FOCUSING ON PROBLEM SOLVING HELPS MOTIVATE OUR TALENTED YOUTH

DR. TITU ANDREESCUUNIVERSITY OF TEXAS AT DALLAS

TANDREESCU@GMAIL .COM

2010 MathComp/MathFun

About the presenter

Since an early age I had a high interest in mathematics competitions

1973, 1974, 1975: I won the Romanian national problem solving contests organized by Gazeta Matematică.

During the 1980s, I served as a coach for the Romanian IMO team

1990 emigrated to the USA

About the presenter

US IMO Team Leader (1995 – 2002)Director, MAA American Mathematics

Competitions (1998 – 2003)Director, Mathematical Olympiad Summer

Program (1995 – 2002)Coach of the US IMO Team (1993 – 2006)Member of the IMO Advisory Board (2002 –

2006)Chair of the USAMO Committee (1996 – 2004)MAA Sliffe Award winner for Distinguished

Teaching

History of math competitions

primary school math competition with 70 participants was held in Bucharest, Romania, as early as 1885

the 1894 Eötvös competition in Hungary is widely credited as the forerunner of contemporary mathematics (and physics) competitions for secondary school students

History of math competitions

The year 1894 is notable also for the birth of the famous mathematics journal KöMaL (an acronym of the Hungarian name of the journal, which translates to High School Mathematics and Physics Journal )

similar development occurred in Hungary’s neighbor, Romania. The first issue of the monthly Gazeta Matematica, was published in September 1895. The journal organized a competition for school students, which improved in format over the years and eventu- ally gave birth to the National Mathematical Olympiad in Romania

History of math competitions

The first International Mathematics Olympiad (IMO) was organized by Romania in 1959. The following countries took part: Bulgaria, Czechoslovakia, German Democratic Republic, Hungary, Poland, Romania, and the Soviet Union (USSR).

USA first participated in 1974More than 100 countries participate in the

IMO today

About the IMO

Each country sends a team of up to six middle school or high-school students, chaperoned by a team leader and a deputy team leader.

The competition is held on two consecutive days; each day, the students have four and a half hours to solve three problems

the six problems are selected by an international jury formed by the national team leaders and representatives of the host country

About the IMO

the problems are rather difficult and solving them requires a significant degree of inventiveness ingenuity, and creativity

each problem is worth seven points (the perfect score is 42 points-see year 1994)

the IMO is a competition for individuals; participants are ranked according to their score and (multiple) individual medals are awarded

scores of participants from each country are totaled and the countries are unofficially ranked, providing grounds for comparison between countries

How does the IMO impact the educational system in a country

IMO imposes high standards, therefore each participating country is trying to constantly improve their mathematics education, the process of selecting and preparing their students

As a consequence, a variety of mathematics competitions and enrichment programs have been developed around the world

Types of contest problems

Multiple-choice, where each problem is supplied with several answers, from which the competitor has to find (or guess, as no justification is required) the correct one

Classical style competitions (such as the IMO) require students to present arguments (proofs) in written form.

Types of competitions

National competitions, such as USAMO, or the Chinese Mathematical Olympiad

Regional Mathematical Olympiads such as the Ibero-American Mathematics Olympiad, or the Asian-Pacific Mathematics Olympiad

Correspondence Exams, such as USAMTS, Tournament of Towns

Competitions ran through the internet, such as Purple Comet

Other team competitions such as Baltic Way

Math competitions in the U.S.

Competitions for elementary and middle school students such as CIE MathComp

MATHCOUNTSAmerican Mathematics CompetitionsThe W.L. Putnam Mathematics Competitions

American Mathematics Competitions

AMC 8AMC 10AMC 12AIMEUSAJMOUSAMO (leading to MOSP and IMO)

Math Competitions are needed

Creates ways to identify mathematical talent Typical school curriculum is aimed towards the

average studentWhat takes place before and after a

competition is meaningful for math education Preparation that takes place and discussions after the

competition ends is importantStudents who take part in math competitions

are steered towards science careers

Olympiad style problems

They are challenging essay-type problemsTo provide correct and complete solutions

require deep analysis and careful argumentThey might seem impenetrable to the novice,

but they can be solved using elementary high school mathematics

Hints for advanced problem solvers

Do not be intimidated! Some of the problems involve complex mathematical ideas, but they can attacked by using elementary techniques, admittedly combined in clever ways

Be patient and persistent! Learning comes more from struggling with problems than from solving them.

Problem solving becomes easier with experience

Success is not a function of cleverness alone

What is an exercise and what is a problem?

The difference between exercises and problems

What is 50% of 2006 plus 2006% of 50?A) 1013.5 B) 1053 C) 1103.3 D) 1504.5

E) 2006

Solution:

€

50

100×2006 +

2006

100×50 = 2006

What is an exercise and what is a problem?

If is written in decimal form, find the sum of its digits.

Solution.Because and , the

given number can be written as = 781250 . . . 0 (25 zeros). The sum of the digits for the decimal representation is 7 + 8 + 1 + 2 + 5 = 23.

€

532325

€

532 = 57525

€

325 = (25)5 = 225

€

57(5 ⋅2)25 = 78125 ⋅1025

Resources available to talented math kids

Participate in competitionsTake on-line classesAttend Math Circles or Math ClubsTake part in Summer ProgramsWork on problems from several books

available for Olympiad training

Mathematical Reflections

Free on-line journal aimed primarily at high school students, undergraduates, and everyone interested in mathematics. Through articles and problems, we seek to expose readers to a variety of interesting topics that are fully accessible to the target audience.

AwesomeMath Summer Program (AMSP) www.awesomemath.org

A three-week intensive summer camp for mathematically gifted students from around the globe

Targeted to bright students who have not yet shone at the Olympiad level, as well as of those who would like to expand what they have learned in other programs

It hones their problem solving skills in particular and further their mathematics education in general

Many of our participants seek to improve their performance on contests such as AMC10/12, AIME, or USAMO

Dates: July 6 – 27 and July 30 – August 20, 2010

Math Rocks!

Available to exceptional Plano ISD students, grades 4 to 7

Will expand from 4 to 8 elementary schools in 2010/2011

Features challenging topics and problem setsExpands mathematical horizons of

participantsDeepens their understanding of mathematicsDevelops important problem solving skills

Metroplex Math Circle (MMC) metroplexmathcircle.wordpress.com

Intended for students who are 14 and older and show a strong desire to go beyond a standard high school curriculum

They can use their experience at MMC to excel in national math competitions or to better prepare them for work at elite universities

Younger students with demonstrated mathematical talents are also welcome to participate in the MMC lectures.

Metroplex Math Circle

Meets in room 2.410 of the Engineering and Computer Sciences building on the campus of the University of Texas at Dallas

Regular sessions are held Saturday afternoons from 2:00 to 4:00 while the university is in sessions

Speakers from all over the country, such as: Richard Rusczyk, Dr. Art Benjamin, Dr. Zumin Feng, Dr. Jonathan Kane, etc

Books

“Mathematical Olympiad Challenges” by Titu Andreescu and Razvan Gelca

“Mathematical Olympiad Treasures” by Titu Andreescu and Bogdan Enescu

“Number Theory: Structures, Examples, and Problems” by Titu Andreescu and Dorin Andrica

“Problems from the Book”, by Titu Andreescu and Gabriel Dospinescu