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## Math Circe Profile: Berkeley

It might be interesting to look at several of the thriving Math Circles around the country.  The Berkeley Math Circle has an excellent site with monthly contest problems available for download.  Berkeley is also an excellent example of the positive impact that a Math Circle can have on its community.  Here is some information from the Berkeley site on their outstanding alumni:

The success of Berkeley Math Circle in identifying and fostering talent is striking. Although the BMC only started in 1998, the 6-member team at the International Math Olympiad (held in Washington, D.C., July 2001) included 3 members from this program:

• Gabriel Carroll, graduated from Harvard as a math major and currently attending the Ph.D. program in Economics at MIT. He has won 2 Gold and 1 Silver medals at the IMO (including one perfect score in 2001), and won the Putnam competition 4 times. He was the grand prize BAMO winner 3 times.
• Tiankai Liu, now at Harvard, won 3 Gold medals at the IMO. He attended the Research Experience for Undergraduates Program at Duluth, Minnesota in Summer 2007.
• Oaz Nir has won 1 gold and 1 silver medal and has graduated from Duke University as a math major.

The three Berkeley Math Circle students contributed to the USA’s second-place finish among over 80 countries at the International Mathematics Olympiad in 2001. In 2002, students from the Berkeley Math Circle and BAMO continued to do exceptionally well in mathematics competitions. Over the years, a number of Berkeley Math Circle students were among the top twelve winners of the USA Math Olympiad, and one was among the five students in the US with a perfect score: Inna Zakharevich (Henry Gunn High School, Palo Alto, Currently a student at Harvard University). Several other Circle students qualified for the summer training program of the US team several years in a row. Evan O’Dorney, only in 8th grade, qualified among the top 24 students for the training of the USA Math Team in Summer 2007. He further won the Grand Prize at BAMO 2007 with a perfect score. Additionally, he has been a three-time finalist and 2007 Champion at the Scripps National Spelling Bee in Washington D.C.

Among other famous alumni of BMC and BAMO, it is worth mentioning Maxim Maydanskiy who tied for first place with Gabriel Carroll at BAMO 2001. Maxim was admitted to UC Berkeley, and upon recommendation from the BMC circle coordinator, Dr. Stankova, his Circle and Olympiad activities played a major role in awarding him the Regent’s scholarship, the most prestigious UC Berkeley scholarship for entering undergraduates. While at UC Berkeley, he also attended the Research Experience for Undergraduates Program at Duluth, Minnesota, and is currently a Ph.D. student in Mathematics at MIT.

## Chengde Feng – Angles and Areas – November 1

Chengde Feng, an instructor at AwesomeMath, father of Zuming Feng and a good friend of the Metroplex Math Circle will come to Dallas this Saturday to lecture on “Angles and Areas.”  To help us prepare, Mr. Feng has sent along this list of facts that every Middle School Student should know.

### Concepts You Need to Know

1.  Angles of a Triangle

• Vertical angles are congruent
• The sum of the measures of the angles of a triangle is 180.
• The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

2. Angles of a Polygon

• The sum of the measures of the interior angles of a n-gon is ${(n-2)180}$
• The measure of each interior angle of a regular n-gon is $\dfrac{(n-2)180}{n}$
• The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360.

3. Triangles

• The length of each side of a triangle must be less than the sum of the lengths of the other two sides.
• If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
• An equilateral triangle has three $60^\circ$ angles.
• Pythagorean Theorem In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

4.  Areas The area of:

• a rectangle equals the product of its base and height.
• a parallelogram equals the product of its base and height.
• a triangle equals half the product of its base and height.
• an equilateral triangle with side length l is $\dfrac{\sqrt{3}}{4}t^2$
• a rhombus equals half of the product of its diagonals.
• a trapezoid equals half the product of the height and the sum of the bases.
• a circle with radius r is $\pi r^2$ and its circumference is $2 \pi r$.
• a sector AOB of circle O with radius r is $\dfrac{m\angle AOB}{360}\pi r^2$

## MEANT Math Contest November 15th

The Malayalee Engineers Association of North Texas is sponsoring a math contest.  To support this initiative, Metroplex Math Circle will suspend its normal November 15th meeting allowing more students to participate.

When MEANT last held a contest one of their winners was a young Zach Abel.  Zach went on to great success at the USAMO and IMO as document in the movie Hard Problems.  Perhaps one of the participants in this year’s MEANT contest will follow in Zach’s footsteps.

We will update the following information as we receive it from MEANT.

MEANT in partnership with Microsoft Corporation, announces a competition in mathematics, open to all 6th to 10th grade students in the Dallas Metroplex.

WHEN : November 15th, 2008, Saturday
TIME : 1:00 PM
VENUE : University of Texas at Dallas (UTD).
School of Management (Room 2.801, 2nd Floor)
800 West Campbell Road, Richardson, TX 75080
Map : (http://www.utdallas.edu/maps/)

Last date to Register : November 11th, 2008.
Registration fee : \$10.00 (Non refundable after Nov 11th, 2008).

## Nova: Hunting the Hidden Dimension

Thank you Mahender Nelakonda for catching the following announcement.

## Next on NOVA: “Hunting the Hidden Dimension”

http://www.pbs.org/wgbh/nova/fractals/

Tuesday, October 28 at 7 p.m. CDT (on KERA)

(Check your local listings as dates and times may vary.  Broadcast in high definition where available.)

You may not know it, but fractals, like the air you breathe, are all around you. Their irregular, repeating shapes are found in cloud formations and tree limbs, in stalks of broccoli and craggy mountain ranges, even in the rhythm of the human heart. NOVA’s Hunting the Hidden Dimension takes viewers on a fascinating quest with a group of renegade mathematicians determined to decipher the rules that govern fractal geometry.

For centuries, fractal-like irregular shapes were considered beyond the boundaries of mathematical understanding. Now, mathematicians have finally begun mapping this uncharted territory. This program highlights a host of filmmakers, fashion designers, physicians, and researchers who are using fractal geometry to innovate and inspire.

Here’s what you’ll find on the companion website:

http://www.pbs.org/wgbh/nova/fractals/mandelbrot.html

Benoit Mandelbrot is a true maverick, as his interview reveals.

### The Most Famous Fractal

http://www.pbs.org/wgbh/nova/fractals/set.html

What exactly is the Mandelbrot set? Find out in this excerpt from the book Fractals: The Patterns of Chaos.

### Design a Fractal

http://www.pbs.org/wgbh/nova/fractals/design.html

Create and save your own wildly colorful fractals using our generator.

### Sense of Scale

http://www.pbs.org/wgbh/nova/fractals/scale.html

Explore the infinite detail of a Mandelbrot set as you zoom to 250,000,000x magnification.

Also, Links & Books, the Teacher’s Guide, the program transcript, and more:

http://www.pbs.org/wgbh/nova/fractals/

## “Hairy Circles” – Recap

Dr. Paul Stanford departed from our recent lectures on applied mathematics and contest preparation to give our students a glimpse into the fascinating world of pure mathematics.  Dr. Stanford is particularly skilled at teaching deep ideas without the need to resort to complex algebra, I’ll attempt in this recap to do his lecture a small bit of justice.

Dr. Stanford began with one of the simplest concepts, points on a plane and arrows connecting those points.  Many of these arrows considered together and connecting a set of points yield all sorts of interesting directed graphs or “digraphs.”  This idea becomes more interesting when restrictions are imposed such as the rule that no two arrows can originate from the same point.  When a sequence of arrows loop back upon themselves they form a shape like a “hairy circle” from the title of the lecture.

Dr. Stanford further restricted the cases with the following rules:  no two arrows can connect to the same point and no symmetry is allowed.    These restrictions yielded only four possibilities including those with only a single point connecting back on itself and the possibility of having no points or arrows at all.

Dr. Stanford then proceeded to show how basic arithmetic could be carried out in each of these four systems by shifting along the chains of arrows and by considering the very special case of the arrow that points back to its own origin.

In the second half of the lecture, Dr. Stanford built upon this foundation as he introduced the Collatz Problem:

Think of a number.

If it is even, divide by two.

Otherwise, triple it and add one.

Repeat.

Does this always reach one?

Dr. Stanford lead the students through multiple examples using positive integers, some of which filled both whiteboards but eventually came back to one.  In fact Dr. Stanford told us that while this process always yields one for numbers from 1 to $2.7 \times 10^{16}$,  nobody has proven that it is true of all positive integers!  Dr. Stanford reminded the students that for mathematicians multiple examples are not “proof” and that numbers even as large as 23 thousand, trillion are not especially large to mathematicians.

However, when Dr. Stanford started the Collatz problem with negative integers, it was common that loops would appear.  Drawn on the board, the chain of calculations was strangely similar to the hairy circles from the previous lecture.  This lead to a discussion of how one could detect when loops appear in an algorithm.  At this point the lecture began to  bridge between pure mathematics and some real problems in computer science.

One method for detecting a loop would be to track and remember every point in the path but this would burden even the most powerful computer.  A more clever method is the “tortoise and the hare” strategy.  This involves sending two runners through the system, one moving twice as quickly as the other.

Dr. Stanford proved how this method would eventually prove the existence of a loop and how a second tortoise could confirm the exact point where the loop begins.

The two lectures formed an exciting and satisfying trip through the world of numbers.  If I have misrepresented or ignored pertinent points from the lecture please feel free to mention them in the comments section below.

Our friends at Art of Problem solving hosted a Math Jam interview with Matt McGann, Associate Director of Admissions and Kiran Kedlaya, Associate Professor of Mathematics at MIT.  Here are a couple of highlights and a link to the full transcript:

Q:  What is MIT’s admission rate?

A: Last year, if memory serves, we received 13,396 applications and admitted 1589 students, for an admission rate of about 11.9%. But remember, just the admission rate tells you very little about the admissions process or the quality of the school.

Q:  Where are the math major students go and what do they do once they graduated from MIT?

A: Our math majors choose a variety of career paths. Some pursue PhDs in math and continue in academic careers; some do likewise in related subjects (physics, computer science). Finance is a popular option, as are various IT-oriented careers.

Q:  What are some of the research opportunities available during the vacations?

A:MIT has an extensive Undergraduate Research Opportunities Program (UROP), through which undergrads can get funding to do research projects with faculty either during vacations or academic terms. The burden is on the student to find a faculty mentor, but many faculty participate in the program. (I’ve advised maybe 10 students in this way.)

Q:  What kinds of things make an undergraduate application “jump off the page” during the MIT admissions process? In other words, what makes someone’s application stand out from the rest of the applicants who are most likely very studious as well?

A:  I know you’re very anxious to have this question answered! It’s a tough question, and one that doesn’t have an easy answer. Lots of things can make an application stand out. A 42 at the IMO would be great, but it can be many, many things. Some students stand out for their personality, or their extra-curricular accomplishments, or for overcoming a challenging situation. But all of these students must have strong academics and an alignment with MIT’s mission and culture. For more detailed thoughts on this, I’d read the blogs at our website, http://mitadmissions.org

Q:  Not everyone gets to IMO. Are USAMO qualifiers also considered for admission?

A:  Of course! And even non-USAMO qualifiers!

## Happy October 23

What is so special about October “23”? Well, as Dr. Paul Stanford has proven in his lectures there are interesting facts about every number! Here is what he has to say about the number 23. If you have any other facts to add or if you see any mistakes in my transcription of Dr. Stanford’s notes please use the comments below.

# 23 is…

The largest number not the sum of distinct powers.

With 23 people in a room, odds are that two share a birthday (better than 50:50.)

Prime, smallest odd prime not a twin.

Woodall number. $23=3 \cdot 2^3-1$

One of the only two numbers that need 9 cubes. (The other is 239.)

$23=1^3+1^3+1^3+1^3+1^3+1^3+1^3+2^3+2^3$

If negatives allowed, $23=2^3+2^3+2^3+(-1)^3$

$23=0 \cdot 0!+1 \cdot 1!+2 \cdot 2!+3 \cdot 3!$

23 is the smallest number of rigid rods that brace a square.

First prime where 23rd roots of unity form cyclotomic integers without unique factorization.

Number of trees with eight nodes.

Factor of $2^{11}-1$

$2^{23}-1$ is composite: $47 \cdot 178481$

Sophie Germain prime: 2(23)+1 also prime.

Wedderburn-Etherington number.

The first pillar prime.