MMPC Honors Top High School Students

A total of 100 Michigan high school students, from 56 different schools, were honored for their achievements on the 40^th Annual Michigan Mathematics Prize Competition at the Awards Day program held on March 1 at Grand Valley State University in Allendale. Karen Novotny (GVSU) is this year's director of the MMPC.

Charles Johnson of the College of William and Mary presented a stimulating talk in the afternoon entitled The Centennial History of the Prime Number Theorem to the award winners and their families and high school mathematics teachers. Camillia Smith (a student from East Lansing High School) spoke at the awards banquet about her 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 1997 contest will take place on May 31 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 page 21).

For a bit of fun, Pam and Clark Wells (GVSU) organized a scavenger hunt for teams of students. While the students were scavenging, the Section's Executive Committee met with their parents and teachers to discuss the MMPC and other issues.

This year's first, second, and third place Gold award recipients were J. Benjamen Hough (H. H. Dow High School), Haiwen Chu (Andover High School), and David Houston (Henry Ford II High School), respectively. The First Level Silver winners were Michael Khoury, Jr. (Brother Rice), Goutam Reddy (Detroit Country Day), and Jonathan Salz (Seaholm); Second Level Silver awards went to Vijay Divi (Troy) and Aram Harrow (East Lansing); and Third Level Silver awards were captured by Noah Levitt (Detroit Country Day), Charles DeZiel (Escanaba), Christian Grostic (Muskegon), and Brian Richardson (Greenhills). An additional 44 students received Bronze awards, and 44 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, II.

Part I of this year's MMPC (a 40-question multiple choice test) was administered to over 15,000 students in October. The top thousand students took Part II in December.

Next year the MMPC will move from Grand Valley to Michigan State. Jerry Ludden will serve as director, at least for the first year. Make sure to plan on helping with the grading next January in East Lansing!

MMPC Top 100 Statistics

· Top Gold winner Ben Hough and third place winner David Houston are seniors; second place winner Haiwen Chu is a junior.

· Among the Silver winners are six seniors, two juniors, and one sophomore.

· Among the Bronze winners are 24 seniors, 12 juniors, five sophomores, two freshmen, and one seventh-grader.

· Twenty-nine seniors, ten juniors and five sophomores took Honorable Mentions.

· There were 12 females among the Top 100 (six Bronze and six Honorable Mentions).

· The highest score was 92.2 out of 100. The cutoff score for scholarships was 62.8. It took a 58 to make it into the Top 100.

· Of the 15,749 students from about 500 schools who took Part I, 989 students from 222 schools qualified for Part II by scoring 22 or higher (out of 40).

40^th MMPC Part II Problems

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

1. An Egyptian fraction has the form 1/n, where n is a positive integer. In ancient Egypt, these were the only fractions allowed. Other fractions between zero and one were always expressed as a sum of DISTINCT

Egyptian fractions. For example, 3/5 was seen as 1/2 + 1/10, or 1/3 + 1/4 + 1/60. The preferred method of representing a fraction in Egypt used the "greedy" algorithm, which, at each stage, uses that Egyptian fraction which eats up as much as possible of what is left of the original fraction. Thus the greedy fraction for 3/5 would be 1/2 + 1/10.

(a,b) Find the greedy Egyptian fraction representations for 2/13 and 9/10.

(c,d) Find the greedy Egyptian fraction representations for 2/(2k+1) and 3/(6k+1), where k is a positive integer.

2. (a) The smaller of two concentric circles has radius one unit. The area of the larger circle is twice the area of the smaller circle. Find the difference in their radii.

(b) The smaller of two identically oriented equilateral triangles has each side one unit long. The smaller triangle is centered within the larger triangle so that the perpendicular distance between parallel sides is always the same number d. The area of the larger triangle is twice the area of the smaller triangle. Find d.

3. Suppose that the domain of a function f is the set of real numbers and that f takes values in the set of real numbers. A real number x₀ is a fixed point of f if f(x₀) = x₀.

(a,b) Let f(x) = m x + b. For which m does f have a fixed point? Find the fixed point, in terms of m and b, when it exists.

(c) Consider the function f(x) = x ² ­ c. For which values of c are there two different fixed points? For which values of c are there no fixed points? In terms of c, find the value(s) of the fixed point(s).

(d) Find an example of a function that has exactly three fixed points.

4. A square based pyramid is made out of rubber balls. There are 100 balls on the bottom level, 81 on the next level, etc., up to 1 ball on the top level.

(a) How many balls are there in the pyramid?

(b) If each ball has a radius of 1 meter, how tall is the pyramid?

(c) What is the volume of the solid that you create if you place a plane against each of the four sides and the base of the balls?

5. We wish to consider a general deck of cards which is specified by a number of suits, a sequence of denominations and a number (possibly 0) of jokers. The deck will consist of exactly one card of each denomination from each suit, plus the jokers, which are "wild" and can be counted as any possible card of any suit. For example, a standard deck of cards consists of 4 suits, 13 denominations, and 0 jokers.

(a) For a deck with 3 suits {a, b, c} and 7 denominations {1, 2, 3, 4, 5, 6, 7}, and 0 jokers, find the probability that a 3-card hand will be a straight. A straight consists of 3 cards in sequence (e.g., a-1, c-3, a-2, but not a-6, b-7, c-1).

(b) For a deck with 3 suits, 7 denominations, and 0 jokers, find the probability that a 3-card hand will consist of 3 cards of the same suit (i.e., a flush).

(c) For a deck with 3 suits, 7 denominations, and 1 joker, find the probability that a 3-card hand will be a straight and also the probability that a 3-card hand will be a flush if dealt at random from such a deck.

(d) Find a number of suits and the length of the denomination sequence that would be required if a deck is to contain 1 joker and is to have identical probabilities for a straight and a flush when a 3-card hand is dealt. The answer that you find must be an answer such that a flush and a straight are possible but not always certain to occur.

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.

Problem 1

"In order to solve the problem, I first had to know if Egyptians were really this picky. So I made telepathic contact with Jo of the Sphinx. He said, 'The answer lies within the pyramid.' Then I realized that the trick is to look at the sequence of triangular numbers. Bet you didn't find it that easy!"

Problem 2

"Also, ­( 5) ­ 2 does not work because it is positive!"

"For both parts I ended up with an unsolvable quadratic equation. I think if I knew how to handle the equations, I'd have the answers."

Problem 3

"A fixed point is a happy point."

"What are you talking about? Too much English; needs more math."

Problem 4

"Ten balls stacked on top of each other20 meters, maybe a tad less if the balls aren't exactly on top of each other."

"h = the height of the period that I failed grievously to find in part (b)"

"My rubber balls do not stick."

"Rubber is not entirely solid, and the weight of the pyramid would compress the ball's height and expand the width."

"V = (3/4) A_b h. This is not the formula for V of a pyramid. It should be!"

Problem 5

"This shows that the number of suits is important in solving 5(d)."

"I refuse to answer this question. Gambling is bad unless I win."