MMPC Honors Top High School Students

A total of 101 Michigan high school students, from 51 different schools, were honored for their achievements on the 44th Annual Michigan Mathematics Prize Competition at the Awards Day program held on March 10 at Albion College. Robert Messer (Albion College) is the director of the MMPC this year.

Joan Hutchinson and Stan Wagon (Macalester College) gave talks to the "Top 100" and their parents and teachers: "How to Color Maps (and Graphs) When You Have Only Two or Three Crayons" and "Unusual Mathematical Models: From Square-Wheel Bikes to Giant Surfaces in Snow". Joshua Boehme, a student from East Lansing High School, spoke at the awards banquet about his experiences last year with the Michigan All-Stars in the ARML competition, in which teams drawn from the Top 100 compete in mathematics problem solving against other top teams from around the country. This year's contest will take place on June 2 at the University of Iowa, and three spring practice sessions around the state will prepare team members. Michigan teams have consistently done well in this event (see report in the Fall Newsletter).

This year's first, second, and third place Gold award recipients were Mike Asmar (Troy), Chenlu Hou (Ann Arbor Huron), and Shailesh Agarwal (Troy), respectively. The First Level Silver winner was Ryan Timmons (Groves); the Second Level Silver awards went to Christopher Battey (Ann Arbor Pioneer), Arrak Bhattacharyya (Dow), Hogyeong Jeong (East Lansing), and Dennis Lu (Detroit Country Day); and Third Level Silver awards were captured by Charles Crissman (Dow), Peter Landry (Dow), and Michael Pierfelice (Allen Park). An additional 39 students received Bronze awards, and 51 received Honorable Mentions. Gold award winner Hou also received a certificate from Women and Math as the highest scoring female in the state.

The top 50 students received nearly $30,000 in scholarships, in amounts ranging from $450 to $2500, thanks to generous funding of the MMPC by corporate and other donors. The Honorable Mention winners received a copy of the book Which Way Did the Bicycle Go? by Joseph Konhauser, Dan Velleman, and Stan Wagon, courtesy of the Michigan Council of Teachers of Mathematics.

Part I of this year's MMPC (a 40-question multiple choice test) was administered to 14,173 students in October. The top 1030 qualified to take Part II in December, and 991 did so.

Make sure to plan on helping to grade the 45th MMPC next January at Albion!

MMPC Top 100 Statistics

44th MMPC Part II Problems

The top 1000 students had 100 minutes to solve these five problems and compete for scholarships and recognition.

  1. José, Luciano, and Plácido enjoy playing cards after their performances, and you are invited to deal. They use just nine cards, numbered 2 through 10, and each player is to receive three cards. You hope to hand out the cards so that the following three conditions hold:
    1. When José and Luciano pick cards randomly from their piles, Luciano most often picks a card higher than José;
    2. When Luciano and Plácido pick cards randomly from their piles, Plácido most often picks a card higher than Luciano ;
    3. When Plácido and José pick cards randomly from their piles, José most often picks a card higher than Plácido.
    Explain why it is impossible to distribute the nine cards so as to satisfy these three conditions, or give an example of one such distribution.
  2. Is it possible to fill a rectangular box with a finite number of solid cubes (two or more), each with a different edge length? Justify your answer.
  3. Two parallel lines pass through the points (0,1) and (-1,0). Two other lines are drawn through (1,0) and (0,0), each perpendicular to the first two. The two sets of lines intersect in four points that are the vertices of a square. Find all possible equations for the first two lines.
  4. Suppose a1, a2, a3, . . . is a sequence of integers that represent data to be transmitted across a communications channel. Engineers use the quantity G(n) = (1 - √3)an - (3 - √3)an +1 + (3 + √3)an +2 - (1 + √3)an +3 to detect noise in the signal. (a) Show that if the numbers a1, a2, a3, . . . are in arithmetic progression, then G(n) = 0 for all n = 1, 2, 3, . . . . (b) Show that if G(n) = 0 for all n = 1, 2, 3, . . . , then a1, a2, a3, . . . is an arithmetic progression
  5. The Olive View Airline in the remote country of Kuklafrania has decided to use the following rule to establish its air routes: If A and B are two distinct cities, then there is to be an air route connecting A with B if either there is no city closer to A than B or there is no city closer to B than A. No further routes will be permitted. Distances between Kuklafranian cities are never equal. Prove that no city will be connected by air routes to more than five other cities.

Out of the Mouths
Budding Mathematicians

This year's MMPC Part II produced the usual collection of humorous remarks which the graders dutifully recorded. Here is a selection from among them.

"It [problem #2] is impossible because no theorem exists that says so."

"Andrew Wiles recently proved that one guy's last theorem. . . . Therefore, one of the two cubes [problem #3] can't be expressed as two smaller cubes."

"The box will never be completely full. It will, however, reach a point where it will appear full to the human eye."

"I know this has nothing to do with problem #3, but please include this sentence as one of the funny remarks you read at the banquet."

"This problem [#3] is very hard to solve without computational aids, if not impossible."

"This [problem #5] is proven by the fact that the ratio of the radius of a circle to its circumference is 6:1."

"Kuklafrania is such a small, remote, underpopulated country that there couldn't possibly be more than 5 routes between each city."

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