Can you solve this question in five minutes without using a calculator?
Compute the number of positive integers n
less than 2004 for which the inequality l n
x/2004 – 1 l < n
/2004 has exactly two integer solutions x.
A team of 15 high school students from the Lehigh Valley sure can, recently placing them 10th out of 88 teams in the country in the American Regions Math League (ARML). The team, led by Don Davis, professor of mathematics at Lehigh, competed on June 5 at Penn State University, one of three sites in the country where teams vied for top honors.
“This was the best performance ever for the local team,” says Davis, who describes his students as “the most elite in the Lehigh Valley.
“Just like an athlete who gets satisfaction out of winning a sports match, we get satisfaction from doing what we know we do best, and that’s math,” Davis says.
Davis has coached the team for 12 years, and is also responsible for selecting the students using his very own math exam. He recruited the top 15 scorers to the ARML team, and is quick to credit their success to practices throughout during the spring and sponsorships from AT&T and Keystone Consulting Engineers.
“We had a very good team this year because we had the best math students from the entire area,” says Gavin McFarland, a junior at Moravian Academy in Bethlehem. “And we all enjoy being in the company of other kids who share our interests.”
Future Lehigh University student Brian Holder of Bethlehem says the competition “is more than just taking a test.”
Holder was home-schooled throughout high school, and began taking college calculus courses at age 15, when he heard about the Lehigh University math exam given by Davis. Holder competed in ARML during all four years of high school, and plans to further expand his mathematical talents by studying math and analytical financing in the fall.
“We like to do well as a team,” says Holder, explaining that the team’s two-day trip to Penn State was competitive and fun.
The Lehigh Valley team beat all-star teams from Massachusetts, Texas, North Carolina and Minnesota, which usually finish in the top 10 and had beaten the LV team for the past 11 years. Other members of the 20-person team (including 5 alternates) include:
Yan Zhang, Piotr Fidkowski, Adam Hallowell, Mark Witmer, Jay Werber (Emmaus), Ameya Velinger, Shiyang Cao (Parkland), Vince Roscioli, Zachary Pierpoint (Freedom), Ranjan Rohatgi, Gavin McFarland, Cyndi Bottomley (Moravian), Sean Keenan (Bethlehem Catholic), Brian Holder (home-schooled), Isha Jain (East Hills), Jason Trigg (Unionville), Maria Monks (Hazleton), Patrick Cesarz (Wyomissing), Shaunak Kishore (Patton), and Tianren Qi (Conestoga). Although the last five students do not live in the Lehigh Valley, they attended practice regularly and chose to be on the LV team over others in their regions.
The ARML test measures pre-calculus, algebra, geometry, and trigonometry skills, and has four parts, three of which include teamwork. Part one is an individual test, and the remaining three parts include a Power Round, Relay, and team questions.
For more information, visit the ARML Web site
. For more on the Lehigh Valley team, e-mail Don Davis
or visit his Web site