## MATH CIRCLES

A kingdom consists of 12 cities located on a one-way circular road. A magician comes on the 13th of every month to cast spells. She starts at the city which was the 5th down the road from the one that she started at during the last month (for example, if the cities are numbered 1-12 clockwise, and the direction of travel is clockwise, and she started at city #9 last month, she will start at city #2 this month). At each city that she visits, the magician casts a spell if the city is not already under the spell, and then she moves on to the next city. If she arrives at a city which is already under the spell, then she removes the spell from the city, and leaves the kingdom until the next month. Last Thanksgiving the capital city was free of the spell. Prove that it will be free of the spell this Thanksgiving as well.*

This question from the 2001 Bay Area Math Olympiad (BAMO) is the sort of math problem tackled by middle and high school students in the Berkeley Math Circle. Virtually unknown in America until recently, mathematical circles for talented school children originated in Hungary more than a century ago and spread over Eastern Europe and Asia. They eventually led to the start of many national and international math contests, including the first International Mathematical Olympiad (IMO) held in Romania in 1959. The United States, which will host the 42nd competition in July, joined the IMOin 1974, and since then its team has performed among the very top of the approximately 80 participating countries.

Math circles are led by mathematicians and teachers trained in solving problems for Olympiads. IMO problems come from various areas of mathematics, many of which are covered in math curricula at secondary schools. The tantalizing problems, however, require not only computational skill but also deep thinking, creativity and the ability to explain each step of the reasoning that leads to a solution.

The middle school years are ideal for nurturing this kind of mathematical talent, and it is then that interested students should begin to study algebra and geometry and to construct proofs. High school is often too late, and even then a good high school mathematics course in the United States generally brings a student no farther along than the 18th century.

Former International Math Olympian for Bulgaria, Zvezdelina ("Zvezda") Stankova-Frenkel, A.B., M.A. '92 is one of the pioneers establishing math circles in the United States. An assistant professor of mathematics and computer science at Mills College, she is a founder of the Berkeley Math Circle and BAMO. "I always wanted to be close to the most talented young kids in the United States and give them the same chance of early encounter and joy with mathematics as I was given in Bulgaria through the math circles," says Stankova-Frenkel, who has coached the U.S. team in preparation for the IMO at the Math Olympiad Summer Program (MOSP) in 1998-2000.

When she joined her Bulgarian middle school math circle in fifth grade, mathematics was just one of her interests along with piano, ballet and poetry. Only three months after joining the circle, she won first prize in a regional competition that included problems in geometry and elementary algebra.

"It was a very creative and competitive atmosphere," she remembers. She entered an elite high school in which many courses were taught in English and was selected for the Bulgarian national mathematics team, winning silver medals in the Mathematics Olympiads in Cuba in 1987 and in Australia a year later. In Australia, her unique solution to a complex problem-"If (a2 + b2)/ 1+ab is an integer, then it is the square of an integer"-made her well known in Bulgaria and in the mathematical world. In 1989, already attending Sophia University, she was among 15 Bulgarian students selected to study in the United States. (She was one of two chosen by the American Embassy for Bryn Mawr, which in turn chose her.)

One of her advisors at Bryn Mawr, Paul Melvin, Rachel C. Hale Professor in the Science and Mathematics, comments, "Zvez enriched our program in many ways, both as a role model for other math majors and, because of her advanced training in Bulgaria and her remarkable mathematical talent, as an active participant in our graduate seminars. In her junior year, she ran an immensely successful course on "Olympiad Problems with Some Theory," which was attended by a mixture of advanced undergraduates, graduate students and faculty."

Nominated by Helen Herrman Professor of Mathematics, Rhonda Hughes, with whom she also worked, Stankova-Frenkel won the Schafer Prize for the top undergraduate woman math student in the country. Hughes also arranged for her to visit many mathematicians at Harvard and Penn when she arrived at BrynMawr.

View from other side
In order to learn about pre-college education in the United States, she worked for certification as a school teacher in Massachusetts while completing her Ph.D. in algebraic geometry at Harvard. "I realized that there is very little, if any, connection between professional mathematicians and secondary school teachers here (not so in Eastern Europe!)," she says. "So, I needed to get on 'the other side of the fence' and see what kind of problems teachers faced every day in school." After receiving her Ph.D in 1997, she also became certified in California, where she was a post-doctoral fellow of the Mathematical Sciences Research Institute at Berkeley and Morrey Assistant Professor of Mathematics at Berkeley.

Stankova-Frenkel did not know Harvard from Bryn Mawr before coming to the United States and says that it took her some time to realize the necessity of women's colleges. "I grew up in a culture that nurtured scientific talent equally in boys and girls from an early age. I never felt slighted or given less of a mathematical chance in Bulgaria based on my gender," she explains. "In fact, for the two years when I was on the Bulgarian IMO team, there were two girls out of six students. Compare this with the U.S. IMO team, which finally had a girl on it (Melanie Wood from Indiana, currently at Duke University) only three years ago! Since starting to teach at Mills two years ago, I have seen many cases of exceptionally bright young women who were indoctrinated with the idea that they would never be good in math. I am proud to say that I have turned a number of cases around and I am sure the same thing happens all the time at Bryn Mawr."