MMPC Honors Top High School Students

A total of 101 Michigan high school students, from 50 different schools, were honored for their achievements on the 42nd Annual Michigan Mathematics Prize Competition at the Awards Day program held on March 6 at Michigan State University. Jerry Ludden (MSU) is this year's director of the MMPC.

Professors Charles MacCluer (MSU) and Andreas Blass (UM) presented stimulating talks in the afternoon to the award winners and their families and high school mathematics teachers, the former on industrial mathematics and the latter on unprovable truths. Michael Khoury, Jr. (a student from Brother Rice High School) spoke at the awards banquet about his experiences last year 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. The 1999 contest will take place on June 5 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 Qian Zhang (Livonia MSC), Michael Khoury, Jr. (Brother Rice), and Brian Schroeder (Okemos), respectively. The First Level Silver winner was Mike Asmar (Troy); the Second Level Silver award went to David Kurtz (Dow); and Third Level Silver awards were captured by Wayland Ni (Okemos), Vivek Shende (Detroit Country Day), Ryan Timmons (Groves), and Chris Wagner (Novi). An additional 41 students received Bronze awards, and 51 received Honorable Mentions. Bronze award winner Anne Kim (Ann Arbor Huron) received a calculator from Women and Math as the state's highest scoring girl.

The top 50 students received over $29,000 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 Ingenuity in Mathematics, 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 about 15,000 students in October. The top 1000 students took Part II in December.

Next year the MMPC will move to Albion College. Make sure to plan on helping with the grading next January!

MMPC Top 101 Statistics

42nd MMPC Part II Problems

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

  1. An organization decides to raise funds by holding a $60 a plate dinner. They get prices from two caterers. The first caterer charges $50 a plate. The second caterer charges according to the following schedule: $500 set-up fee plus $40 a plate for up to and including 61 plates; $2500 log10(p/4) for p > 61 plates.
    1. For what number of plates N does it become at least as cheap to use the second caterer as the first?
    2. Let N be the number you found in (a). For what number of plates X is the second caterer's price exactly double the price for N plates?
    3. Let X be the number you found in (b). When X people appear for the dinner, how much profit does the organization raise for itself by using the second caterer?
  2. Let N be a positive integer. Prove the following:
    1. If N is divisible by 4, then N can be expressed as the sum of two or more consecutive odd integers.
    2. If N is a prime number, then N cannot be expressed as the sum of two or more consecutive odd integers.
    3. If N is twice some odd integer, then N cannot be expressed as the sum of two or more consecutive odd integers.
  3. Let S = 1/12 + 1/22 + 1/32 + 1/42 + . . . .
    1. Find, in terms of S, the value of 1/22 + 1/42 + 1/62 + 1/82 + . . . .
    2. Find, in terms of S, the value of 1/12 + 1/32 + 1/52 + 1/72 + . . . .
    3. Find, in terms of S, the value of 1/12 - 1/22 + 1/32 - 1/42 + . . . .
  4. Let {P1, P2, P3, . . .} be an infinite set of points on the x-axis having positive integer coordinates, and let Q be an arbitrary point in the plane not on the x-axis. Prove that infinitely many of the distances |PiQ | are not integers.
    1. Draw a relevant picture.
    2. Provide a proof.
  5. Point P is an arbitrary point inside triangle ABC. Points X, Y, and Z are constructed to make segments PX, PY, and PZ perpendicular to AB, BC, and CA, respectively.
    1. If triangle ABC is an equilateral triangle, prove that x + y + z does not change regardless of the location of P inside triangle ABC.
    2. If triangle ABC is an isosceles triangle with |BC | = |CA|, prove that x + y + z does not change when P moves along a line parallel to AB.
    3. Now suppose that triangle ABC is scalene. Prove that there exists a line for which x + y + z does not change when P moves along this line.

Out of the Mouths of 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, culled by Exam Committee Chair Renate McLaughlin (UM-Flint).

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