A total of 103 Michigan high school students, from 43 different schools, were honored for their achievement on the 45th Annual Michigan Mathematics Prize Competition at the Awards Day program held on March 2 at Albion College. Robert Messer (Albion C) is completing the final year of his three-year term as director of MMPC. David Redman (Delta C) will be the director next year for the 46th MMPC.
Aparna Higgins (U of Dayton) gave a talk “Defending the Roman Empire” and Joseph Gallian (U of Minn–Duluth) spoke on “Breaking Drivers’ Licence Codes”. Ryan Timmons, the second-place Gold Award winner, spoke to the banquet about his participation in the Michigan All-Star Math Team and the ARML competition. This year’s Midwestern ARML, in which teams drawn from the MMPC top 100 compete against teams from around the country, will be held June 1 at the University of Iowa in Iowa City.
The first-place Gold Award winner and Ford Motor Company Scholar is Robert Hough (Dow High School). The second-place award went to Ryan Timmons (Groves High School) and Christopher Cunningham (Bay City Central High School), and Thitidejõ Ujjathammarat (Cranbrook Kingswood School) tied for third-place Gold. The first-level Silver Award went to Peter Landry (Dow High School); Shailesh Agarwal (Troy High School) and Charles Crissman (Dow High School) tied for second-level Silver; and third-level went to a group of five students: Craig Chasseur (Saginaw Arts and Sciences Academy), Matt Elsey (Harrison High School), Dennis Lu (Detroit Country Day), Dino Sejdinovic (Cranbrook Kingswood School), and Michael Zajac (Dow High School). An additional 53 Bronze Awards were given, and 24 students received Honorable Mentions.
The top 50 students received over $30,000 in scholarships in amounts ranging from $450 to $2500. Thanks go to the corporate and other donors to the MMPC scholarship fund. The Honorable Mention winners received a copy of the book Mathematical Treks, by Ivars Peterson, courtesy of the Michigan Council of Teachers of Mathematics.
Part I of the MMPC is a 40-question multiple choice test, which this year was administered on October 10. The top 997 scorers were invited to take Part II on December 5 at Albion. There were 969 participants.
The exams are available in PDF format at the MMPC Web site: http://www.albion.edu/math/mmpc. Follow the link to “Previous MMPC Exams”.
- Top Gold winner Robert Hough is a junior, having been a Bronze winner as a sophomore and a Silver winner the previous year. Ryan Timmons, a senior, was the second-place Gold Award winner. He makes his sixth straight appearance in the Top 100. Two students tied for the third-place Gold Award: Christopher Cunningham, a senior, and Thitidej Ujjathammarat, a senior.
- Of the eight Silver winners, six are seniors and two are juniors, including Silver winners Craig Chasseur and Dino Sejdinovic.
- Among the 39 Bronze winners are 24 seniors, 9 juniors, and five are sophomores.
- Twenty-six seniors, 19 juniors, four sophomores, three freshman, and one eighth grader took Honorable Mentions.
- About 45% of the original contestants were female, as were about 23% of those who qualified for Part II. There were 13 young women among the Top 103 (including four Bronze Award winners).
- The total score for the competition is the sum of the Part I points (out of 40) and 1.2 times the Part II points (out of 50). The highest score was 91.6 out of 100. The cutoff score for scholarships was 49.5. It took a 48.2 to make it into the Top 103.
- The cut-off score to qualify for Part II this year was 24.
The top 1000 students had 100 minutes to solve these five problems and compete for scholarships and recognition.
- A clock has a long hand for minutes and a short hand for hours. A placement of those hands is natural if you will see it in a correctly functioning clock. So, having both hands pointing straight up toward 12 is natural and so is having the long hand pointing toward 6 and the short hand half-way between 2 and 3. A natural placement of the hands is symmetric if you get another natural placement by interchanging the long and short hands. One kind of symmetric natural placement is when the hands are pointed in exactly the same direction.
Are there symmetric natural placements of the hands in which the two hands are not pointed in exactly the same direction? If so, describe one such placement. If not, explain why none are possible.- Let m/n be a fraction such that when you write out the decimal expansion of m/n it eventually ends up with the four digits 2001 repeated over and over and over. Prove that 101 divides n.
- Consider the following two questions:
Question 1: I am thinking of a number between 0 and 15. You get to ask me seven yes-or-no questions, and I am allowed to lie at most once in answering your questions. What seven questions can you ask that will always allow you to determine the number? Note: You need to come up with seven questions that are independent of the answers that are received. In other words, you are not allowed to say, “If the answer to question 1 is yes, then question 2 is XXX; but if the answer to question 1 is no, then question 2 is YYY.”
Question 2: Consider the set S of all seven-tuples of zeros and ones. What sixteen elements of S can you choose so that every pair of your chosen seven-tuples differ in at least three coordinates?
- These two questions are closely related. Show that an answer to Question 1 gives an answer to Question 2.
- Answer either Question 1 or Question 2.
- You may wish to use the angle addition formulas for the sine and cosine functions: sin(a+b) = sin a cos b + cos a sin b, cos(a+b) = cos a cos b - sin a sin b.
- Prove the identity (sin x) (1+2 cos 2x)=sin(3x)
- For any positive integer n, prove the identity (sin x)(1 + 2 cos 2x + 2 cos 4x + ... + 2 cos 2nx) = sin((2n+ 1)x).
- 5. Define the set Ω in the xy-plane as the union of the regions bounded by the three geometric figures triangle A with vertices (0.5,1.5), (1.5,0.5) and (0.5,-0.5); triangle B with vertices (–0.5,–1.5), (–1.5,–0.5), and (–0.5,0.5); and rectangle C with corners (0.5,1.0), (–0.5,–1.0), and (0.5,–1.0).
- Explain how copies of Ω can be used to cover the xy-plane. The copies are obtained by translating Ω in the xy-plane, and copies can intersect only along their edges.
- We can define a transformation of the plane as follows: map any point (x,y) to (x + G, x + y + G), where G = 1 if y < –2x, G = –1 if y > –2x, and G = 0 if y = 2x. Prove that every point in Ω is transformed into another point in Ω, and that there are at least two points in Ω that are transformed into the same point.
Here is a collection of the humorous remarks of contestants in MMPC Part II recorded by graders.
“Hello Mr./Mrs. Judge—Be kind—I could use some $”
“It is difficult to prove a statement I don’t understand”
Regarding #3: “Question 2 is the offspring of Question 34 who is the sister of Question 15 being the third cousin (twice removed) of Question 66 …”
One student offered a ‘proof by injunction’ on #4.
“Hey! Look! A nose plug (Ω)”
“What do you call Santa's elves? Subordinate clauses.”
“I don’t need to prove it … I believe you.”
On #2: “Well, you look at the number, then you do some math.”
On #3: “This is tricky, I haven’t taken precalculus in months.”
“This is hard. Why am I doing this? I hate math.”
“I hope this isn't the test to get to utopia because if it is … I'm screwed.”
Answer to several problems: “42” [c.f. Hitchhiker’s Guide to the Galaxy, by Doug Adams]
“I haven’t taken trigonometry, I have no clue what I am doing here. I guess I am just trading convenience with points—I give one less paper to grade and you give me some points.” [On a blank #4.]
“I’m only in the 8th grade. Please don’t try to hurt me.”
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