A total of 103 Michigan high school students, from 53 different schools, were honored for their achievements on the 41

^{st}Annual Michigan Mathematics Prize Competition at the Awards Day program held on February 28 at Michigan State University.Jerry Ludden(MSU) is this year's director of the MMPC.Professors

Mary WinterandSheldon Newhouse(MSU) presented stimulating talks in the afternoon to the award winners and their families and high school mathematics teachers. Both presentations dealt with geometry, fractals, and chaos, and the students also had an opportunity for some hands-on work with relevant software.Vijay Divi(a student from Troy 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 1998 contest will take place on May 30 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 FallNewsletter).This year's first, second, and third place Gold award recipients were

Michael Khoury, Jr.(Brother Rice),Robert Easton(Lapeer West), andVijay Divi(Troy), respectively. First Level Silver winners wereRobert Lillibridge(Livonia MSC) andQian Zhang(Livonia MSC); the Second Level Silver award went toGoutam Reddy(Detroit County Day); and Third Level Silver awards were captured byHaiwen Chu(Andover),Betsy Huebner(Grosse Pointe North),Ryan Lang(L'Anse Creuse), andThomas Chambers, III(Royal Oak Kimball). An additional 46 students received Bronze awards, and 47 received Honorable Mentions.The top 56 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 Gems, III.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 stay at MSU. Make sure to plan on helping with the grading next January 9 in East Lansing!

MMPC Top 103 Statistics

- Top Gold winner Michael Khoury, Jr., is a junior. Second place winner Robert Easton and third place winner Vijay Divi are seniors.
- Among the seven Silver winners are four seniors, two juniors, and one sophomore.
- Among the 46 Bronze winners are 22 seniors, 15 juniors, five sophomores, two freshmen, and two eighth-graders.
- Twenty-five seniors, 14 juniors and eight sophomores took Honorable Mentions.
- There were 12 females among the Top 103 (one Silver, seven Bronze, and four Honorable Mentions).
- The highest score was 92 out of 100. The cutoff score for scholarships was 65. It took a 60 to make it into the Top 103.
- Of the roughly 15,000 students from about 500 schools who took Part I, 1050 students from 157 schools qualified for Part II by scoring 21 or higher (out of 40).

## 41

^{st }MMPC Part II Problems

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

- It can be shown in Calculus that the area between the
x-axis and the parabolay=k x^{2}(ka positive constant) on the interval 0<=x<=aisk a^{3}/ 3.

- Find the area between parabola
y= 4x^{2}and thex-axis for 0 <=x<= 3.- Find the area between parabola
y= 5x^{2}and thex-axis for -2 <=x<= 4.- A square 2 by 2 dartboard is situated in the
xy-plane with its center at the origin and its sides parallel to the coordinate axes. Darts that are thrown land randomly on the dartboard. Find the probability that a dart will land at a point of the dartboard that is nearer to the point (0,1) than to the bottom edge of the board.- When two rows of a determinant are interchanged, the value of the determinant changes sign. There are also certain operations which can be performed on a determinant which leave its value unchanged. Two such operations are changing any row by adding a constant multiple of another row to it, and changing any column by adding a constant multiple of another column to it. Often these operations are used to generate lots of zeroes in a determinant in order to simplify computations. In fact, if we can generate zeroes everywhere below the main diagonal in a determinant, the value of the determinant is just the product of all the entries on that main diagonal. [An example was given.] Transform these determinants into determinants with zeroes everywhere below the main diagonal, and find their values:
(a) 1 3 -1 (b) 0 1 2 3 4 7 2 1 0 1 2 3 -6 5 2 1 0 1 3 2 1 0- In Pascal's triangle, the entries at the ends of each row are both 1, and otherwise, each entry is the sum of the two entries diagonally above it:
Row number 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1The triangle gives the binomial coefficients in expansions like (a+b)^{3}= 1a^{3}+ 3a^{2}b+ 3ab^{2}+ 1b^{3}.

- What is the sum of the numbers in row #5 of Pascal's triangle?
- What is the sum of the numbers in row #
nof Pascal's triangle?- Show that in row #6 of Pascal's triangle, the sum of all the numbers is exactly twice the sum of the first, third, fifth, and seventh numbers in the row.
- Prove that in row #
nof Pascal's triangle, the sum of all the numbers is exactly twice the sum of the numbers in the odd positions of that row.- The product of several terms is sometimes described using the symbol , which is capital pi, the Greek equivalent of p, for the word "product". For example, the symbol P
_{k=1..4}(2k+1) means the product of numbers of the form (2k+1), fork= 1,2,3,4. Thus it equals 945.

- Evaluate as a reduced fraction P
_{k=1..10}k/(k+2).- Evaluate as a reduced fraction P
_{k=1..10}(k^{2}+ 10k+ 17)/(k^{2 }+ 14k+ 41).- Evaluate as a reduced fraction P
_{k=1..}(k^{3}1)/(k^{3}+ 1).

- In right triangle
CAB(right angle atA), the medianAF, the angle bisectorAE, and the altitudeADdivide the right angleAinto four equal angles, whereF,E, andDoccur in that order on the line segment fromCtoB. IfAB= 1, find the area of triangleAFE.- If in any triangle, an angle is divided into four equal angles by the median, angle bisector, and altitude drawn from that angle, prove that the angle must be a right angle.

## 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 ChairMike Merscher(LTU)."The area of the triangle is 148 pounds."

"The answer is between 0 and 7/9. That should be worth 4 points credit."

"I should know this, but we skipped that chapter."

"You obviously believe this is true, and that's okay. I just don't understand why you need

mysupport.""This seems hard, but it is actually very easy, because the product of an infinite amount of numbers is zero."

"This is a big math test. A big math test would not ask us to prove that a false statement is true. Therefore the statement is true."

"The angle bisector divides the angle into two equal but quite different angles."

"I have a wonderful proof, and I'd like to show it to you, but I have to go to Lit class."

"The dart will probably have a low probability of landing nearer the point than the bottom edge of the dartboard, because most people tend to shoot right at where they want the dart to land and neglect to consider that gravity pulls down on the dart at 9.8 meters per second per second."

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