has a natural point of view, such as a time or place where something is happening. Step back and try a different point of view. This could mean recasting an algebra problem as one in geometry, or vice versa.

The next step after choosing a strategy is to find a suitable tactic. For example, suppose you live in a cabin that is two miles north of a river that runs east and west, and your grandma’s cabin is 12 miles west and 1 mile north of your cabin. Every day you go to visit grandma, but first you stop by the river to get fresh water for her. What is the length of the route that has the minimum distance?

You start with the “draw a picture” strategy. Once you have something to look at, you realize that the “symmetry” tactic will give you the shortest distance. Here’s how it works: Imagine an alternate you on the same errand but on the south side of the river, in a mirror image of the situation on the north side. By drawing a line connecting your real cabin with the alternate grandma’s cabin, and another line connecting the real grandma’s cabin and the one belonging to the alternate you, you find an intersecting point at the river that is the perfect place to stop.

On some problems you also need a special-purpose technique—a tool. For example, the 10-year-old Gauss’s trick of pairing numbers in the earlier example is a tool whose underlying idea—symmetry—can be applied to a wide range of problems. You learn the strengths, as well as possible pitfalls, of such tools.

Prepare for an Exhilarating Experience

Professor Zeitz compares this systematic approach to problem solving—in which you deploy strategies, tactics, and tools—to the mountaineer’s quest to reach the top of a high peak. The mountain may seem insurmountable, but there is always a way to conquer it by proceeding one step at a time.

Looking at an impressive mountain, you can’t but feel a sense of awe at the prospect of climbing it. Math problems, too, can produce this same reaction. But don’t be daunted: You are more ready than you think. So sharpen your pencil, get some paper, and prepare for the exhilarating experience of The Art and Craft of Mathematical Problem Solving.

Dr. Paul Zeitz is Professor of Mathematics at the University of San Francisco. He majored in history at Harvard and received a Ph.D. in Mathematics from the University of California, Berkeley, in 1992, specializing in ergodic theory.

One of his greatest interests is mathematical problem solving. He won the USA Mathematical Olympiad (USAMO) and was a member of the first American team to participate in the International Mathematical Olympiad (IMO) in 1974. Since 1985, he has composed and edited problems for several national math contests, including the USAMO. He has helped train several American IMO teams, most notably the 1994 “Dream Team,” which, for the first—and heretofore only—time in history achieved a perfect score. He founded the San Francisco Bay Area Math Meet in 1994 and cofounded the Bay Area Mathematical Olympiad in 1999. These and other experiences led him to write The Art and Craft of Problem Solving (1999; second edition, 2007).

He was honored in March 2002 with the Award for Distinguished College or University Teaching of Mathematics by the Northern California Section of the Mathematical Association of America (MAA), and in January 2003, he received the MAA’s national teaching award, the Deborah and Franklin Tepper Haimo Award.

January 23 – Dr. Andreescu – A Few Brain Teasers

Metroplex Math Circle will start its 2010 Spring season this Saturday with a lecture by Dr. Andreescu himself.  Dr. Andreescu’s problem sets are always entertaining and challenging.  However, as students prepare for the AMC 10 and 12 it is particularly helpful to learn from a former director of the AMC.

MEANT Spring 2010 Results

Congratulations to all of the students who took part in the recent MEANT Math Olypiad at UTD.  We saw many math circle participants and they featured prominently among the announced winners:

Group A (Grades 5 & 6)
——–
1st Place – Vivian Zhou(6th Grade, Rice Middle School)
2nd Place – Youngwoo Sohn(6th Grade, St.Mark School of Texas)
3rd Place – Nikitha Vicas(5th Grade, MIS)

Group B (Grades 7 & 8 )
——–
1st Place – Niranjan Balachandar(7th Grade, Frankford Middle School)
2nd Place – Victor Zhou(8th Grade, Rice Middle School)
3rd Place – Robert Tung(8th Grade, Rice Middle School)

Group C (Grades 9 & 10)
——–
1st Place – Daniel Huang(9th Grade, Shepton High School)
2nd Place – Hongzhi Zhao(10th Grade, Jasper High School)
3rd Place – Jonathan Zhu(10th Grade, Jasper High School)

These students will be guests at the MEANT banquet where they will be recognized and given prizes.

First Annual UT Arlington Math Competition

UT Arlington will be hosting a Math Competition that should be of interest to many of our Math Circle participants.  Here are the details:

First Annual UT Arlington Math Competition

Saturday, February 6, 2010, 9:00 am – 12:00 pm
Pickard Hall, Room 110
The University of Texas at Arlington

Who Participates?
Both middle-school and high-school students are welcome to participate in the competition. There is no registration fee. If you plan to attend, please indicate so on your Calculus Bowl online registration form. If you plan not to attend the Calculus Bowl, but to participate in the UT Arlington Math Competition, please contact Dr. Dimitar Grantcharov (grandim@uta.edu, 817-272-1148).

What is the UT Arlington Math Competition?
The competition will contain two parts: one part with about 20 multiple-choice problems, and another part containing one essay-type problem. Calculators will not be allowed. The multiple-choice problems will be at a level of compatible to the American Mathematical Contest 10 (AMC 10), while the essay-type problem will be a bit more challenging. A few sample problems are provided below.

• Which of the following numbers is a perfect square?
(A) 98!99! (B) 98! 100! (C) 99! 100! (D) 99!101! (E) 101!101!
• A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle?
• A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players A, B, and C start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game?

What is the Math Battle?
The Math Battle is an exciting two-team problem-solving competition. Each of the two teams will receive a list of problems in advance. A jury and the teams will discuss the solutions of the problems in the afternoon.

Prizes?
The student with the highest score on the UT Arlington Math Competition will be awarded a \$100 gift card. The top five contestants will be awarded medals and trophies.

AwesomeMath 2010 Early Application

AwesomeMath, Dr. Andreescu’s highly regarded summer camp for mathematically gifted students has just released its test and forms for early admission.  Early registration is due by February 5th.

For those of you who are unfamiliar with the camp here are some details for 2010:

What? A three-week intensive summer camp for mathematically gifted students from around the globe. An initiative in response to parents and teachers of bright students who have not yet shone at the Olympiad level, as well as of those who would like to expand what they have learned in programs such as MATHCOUNTS. These talented students wish to hone their problem solving skills in particular and further their mathematics education in general. Many of our participants seek to improve their performance on contests such as AMC10/12, AIME, or USAMO.

Why? To offer gifted math students the opportunity to attend a high-quality summer program.

When? July 6 – 27, 2010 and July 30 – August 20, 2010.

Who? Around 80 students in grades 6-11, distinguished faculty, a large group of mentors and assistants.

Where? At the University of Texas at Dallas (7/6-7/27) and the University of California, Santa Cruz (7/30-8/20).

NACLO 2010

Our good friend and 2009 lecturer, Dr. Vincent Ng, will once again host the NACLO competition at UTD this year.  Computational Linguistics is a fascinating field which shares a lot in commons with mathematical problem solving.  In fact most of the students who make the national team are also highly accomplished math problem solvers.  No prior experience is necessary but the prep sessions that Dr. Ng may offer are not to be missed!  Here is his announcement to past participants:

As you may know already, we are organizing NACLO again on the UT Dallas campus this year. Just like last year, we will have an open round (on Feb 4th) and an invitational round (on March 10) [just check out www.naclo.cs.cmu.edu for details].

Registration is open until noon, Feb 3 (PST), but I encourage you to sign up early if you are interested. (I noticed that some of you have already signed up.) If there is sufficient interest, we will be organizing 1-2 problem sessions to solve some of the past NACLO problems together at the end of January (most likely on Jan 24 or Jan 31 or both).

If you have any questions, feel free to let me know. I hope to see you again in this year’s NACLO!

Best,
Vincent Ng
Dallas Site Coordinator
NACLO 2010

MEANT Contest – January 16

The Malayalee Engineers’ Association of North Texas (MEANT) will hold their math contest Saturday, January 16th at UTD.  This was a well attended contest last year with some challenging and interesting problems.  Please visit the MEANT web site for more information:

• When: 1:00 PM on Saturday, January 16th 2010
• Where:  UTD School of Management, Room 1.107, 800 West Campbell Road, Richardson TX
• Test Levels:  Grades 6 through 10th

AMC 8 2009 Perfect Scores

The perfect scores for the recent AMC 8 have been posted.  Congratulations to all of the Texas students listed below, but we are especially proud of our own Metroplex Math Circle attendee, Niranjan Balachandar!

NIRANJAN BALACHANDAR, Frankford MS, Dallas

BRIAN XU, Harmony Sch of Excellence, Houston

PAUL CRUZ, Harmony Sch of Excellence, Houston

GENE HSU, First Colony MS, Sugar Land

MITCHELL HWANG, First Colony MS, Sugar Land

ZACHARY TU, Fort Settlement MS, Sugar Land

Telescoping Sums and Products (Example 2)

This time I will provide both the problem and solution in  the continuing series from Dr. Andreescu’s lecture on Telescoping Sums and Products.

Example 2.  Evaluate

$\displaystyle\sum^{n}_{k=2}k!(k^2+k+1)$

Example 2.  Solution

We can write

$\displaystyle\sum^{n}_{k=1}k!(k^2+k+1)=\displaystyle\sum^{n}_{k=1}[(k+1)^2-k)]k!$

$=\displaystyle\sum^{n}_{k=1}[(k+1)!(k+1)-k!k]$

$=(n+1)!(n+1)-1$