Results of the First NES/MAACollegiate Mathematics Contest

The Northeastern Section of the Mathematical Association of America held its first Collegiate Mathematics Competition on November 17th, 2006 at the Fall Meeting at Sacred Heart University.   Eight teams consisting of a total of 22 undergraduates worked hard for two hours on the problems.  The teams were from Dartmouth College, Fairfield University, the University of Hartford, Norwich University, Providence College, Roger Williams University, Sacred Heart University, and Wheaton College.  These pioneering competitors deserve our applause.

The team from Dartmouth College -- Yong Su, Yangyang Liu, and Tom Kern -- won the competition with a score of 68 out of a possible 70 points and took home individual copies of Mathematica for Students and split a $100 prize.  The team from Providence College -- Jaclyn Scholl, Deanna Dupuis, and Bernadette Boyle -- came in second place and split a $75 prize, while the team from Norwich University -- Matt Burger and John Walthour -- came in third place and divided a $50 prize.

Special thanks go to Jason Molitierno and Sarah Novotny at Sacred Heart for their constant on-site help.  Thanks to Ed Sandifer for his assistance during the competition.  Thanks also to Ben Wilson at Wolfram Research for his and Wolfram's generosity in donating the copies of Mathematica for Students.

Here are the problems in the contest.

1. The figure below shows a circle with radius 1 inscribed in the parabola y = x2. Find the center of the circle.

(Thee was a graph here which I have omitted.- The Editor.)

2. Which number is bigger: e^p or p^e? (Your calculator will of course tell you the answer; You need to prove it.)

3. If x, y, and z are positive numbers, prove that

4. The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o’clock?

5. Let A be a square matrix and suppose that there exist positive integers m and n such that A^m = I and Aⁿ ¹ I. Calculate det(I + A + A² + · · ·A^m-1).

6. Can a group be a union of two proper subgroups?

7. A chicken and a half can lay an egg and a half in a day and a half. How long will it take for two chickens to lay 32 eggs?