May 24, 2008
Six congruent circles are arranged inside a larger circle so that each small circle is tangent to two other small circles and is tangent to the large circle. The radius of the large circle is 2007 centimeters. Find the radius of the small circles.
Solving this kind of problem helped Colin Sandon and David Rolnick earn slots on the 12-member USA Mathematical Olympiad (USAMO) team. Sandon, 18, who tied for first place, and Rolnick, 16, are Vermont's first duo in at least a decade to make the team, according to the MAA.
Sandon was discovered by the Vermont State Math Coalition when he was in first grade. Engineers and physicists from IBM then tutored him because his capacity for mathematics exceeded that of his teachers. Sandon finished pre-calculus in sixth grade and began taking courses at the University of Vermont three years ago. His course load includes calculus, linear algebra, graph theory, and number theory.
Sandon, a senior, and Rolnick, a junior, have been accepted by the Massachusetts Institute of Technology. "I'm kind of nervous, because I've never been away from home for more than a month, and MIT will be my home for the next four years," Sandon told the Burlington Free Press. "On the other hand, I'll get to meet new people there and take more challenging classes," he indicated.
Rolnick enjoys strategy games, as well as hiking, tennis, word play, writing, listening to classical composers such as Beethoven, and studying moths. He finds it easy to talk about his love for geometry.
"I love the way that things that are true, really are true," Rolnick said. "If you have a triangle, and you join the vertices to the midpoints of the opposite side, you come up with three lines. Those lines will come to a point—those three lines will always meet—and I find that very beautiful."
Problem-solving skills become more and more important as students advance through the competitions that lead to the USAMO. "For all the problems, there is a certain amount of thinking and puzzling that is absolutely necessary," Rolnick said. But Olympiad problems are especially difficult. The Olympiad "is meant to be hard, even for professional mathematicians," he added.
Retired Burlington High School teacher Anthony Trono, 80, who trained Sandon and Rolnick, created the sample problem, above, as part of an exam designed to test for up-and-coming math whizzes. Like the problems on the exams of the American Mathematics Competitions, it involves no mathematics beyond the pre-calculus level.
"Some of these problems aren't even algebra," Trono said. "It's just arithmetic, but you gotta use your head to solve them." The answer to the problem given above is 669 centimeters.
Students, he added, "usually have to prove something is true, derive some kind of formula, or solve a very, very complex problem."
On June 7-8, Sandon, Rolnick, and the 10 other USAMO winners will take the selection test for the U.S. International Mathematical Olympiad team, which has six members. An awards ceremony for the USAMO winners will be held at the MAA Headquarters on June 8 and at the National Academy of Sciences and U.S. Department of State on June 9.
Source: Burlington Free Press, May 23, 2008.