Archive for the ‘math circle’ Category
Please join us this Saturday as Adrian Andreescu leads the group through a selection of some of the most interesting AMC 8 problems.
And while you attend, if you meet the requirements below you can sign up for the AMC 8 according to the following instructions:
Hooray, it’s getting close to AMC 8 time again! I’m handling signups differently this year. First, here are the particulars for the test:
- When: November 19th at 5pm SHARP
- Where: The Davis Library
- Requirements: Must be in 8th grade or below and current school does not offer the test.
- Cost: $10 (The Davis library charges $50 for the program room, so I’ve had to increase the cost to cover all related fees)
- What to bring: number 2 pencils, protractor, compass (scratch paper will be provided). NO calculators
If you meet the above requirements, then registration for the test will happen in person on November 2nd. Here are the particulars to sign up:
- When: November 2nd starting at 4pm.
- Where: UTD after math circle, http://metroplexmathcircle.wordpress.com/about/directions-and-times/
- Requirements: I will take a short time to discuss math resources and answer questions, then you can fill out the registration form
- What to bring: Please bring the $10 fee for the test.
If you have any questions, feel free to email me.
We are pleased to announce that the first topic of our 2013-2014 Metroplex Math Circle will be Pólya-Burnside Enumeration in Combinatorics, presented by our own Adithya Ganesh on September 14, 2013.
Burnside’s lemma from group theory has a broad scope of application in combinatorial enumeration problems. Pólya’s enumeration theorem, which generalizes Burnside’s lemma using generating functions, provides a remarkable framework to easily solve counting problems in which we want to regard two entities as equivalent under some symmetry.
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.
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.
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?
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!