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) andAndreas 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), andBrian Schroeder(Okemos), respectively. The First Level Silver winner wasMike Asmar(Troy); the Second Level Silver award went toDavid Kurtz(Dow); and Third Level Silver awards were captured byWayland Ni(Okemos),Vivek Shende(Detroit Country Day),Ryan Timmons(Groves), andChris Wagner(Novi). An additional 41 students received Bronze awards, and 51 received Honorable Mentions. Bronze award winnerAnne 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!

- Top Gold winner
Qian Zhangis a junior. Second place winnerMichael Khoury, Jr.and third place winnerBrian Schroederare seniors.- Among the six Silver winners, three are juniors, two are sophomores, and
Ryan Timmonsis a freshman (this is his third year in the Top 100).- Among the 41 Bronze winners are 20 seniors, 12 juniors, four sophomores, and five freshmen.
- Twenty-two seniors, 18 juniors, five sophomores, two freshmen, and four students whose class was not reported took Honorable Mentions.
- There were 13 females among the Top 101 (three Bronze and 10 Honorable Mentions).
- The highest score was 88.2 out of 100. The cutoff score for scholarships was 64.8. It took a 57 to make it into the Top 101.
- The cut-off score to qualify for Part II this year was 23.

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

- 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.

- For what number of plates N does it become at least as cheap to use the second caterer as the first?
- 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?
- 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?
- Let N be a positive integer. Prove the following:

- If N is divisible by 4, then N can be expressed as the sum of two or more consecutive odd integers.
- If N is a prime number, then N cannot be expressed as the sum of two or more consecutive odd integers.
- If N is twice some odd integer, then N cannot be expressed as the sum of two or more consecutive odd integers.
- Let S = 1/1
^{2}+ 1/2^{2}+ 1/3^{2}+ 1/4^{2}+ . . . .

- Find, in terms of S, the value of 1/2
^{2}+ 1/4^{2}+ 1/6^{2}+ 1/8^{2}+ . . . .- Find, in terms of S, the value of 1/1
^{2}+ 1/3^{2}+ 1/5^{2}+ 1/7^{2}+ . . . .- Find, in terms of S, the value of 1/1
^{2}- 1/2^{2}+ 1/3^{2}- 1/4^{2}+ . . . .- Let {P
_{1}, P_{2}, P_{3}, . . .} 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 |P_{i}Q | are not integers.

- Draw a relevant picture.
- Provide a proof.
- 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.

- If triangle ABC is an equilateral triangle, prove that x + y + z does not change regardless of the location of P inside triangle ABC.
- 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.
- 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.

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).

- "Does Problem #1 mean you're charging us for this year's awards banquet?"
- "Assume that (3 + 5) / 4 is an integer."
- "On Problem #3 you need to express S as a fraction using common denominators."
- "Between any two points in the plane there are an infinite amount of subpoints."
- "I hope the pain of my total defeat has satisfied you as much as it has satisfied me. Next year I shall return with a vengeance. Be prepared for your highest score ever."
- "On Problem #5, x + y + z doesn't change because you said so. And I believe you because you are all math geniuses and are too nice to tell me a falsehood."
- "My friend told me we could make a living off MMPC. I guess it's always harder in a new business at first."

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