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AMC 8 Preparation Handouts

On October 20th Dr. Titu Andreescu spoke to the Metroplex Math Circle at UT Dallas and shared a number of interesting problems.  The video below will give people some sense of what a math circle session can be.  We are still experimenting with our video and streaming technology so we appreciate your patience and are grateful for your tips to improve the quality of these videos which will be posted to our YouTube channel.

Please note that every Metroplex Math Circle speaker is encouraged to choose the topic and format that best suits their style and passions.  This means that every session is somewhat different and you are encouraged to read the description of each session posted here or on our FaceBook and Google+ pages when you make your decision to attend.

Telescoping Sums and Products (Example 1)

While we wait for the Metroplex Math Circle to continue again in mid January, I will be posting a series of problems and solutions from our last session. Those who were fortunate enough to attend learned useful techniques for solving problems common in math contests.

Telescoping Sums and Products

The telescoping sums and products idea is used to solve many problems involving sums or products in algebra. For problems involving sums, the idea is to use identities, to write the sum in the form

$\displaystyle\sum^{n}_{k=1}[F(k+1)-F(k)]$

and then cancel out terms to get F(n+1)-F(1). Sometimes the desired identity is hard to find, but basically you are searching for it in the recursive form of the sequence, or you can look foor the “conjugates” for the terms you have. The first example is classical. You Certainly know these formulas.

$\displaystyle\sum^{n}_{k=1}k=\dfrac{n(n+1)}{2}$

$\displaystyle\sum^{n}_{k=1}k^2=\dfrac{n(n+1)(2n+1)}{6}$

$\displaystyle\sum^{n}_{k=1}k^3=\left[\dfrac{n(n+1)}{2}\right]^2$

What about $\displaystyle\sum^{n}_{k=1}k^4$?

Example 1.  Prove that

$\displaystyle\sum^{n}_{k=1}k^4=\dfrac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$

Dr. Bennette Harris – Problem Set 2

Here is the continuation of Dr. Bennett Harris’ problems to warm us up before his lecture on September 27th. Congratulations to Dominic for being the first to answer the previous set of problems in the comments. Another happy discovery is the fact that WordPress, which hosts this site, supports the use of $\LaTeX$ code!

Problems

A solution for each of the following should either give the correct answer, or a technique for determining the answer in reasonable time with the assistance of a calculator.

6. Crack this secret message: “uifsfaxjmmacfabmnptuaopaqsppgtaupebz”

7. Code the alphabet a=0, b=1, …, z=25, space=26. We will encrypt each letter of the alphabet by mapping it to the letter 3 farther along in the alphabet. Thus, a maps to d, b maps to e, and so forth. The message “a bat is fat” would be encrypted as “dcedwclvciodw” in this scheme. Find a mathematical function f(x) such that y=f(x) gives the correct encryption for every value x = 0,1,2,…,26. Test your function by encrypting “the toy boat floats”.

8. The purpose of these two questions is to provide points of comparison for the questions that follow:
How many particles are there in the known universe?
How many microseconds have elapsed since the beginning of the universe?

9. How long (approximately will do) would it take to calculate $10000^{1234567890}$ at 1,000,000 calculations per second? How many digits does the answer have? If p is 200 digits long, and if q is 200 digits long, about how many digits are there in pq?

10. The number $2^{521}$ – 1 is known to be prime. Estimate how long it would take a computer to demonstrate this by repeated division, at 1,000,000 divisions per second.

Dr. Bennette Harris – Problem Set 1

This Saturday’s (9/27) Math Circle lecture by Dr. Bennett Harris on “Computer Data Encryption – Decrypted” promises to be a great combination of number theory and applied math. To warm up the students, Dr. Harris forwarded some problems that I will post in two parts. Feel free to offer solutions in the comments or to just work them on your own. Full solutions will be made available at the next Math Circle.

Problems

A solution for each of the following should either give the correct answer, or a technique for determining the answer in reasonable time with the assistance of a calculator.

2. What is the largest number you must test to demonstrate that 83 is prime?

3. What is the smallest 5-digit prime? How could you find this number?

4. Is 1234567890 prime? What about 123456789?

5. Assume an alphabet with 26 letters (plus a blank space). A substitution cipher is a one-to-one mapping of this alphabet onto itself. How many such substitution ciphers are there?

Problem Solving Books

In addition to being the subject of books like Count Down, the Director of Metroplex Math Circle, Dr. Titu Andreescu is also the author of multiple books on problem solving. These books draw on his many years of experience as the director of AMC, coach of the US International Math Olympiad team and author of many contest problems.

To help the Metroplex Math Circle community we have created an Amazon List with some of Dr. Andreescu’s currently available books. In addition to Dr. Andreescu’s books for experienced problem solvers we have also included some books and resources on the list for students just starting into problem solving.

Not only does Metroplex Math Circle benefit from Dr. Andreescu himself, but many of his co-authors are also friends of MMC and frequent lecturers.

Following are the author descriptions from the book 104 Number Theory Problems: From the Training of the USA IMO Team:

Titu Andreescu received his Ph.D. from the West University of Timisoara, Romania. The topic of his dissertation was “Research on Diophantine Analysis and Applications.” Professor Andreescu currently teaches at The University of Texas at Dallas. He is past chairman of the USA Mathematical Olympiad, served as director of the MAA American Mathematics Competitions (1998–2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993–2002), director of the Mathematical Olympiad Summer Program (1995–2002), and leader of the USA IMO Team (1995–2002). In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the world’s most prestigious mathematics competition. Titu co-founded in 2006 and continues as director of the AwesomeMath Summer Program (AMSP). He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a “Certificate of Appreciation” from the president of the MAA in 1995 for his outstanding service as coach of the Mathematical Olympiad Summer Program in preparing the US team for its perfect performance in Hong Kong at the 1994 IMO. Titu’s contributions to numerous textbooks and problem books are recognized worldwide.

Dorin Andrica received his Ph.D. in 1992 from “Babes-Bolyai” University in Cluj-Napoca, Romania; his thesis treated critical points and applications to the geometry of differentiable submanifolds. Professor Andrica has been chairman of the Department of Geometry at “Babes-Bolyai” since 1995. He has written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. He is an invited lecturer at university conferences around the world: Austria, Bulgaria, Czech Republic, Egypt, France, Germany, Greece, Italy, the Netherlands, Portugal, Serbia, Turkey, and the USA. Dorin is a member of the Romanian Committee for the Mathematics Olympiad and is a member on the editorial boards of several international journals. Also, he is well known for his conjecture about consecutive primes called “Andrica’s Conjecture.” He has been a regular faculty member at the Canada–USA Mathcamps between 2001–2005 and at the AwesomeMath Summer Program (AMSP) since 2006.

Zuming Feng received his Ph.D. from Johns Hopkins University with emphasis on Algebraic Number Theory and Elliptic Curves. He teaches at Phillips Exeter Academy. Zuming also served as a coach of the USA IMO team (1997-2006), was the deputy leader of the USA IMO Team (2000-2002), and an assistant director of the USA Mathematical Olympiad Summer Program (1999-2002). He has been a member of the USA Mathematical Olympiad Committee since 1999, and has been the leader of the USA IMO team and the academic director of the USA Mathematical Olympiad Summer Program since 2003. Zuming is also co-founder and academic director of the AwesomeMath Summer Program (AMSP) since 2006. He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1996 and 2002.

Problems – Geometric Combinatorics

Last semester we were very fortunate to have Dr. Tatiana Shubin join us and present on Geometric Combinatorics. Following are some of the problems she presented:

1. Prove that if each three of n points in a plane can be enclosed in a circle of radius 1, then all n points can be enclosed in such a circle.

2. Given 7 lines in a plane, if no two of them are parallel to one another prove that there exists a pair of lines with the angle between them less than 26 degrees

3. How many acute inner angles can a convex n-gon have?

If you would care to answer them in the comments please do. These are just a sampling of the interesting problems and topics to be found at Metroplex Math Circle.