A total of 102 Michigan high school students, from 59 different schools, were honored for their achievements on the 43rd Annual Michigan Mathematics Prize Competition at the Awards Day program held on March 4 at Albion College.

Robert Messer(Albion College) is the director of the MMPC this year.

Donald Saari(Northwestern University) gave two talks to the Top 100 and their parents and teachers: "The Mathematics of Voting: Do We Inadvertently Choose Badly?" and "Weird and Unexpected Behavior in Mathematical Astronomy".Qian Zhang, a student from Livonia MSC Program, 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 3 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 Program),Ryan Timmons(Groves), andMike Asmar(Troy), respectively. The First Level Silver winners wereMatt Orians(East Lansing) andChris Wagner(Novi); the Second Level Silver awards went toDennis Lu(Detroit Country Day) andJoseph Nievelt(Dondero); and Third Level Silver awards were captured byRobert Hough(Dow),Lee Keeram(Ann Arbor Huron),Vivek Shende(Detroit Country Day), andWen Shi(Andover). An additional 41 students received Bronze awards, and 50 received Honorable Mentions. Bronze award winnerChandra Clarke(Cranbrook Kingswood) received a certificate from Women and Math as the highest scoring female in the state.The top 52 students received over $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 Mathematical Cranks by

Underwood Dudley, 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 13,838 students in October. The top 1028 qualified to take part II in December, and 993 did so.

Make sure to plan on helping to grade the 44th MMPC next January at Albion!

- Top Gold winner
Qian Zhang, who repeats as top prize winner, is a senior. Second place winnerMike Asmaris a junior, and third place winnerRyan Timmonsis a sophomore (in his fourth year in the Top 100).- Of the eight Silver winners, one is a freshman, one is a sophomore, one is a junior, and the rest are seniors.
- Among the 41 Bronze winners are 24 seniors, nine juniors, and eight sophomores.
- Twenty-nine seniors, 13 juniors, seven sophomores, and one freshman took Honorable Mentions.
- There were seven females among the Top 102 (including one Bronze winner).
- The highest score was 94.6 out of 100. The cutoff score for scholarships was 58.4. It took a 50.8 to make it into the Top 102.
- The cut-off score to qualify for Part II this year was 21.

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

- The final Big 10 standings for the 1996 Women's Softball season were 1-Michigan, 2-Minnesota, 3-Iowa, 4-Indiana, 5-Michigan State, 6-Purdue, 7-Northwestern, 8-Ohio State, 9-Penn State, and 10-Wisconsin. (Illinois does not participate.) When you compare the 1996 final standings to the final standings for the 1999 season, you find that the following pairs of teams changed order relative to each other from 1996 to 1999 (there are no ties, and no other pairs changed places): (Iowa, Michigan State), (Iowa, Penn State), (Indiana, Michigan State), (Indiana, Purdue), (Indiana, Northwestern), (Indiana, Ohio State), (Indiana, Penn State), (Indiana, Wisconsin), (Michigan State, Penn State), (Purdue, Northwestern), (Purdue, Penn State), (Purdue, Wisconsin), (Northwestern, Penn State), (Northwestern, Wisconsin), (Ohio State, Penn State), (Ohio State, Wisconsin). Determine as much as you can about the final Big 10 standings for the 1999 Women's Softball season. If you cannot determine the standings, explain why you do not have enough information. You must justify your answer.
- (a) Take as given that any expression of the form
A sin t + B cos t, whereA > 0, can be put in the formC sin(t+D), whereC > 0and-pi/2 < D < pi/2. DetermineCandDin terms ofAandB.

(b) For the values ofCandDfound in part (a), prove thatA sin t + B cos t = C sin(t+D).

(c) Find the maximum value of3 sin t + 2 cos t.- A 6-by-6 checkerboard is completely filled with 18 dominoes (blocks of size 1-by-2). Prove that some horizontal or vertical line cuts the board in two parts but does not cut any of the dominoes.
- (a) The midpoints of the sides of a regular hexagon are the vertices of a new hexagon. What is the ratio of the area of the new hexagon to the area of the original hexagon?

(b) The midpoints of the sides of a regularn-gon (n > 2) are the vertices of a newn-gon. What is the ratio of the area of the new n-gon to that of the old? For both parts, justify your answer and simplify as much as possible.- You run a boarding house that has 90 rooms. You have 100 guests registered, but on any given night only 90 of these guests will show up. You don't know which 90 it will be, but they'll arrive for dinner before you have to assign rooms for the night. You want to give out keys to your guests so that for any set of 90 guests, you can assign each to a private room without any switching of keys.

(a) You could give every guest a key to every room. But this requires 9000 keys. Find a way to hand out fewer than 9000 keys so that each guest will have a key to a private room.

(b) What is the smallest number of keys necessary so that each guest will have a key to a private room? Describe how you would distribute these keys and assign the rooms. Prove that this number of keys is as small as possible.

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).

- Student asks for extra credit for relabeling the teams: Wolverines, Golden Gophers, Nittnay Lions, Spartans, Hawkeyes, Badgers, Wildcats, Boilermakers, Buckeyes, and Hoosiers.
- "Because of my religious convictions I cannot deal with sin [problem #2]. Sorry."
- "I liked the sports problem better. I doubt anyone has ever cheered for a trig function. That's why we have the Detroit Lions instead of the Detroit Cosines."
- "Assume B = 0 [problem #2], because it was not stated otherwise."
- A haiku for Problem 2: "Ah, cosecant are/ You jealous of the worn down/ Sin and cos buttons"
- "The question [problem #3] is not Y2K compliant; the proof may be miscalculated in my brain."
- "Well, I'm not sure how the numbers work on a set of dominoes, so I don't think this answer will be right. I haven't played for at least 8 years."
- "I am not sure how this [problem #3] could be proved except by a computer, as in high-profile problems such as the 4-color problem."
- "I don't know what you mean by private room [problem #5]. How can a room that 99 other people have keys to be considered a private room?"
- "Make all the locks on the doors the same so that one key will work for each of them, then give one key to each person. So minimum number = 100."

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