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## December 13, 2014 – Dr. Jon Kane – Counting Problems

Suppose one afternoon a theater sells 10 adult tickets and 15 child tickets. In how many different orders can those tickets be sold? What if no more than two child tickets are sold between each sale of adult tickets? How about if 10 adult tickets and 10 child tickets are sold but as the tickets are sold, the number of child tickets sold is never allowed to exceed the number of adult tickets sold? We will discuss many variations of these types of problems and draw parallels to many other kinds of counting problems.

## December 6, 2014 – Gabriel Dospinescu – What do powers of 2 and primes have in common?

Join us this Saturday as we journey together to discover the answer to this question, “What do powers of 2 and primes have in common?”  We will start by discussing a problem de Polignac posed in 1849 that raised the question, “Which numbers can be written as the sum of a prime and of a power of 2?”  The conjecture was refuted by Paul Erdos in 1950 with the following theorem: there is an infinite arithmetic progression of odd numbers which cannot be written as the sum of a prime and of a power of 2.  After we reflect on Erdos’ theorem, we will compare the result with Romanov’s theorem which states that a positive proportion of positive integers are the sum of a prime and of a power of 2.

Gabriel Dospinescu, co-author of Problems from the Book and Straight from the Book, is an associate professor at l’Ecole Normale Supérieure de Lyon.

**Special Note:  While our MMC students enjoy this enlightening talk, parents can learn more information about Stanford University Online High School (OHS).  The Director of Admissions for the OHS, Claire Goldsmith, will be in the area offering presentations at local private/public schools and will stop by the math circle for the benefit of our community.

Stanford Online High School serves academically gifted and talented students in grades 7 – 12 from around the world. Students partake in interactive, real-time online seminars taught by expert instructors and are placed by ability, not grade level. Students also have access to a vibrant student life with over 50 clubs and organizations as well as local and regional meet ups. Students can attend full-time and graduate from Stanford OHS or enroll part-time, choosing courses from a catalog of over 60, many of which are AP or University level. For more information visit the school’s website https://ohs.stanford.edu/; or to speak with an Admission Officer, email ohsadmissions@stanford.edu.
Claire Goldsmith is Director of Admission and Financial Aid at Stanford OHS and also oversees external relations for Stanford Pre-Collegiate Studies. She has also run student life and taught History at Stanford OHS. Previously, Claire taught English and French and coached debate at the Harvard-Westlake School in Los Angeles. She graduated magna cum laude in History and Literature from Harvard College and holds a Masters in Policy, Organization, and Leadership Studies from Stanford University Graduate School of Education.

## November 22, 2014 – Razvan Gelca – Do you really know the Pythagorean Theorem?

Please join us this Saturday for a discussion about the Pythagorean Theorem.  We will look at this famous theorem from various angles and illustrate with interesting applications.

## November 15, 2014 – Jacob Cordeiro – Fun with Computational Linguistics

Jacob Cordeiro will discuss the NACLO (North American Computational Linguistics Olympiad), and the skill which you can pick up in order to take part in this clever and fun competition. We’ll solve some favorite problems in group discussions, and learn the logic behind computational linguistics. Linguistic experience is by no means required–there will be something for all ages and all levels of skill.

## November 8, 2014 – Dr. Mirroslav Yotov – “Soccer balls, Doughnuts, Pretzels… or how do we distinguish between shapes of surfaces in 3D.”

It is known how important it is to be able to “recognize” shapes in 3D. A way to achieve this is by using concepts of modern topology (a part of mathematics studying shapes). In the talk, in a down to earth form, we introduce important concepts from Algebraic Topology and show examples of how they work. We explain also how these can be used in identifying the shape of objects in 3D.

## November 1, 2014 – Dr. Titu Andreescu – Problem Solving Session

Come join us November 1st and hone your skills for this year’s problem solving season.  AMC 8 is this month, making it a great time to delve into various interesting problems.  Combinatorics, Number Theory, Geometry, and Algebra problems will be presented with multiple difficulty levels to challenge and delight our math circle patrons.  Whether you are new to problem solving or looking for some additional challenges, this is a great circle for working with peers and under the instruction of Dr. Titu Andreescu, who has been coaching and educating mathletes for over 30 years.

## October 25, 2014 – Oleg Mushkarov – “Napoleon Polygons”

In this talk we discuss a new geometric characterization of the so called Napoleon n-gons characterized by the property that the centers of the regular n-gons erected outwardly on its sides are vertices of a regular n-gon. As a consequence, we obtain a new proof of the well-known theorem of Barlotti-Greber that an n-gon is Napoleon if and only if it is affine-regular. Moreover, we generalize this theorem by obtaining an analytic characterization of the n-gons leading to a regular n-gon after iterating the above construction k  times.
The talk is based on a joint work with Prof. T. Andreescu, University of Texas at Dallas and
Prof. V. Georgiev, University of Pisa, Italy accepted for publication in American Mathematical Monthly.

Later in the session, you will learn more about the High School Students Institut of Mathematics and Informatics established in 2000 by the Union of Bulgarian Mathematicians on the occasion of the World Year of Mathematics.