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## 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.

## 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!

## February 2, 2013 – Dr. Branislav Kisačanin – “A Tour of Mathematical Functions”

Parabola or catenary?

In this lecture we will retrace the steps of Archimedes, Newton, Euler, and other great mathematicians and learn about important mathematical functions, their properties, history, and applications. We will look at several interesting competition topics that often show up on AMC 10/12, related to exponential, logarithmic, trigonometric, hyperbolic, and other functions. We will also see how these functions turn up in solutions of some fundamental problems in math, physics, and engineering. We will have fun using them to draw important curves: cicloids, cardioids, catenaries, circles, ellipses, hyperbolas, parabolas, and will discover which one of them is brachistochrone and which one is tautochrone.

## October 6, 2012 – Dr. Branislav Kisačanin – “First Encounter with Calculus”

Last year one of our most popular sets of lectures were on Calculus.  Take this opportunity to hear, Dr. Branislav Kisačanin, introduce even our younger students to this beautiful topic.  Here is a description of the talk in his own words:

In this session we will look at calculus and try to dispel the mystery that often surrounds it. We will begin by looking at examples of limiting processes, and show for example that as x approaches 0, the value of $\frac{\sin x}{x}$ approaches 1. Based on several other examples of limits, we will be able to determine line equations for tangents of various curves. This will bring us to the doorstep of the first derivative.

We will look at other interpretations of the first derivative and also look at how it is applied to maximization and minimization, Newton’s method of tangents for solving equations, etc. In the second hour we will look at another major part of calculus, integration, and will show how integration is used to compute areas and volumes. We will also look at the methods used by Archimedes to compute the volumes of solid spheres, which are remarkably similar to modern integration. Of course, there is much more to calculus than we can learn in one day, but this will be a good start!

## February 18, 2012 – Dr. Branislav Kisačanin – “Introduction to Graph Theory”

February 18, 2012 – Dr. Branislav Kisačanin – “Introduction to Graph Theory”

Graph theory was born when Leonhard Euler solved the Seven Bridges of Koenigsberg problem and has since grown into a mathematical discipline with beautiful theoretical results and with applications in disciplines like theory of dynamic systems, electrical engineering, computer vision, neuroscience, and social networking.
In this lecture we will learn about the fascinating early history of the graph theory, discuss Eulerian and Hamiltonian paths, node and edge coloring, and look at other important properties of graphs, such as planarity. We will even work on several problems like this one from the Sixth IMO (Moscow, 1964):
Each of 17 students talked with every other student. Each pair of students talked about one of three different topics. Prove that there are three students that talked about the same topic among themselves.

## December 3, 2011 – Dr. Branislav Kisačanin – “Solving Recursions”

Sequences of numbers are often defined using a recursive relation and initial conditions, for example, the sequence of Fibonacci numbers is defined with F_{n+2} = F_{n+1} + F_{n} and initial conditions F_{1} = 1, F_{2} = 1. In this lecture we will see (a) how to solve various types of recursions (b)How to determine various sequence properties of number sequences directly from the recursions, and (c) how this knowledge can come in handy in many competition problems as well as in the study of computer science.

## October 22, 2011 – Dr. Branislav Kisačanin – “Second Encounter with Calculus”

Following his very successful lecture on October 1, Dr. Kisačanin will continue his introduction to calculus.  Students who have already studied some calculus will gain a deeper insight into this beautiful and useful tool.  Those who have not yet studied calculus should not be put off, however, as Dr. Kisačanin is very skilled at explaining mathematical concepts so that they can be appreciated by a broad audience.

## October 1, 2011 – Dr. Branislav Kisačanin – “First Encounter with Calculus”

Many of our students have started taking Calculus or are curious about what it’s all about.  Take this opportunity to hear one of our favorite lecturers, Dr. Branislav Kisačanin, introduce even our younger students to this beautiful topic.  Here is a description of the talk in his own words:

In this session we will look at calculus and try to dispell the mistery that often surrounds it. We will begin by looking at examples of limiting processes, and show for example that as x approaches 0, the value of $\frac{\sin x}{x}$ approaches 1. Based on several other examples of limits, we will be able to determine line equations for tangets of various curves. This will bring us to the doorstep of the first derivative.

We will look at other interpretations of the first derivative and also look at how it is applied to maximization and minimization, Newton’s method of tangents for solving equations, etc. In the second hour we will look at another major part of calculus, integration, and will show how integration is used to compute areas and volumes. We will also look at the methods used by Archimedes to compute the volumes of solid spheres, which are remarkably similar to modern integration. Of course, there is much more to caclulus than we can learn in one day, but this will be a good start!

## Dr. Branislav Kisačanin – Geometry of Triangles

This Saturday, February 26th Dr. Kisačanin will return for another of his fantastic lectures.  Anyone preparing for the AMC, AIME or USAMO should plan to attend.  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.

Computing a correct answer is sufficient for most grade school work, but proving that something must be true is one of the most important skills our students can learn.  This ability has immediate benefits for such important tests as USAMTS or the USAMO as well as teaching the skills of reasoning and communication that can help in any profession.

We are very fortunate to once again have a lecture by Dr. Branislav Kisačanin.  Here is his description of the upcoming session:

Introduction to Mathematical Proofs
In this lecture we will see examples of common proof techniques: direct proofs, proofs by induction, and proofs by contradiction. The students should have their pencils ready, as they will get a hands-on experience solving and proving mathematical problems.
In addition to a few more problems of historical significance, we will work on relatively simple problems that illustrate various proof techniques. In order to make things more interesting, we will let the students figure out which technique to use for each problem.