Dr. Branislav Kisačanin – Geometry of Triangles

This Saturday, April 20th Dr. Kisačanin will return for another of his fantastic lectures.  Triangles factor into almost every math contest in addition to being endlessly fascinating objects in themselves.  Here is Dr. Kisačanin’s description of the session with links to resources:

In this talk about geometry of triangles we will see two different proofs of Stewart’s theorem, derive formulas for important cevians, and solve several interesting geometric problems.

We will also look at other important points in triangles (Fermat point, centers of excircles, …) and look at the Euler line, the nine-point circle, and related problems.

http://en.wikipedia.org/wiki/Stewart%27s_theorem
http://en.wikipedia.org/wiki/Fermat_point
http://en.wikipedia.org/wiki/Excircle
http://en.wikipedia.org/wiki/Euler_line
http://en.wikipedia.org/wiki/Nine_point_circle

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April 13, 2013 – Dr. Paul Stanford – Primes and Misdemeanors

For 2ⁿ – 1 to be prime we also need n itself to be prime, but that is not sufficient. For example, 2¹¹ – 1 is composite even though 11 is prime.  However, if you look at tables of Mersenne primes it is interesting to note that if you start with 2 and use that to make a new number 2ⁿ – 1 with n = 2 you get 3, then recycling the 3 you get 7, use n = 7 and you get 127, another prime! How long could this go on?

Let f(n) = 2ⁿ – 1.  The iterations you get, starting from 2, are f⁰(2) = 2, f¹(2) = 3, f²(2) = 7, f³(2) = 127, f⁴(2) = 1701411834604692317316873037158884105727.

It turns out these are all prime!  But what of the next one???  Well, it may be a very long time before any of us know. The largest value of n for which 2ⁿ – 1 is known to be prime is n = 57885161, and that after a concerted effort using volunteers from around the globe.  Not much chance of answering this one in our lifetimes, unless some really new idea arrives.Before you make a hasty conjecture (as has already been done), a cautionary piece of history is in order.  If you define a new function to iterate you get some other interesting numbers.  Let g(n) = n² – 2n + 2. Then the iterates are g⁰(3) = 3, g¹(3) = 5, g²(3) = 17, g³(3) = 257, g⁴(3) = 65537, and all of these are prime!  So, with forgivable excitement, the conjecture was made that all of these will be prime, especially as the next one, g⁵(2) = 4294967297, was much too large at the time for mere mortals to conceive of factoring with their bare hands.

None, that is, until Euler combined his genius with an impish disbelief in Fermat’s conjecture to discover that g⁵(2) = 4294967297 = 641 * 6700417.  And since then we have found many more composite Fermat numbers, and no further Fermat primes, leading to the complementary conjecture that all the rest are composite!  It seems that we never learn to be humble around these things…

It takes a larger number to be “forever beyond reach” these days.  Rather than the now puny 4294967297 we cower before f⁵(2) = 2¹⁷⁰¹⁴¹¹⁸³⁴⁶⁰⁴⁶⁹²³¹⁷³¹⁶⁸⁷³⁰³⁷¹⁵⁸⁸⁸⁴⁴¹⁰⁵⁷²⁷ – 1, and who can blame us?

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March 2, 2013 – Dr. Titu Andreescu – More Interesting Problems

With the AMC tests behind us and the AIME to come, don’t miss this opportunity to hear Dr. Andreescu talk about some of his favorite problems!

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February 16, 2013 – Dr. Branislav Kisačanin – “A Tour of Mathematical Functions II”

In a follow up lecture on mathematical functions, we will explore more stories and problems related to polynomials, trigonometric functions, and functional equations. Furthermore, we will dive deeper into the original historical context of functions – curves such as cycloids, cardioids, catenaries, circles, ellipses, hyperbolas, parabolas. Finally, we will try to understand why exponential and trigonometric functions turn up in solutions of so many fundamental problems in math, physics, and engineering. Come and learn with us!

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January 19, 2013 – Ivan Borsenco – “Exploring the world of Probability”

MIT student and four time International Mathematical Olympiad participant, Ivan Borsenco, will return to the Metroplex Math Circle this week!

Ivan will introduce the classical probability theory. There will be many interesting examples and several unexpected results. Students will solve a few mathematical paradoxes, find out how to build simple probabilistic models, and have lots of fun.

A deep understanding of probability is not only useful for contest preparation, but is critical for anyone planning a career in science or business.

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December 15, 2012 – Dr. Titu Andreescu – “Some of my Favorite Problems”

This next meeting will be our last before the holiday break at UTD and the end of the Fall semester.  Students preparing for the 2013 AMC 10 and 12 will not want to miss this session when Dr. Andreescu share some of his favorite problems and approaches for solving them.

All students should make a serious effort to register and prepare for the AMC 10 or 12 exam.  There is no limitation on how young a student can participate and no restriction on how many years you take the exam until you reach 10th and 12th grades respectively.  The AMC results are requested by many elite colleges to differentiate among the many applications they see with 800 SAT scores.

Here is some additional information about this year’s AMC 12:

The AMC 12 is a 25 question, 75 minute multiple choice examination in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts. Calculators are not allowed starting in 2008. For the 2012-2013 school year there will be two dates on which the contest may be taken: AMC 12A on Tuesday, February 5, 2013 , and AMC 12B on Wednesday, February 20, 2013 .

 

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Mathew Crawford – “Exploring Recursion” — Replay

While Metroplex Math Circle is on Thanksgiving break please enjoy this replay of the excellent talk given by Mathew Crawford in our last session:

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November 10, 2012 – Mathew Crawford – “Exploring Recursion”

This weekend we have a very special guest lecturer.  Mathew Crawford, who will be well known to many MMC attendees as the author of the popular AOPS titles:  Introduction to Number Theory and  Intermediate Algebra.   Mr. Crawford will be bringing with him 50 copies of problem materials which will be available only to the first 50 families to join us.  If you are unfamiliar with Mathew Crawford and his extensive work, the following information comes from the AOPSWiki:

Mathew Crawford is the founder and CEO of MIST Academy, a school for gifted and talented students, headquartered in Birmingham, Alabama. Crawford won numerous national math championships as a student before attending Washington University in St. Louis on a Compton Fellowship where he studied mathematics and worked on the Human Genome Project at the Institute for Biomedical Computing. After spending several years on Wall Street and eventually running a finance operation from the basement of his apartment, Crawford founded his first education company in 2001, Universal Set Educational Resources, with childhood friend Cameron Matthews. In 2003, Crawford became the first employee of Art of Problem Solving where he helped to write and teach most of the online classes during the first three years of the AoPS online school.

His competition achievements include:

• National MathCounts written test champion in 7th grade (perfect score of 46) and second place in 8th grade (score of 44).
• Two-time perfect scorer on the AHSME.
• Perfect score on the AIME as a freshman.
• Three-time invitee to the Mathematical Olympiad Summer Program.
• Member of a top 4 Putnam team.
• Youngest winner of the National Mu Alpha Theta convention.
• Only 5-time winner of the Alabama State Written Examination (Algebra II/Trig once, Comprehensive four times).
• Twice among ARML high scorers (tie-breakers) and Zachary Sobol Award winner.

Crawford also writes competition problems and performs duties for many math competitions:

• USAMTS problem writer and grader (2004-2006)
• iTest head test writer (2007 and 2008)
• Birmingham and Alabama MATHCOUNTS coordinator
• Mu Alpha Theta test writer and proof reader
• Co-coach of the Missouri ARML team (1996,1997)
• Coach of the San Diego ARML team (2005,2006)
• Coach of the Alabama ARML team (2008, 2010-present)
• Headed up the grading of the Power Round for the Georgia ARML site (2010)
• San Diego Math League test writer and problem writer for the San Diego Math Olympiad (2004-6)

His first book, Introduction to Number Theory was published by AoPS in June, 2006. He is also coauthor of the Intermediate Algebra text, which came out in April, 2008.

Crawford’s user page can be found here.

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November 3, 2012 – Dr. Titu Andreescu – More Interesting Problems

We are very fortunate this week to have Dr. Titu Andreescu return to present some more challenging problems and strategies for solving them.

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October 27, 2012 – Cosmin Pohoata – Everything about Symmedians!

Cosmin Pohoata will return to the Metroplex Math Circle this week to give a talk on symmedians.  This will be an excellent lecture particularly for some of our more advanced students and those preparing for mathematical contests where symmedians can be a powerful part of their tool kits.

Abstract: We will introduce symmedians from scratch and prove the entire collection of interconnected results that characterize them.  Applications from contests around the world will also be presented if time permits.

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