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The Mathematical Association of America

AMericAn MATheMATics coMpeTiTions

2006
22nd Annual

MAA AMC 8

Summary of Results and Awards

Learning Mathematics
Through Selective Problem Solving

Examinations prepared by a subcommittee of the American Mathematics Competitions and administered by the office of the Director

The
American Mathematics competitions

are sponsored by

e Mathematical Association of America and e Akamai Foundation

Contributors:

American Mathematical Association of Two Year colleges

American Mathematical society
American society of pension Actuaries
American statistical Association
Art of problem solving
Awesome Math canada/UsA Mathcamp canada/UsA Mathpath casualty Actuarial society clay Mathematics institute

institute for operations research and the Management sciences
L. G. Balfour company

Mu Alpha Theta national Assessment & Testing national council of Teachers of Mathematics pedagoguery software inc. pi Mu epsilon society of Actuaries
U.s.A. Math Talent search W. h. Freeman and company

Wolfram research inc.

Table Of COnTenTs

2006 IMO T e am with their medal s . .................................................................. 2

Report of the Director..........................................................................................3

I. Introduction.................................................................................................... 3 II. General Results............................................................................................. 3 III. Statistical Analysis of Results........................................................................ 4 Table ꢀ - School & Student Registrations & Averages ..................................... 4 Table 2 - Participation & Average Score by Grade Levels................................. 4 IV. School Plaque/Nameplates........................................................................... 4 V. 2006 State Top Student Plaque..................................................................... 5 VI. Area of Participation.................................................................................... 5 VII. Date and Format of the 2007 AMC 8........................................................ 5 VIII. Acknowledgement..................................................................................... 6

Janet Sylvester................................................................................................... 6

IX. State Directors, Sponsors, Donors and Committees..................................... 6 X. Documentary Film - USAMO and IMO ...................................................... 7 XI. Contributions of the University of Nebraska................................................ 7 XII. Final Comments......................................................................................... 7

Regions of the aMC 8..........................................................................................8 Message from the aMC 8 Chair...........................................................................9 Message from the CaMC Chair.........................................................................10 sliffe awards for Distinguished Mathematics Teaching in the Middle school ....11

2006 AMC 8 Edyth May Sliffe Award Winners................................................ ꢀ2

Current & Former Sliffe Winners Attending Reception...................................... 15 2005 Sliffe Winners Attending Reception ........................................................ 15 2006 Sliffe Awards Reception, Miscellaneous Pictures........................................ 16

american Mathematics Contest 10 and american Mathematics Contest 12.......17 aMC Committees..............................................................................................18

American Mathematics Contest 8 Examination Committee ............................. ꢀ8 Committee on the American Mathematics Competitions................................. ꢀ8 CAMC Executive Committee........................................................................... ꢀ8 Advisory Panel.................................................................................................. ꢀ9

state Directors aMC 8.......................................................................................21 sponsors and Donors .........................................................................................22

Sponsors ........................................................................................................... 22 Contributors..................................................................................................... 22

MAA AMC Offic e . ......................................................................................... 22

school Honor Roll .............................................................................................23

United States .................................................................................................... 23 International ................................................................................................... 27

school Merit Roll...............................................................................................29

United States .................................................................................................... 29 International..................................................................................................... 35

Scanning the answer forms . . ............................................................................ 36

Perfect scores .....................................................................................................37

United States .................................................................................................... 37 International Students ...................................................................................... 43

student achievement Roll .................................................................................48

United States .................................................................................................... 48

Missouri State Awards for 2005-2006 ............................................................. 55

International..................................................................................................... 56

Distinguished Honor Roll..................................................................................60

United States .................................................................................................... 60

ꢀ

Table of Contents continued

AMC affiliates gather at the Winter Joint Math Meetings in New Orleans. ........ 73

International..................................................................................................... 74 Late Participation ............................................................................................. 88

Affiliated Middle School to Jilin University   ..................................................... 88

Honor Roll.........................................................................................................89

United States .................................................................................................... 89 International................................................................................................... ꢀ32 Late Participation ........................................................................................... ꢀ53

Item Difficulty .................................................................................................154

Table 3............................................................................................................ ꢀ54 Graph 3.......................................................................................................... ꢀ55

Results by Gender, 2006 aMC 8......................................................................156

Table 4............................................................................................................ ꢀ56 Graph 4.......................................................................................................... ꢀ57

Results by Grade level .....................................................................................158

Table 5............................................................................................................ ꢀ58 Graph 5.......................................................................................................... ꢀ59

score Comparison between 2005 and 2006 .....................................................160

Table 6.............................................................................................................160

Graph 6.......................................................................................................... ꢀ6ꢀ

Rank listings ...................................................................................................162

Table 7 - Schools by State/Province/Country.................................................. ꢀ62 Table 8 - Student Enrollment by State/Province/Country............................... ꢀ62

Index................................................................................................................163

2007 aMC 8 school Registration-nov. 13, 2007...........................................................169

2006 IMO T e am with their medals

from left: Alex Saltman (deputy leader), Ryan Ko, Zuming Feng (Leader), Zach Abel, Yi Sun, Steven Dunbar (AMC Director), Alex Zhai, Arnav Tripathy and Zeb Brady.

2

RePORT Of THe DIReCTOR

I. InTroduCTIon

I am pleased to announce that over ꢀ47,000 students from more than 2,200 schools participated in the 2006 AMC 8 contest. Many dedicated mathematics teachers provided an opportunity for students to participate in this national exam, and the Committee created an exam which was both “exciting and challenging”. A hearty thanks to all of you for continuing to recognize the fact that the AMC 8 plays a complementary role in support of the junior high/middle school mathematics curriculum.

II. General resulTs

is summary includes a listing of the results and awards associated with the 22nd annual AMC 8, which was held on Tuesday, November ꢀ4, 2006. PLEASE NOTETHATTHE 2007 AMC 8 WILL BE HELD ONTUESDAY,

NOVEMBER ꢀ3th.

From the responses received from many of the contest managers, it is evident that the examination was once again well received by the students and teachers.

A comparison of the 2006 statistics shows that this year’s contest was comparable to the 2005. A total of 293 perfect scores were recorded (79 in 2005) while the national average moved to ꢀ0.ꢀ4 (9.835 in 2005). e AMC 8 Committee is in its final stage of preparation of the 2007 exam and I expect it to be similar to the 2006 edition. e one notable change will be a “No Calculator” rule for the AMC 8, AMC ꢀ0 and AMC ꢀ2 contests next school year. A four-year statistical comparison of results is given in Section III.

We sincerely hope that your school administrators will continue to support your future requests for funds to enable your students to participate in this academic activity which is consistent with the standards set by the National

Council of T e achers of Mathematics.

On behalf of the AMC 8 committee, I express my appreciation to all of the school contest managers and their associated teachers, school counselors and administrators for providing an opportunity for their students to participate in the 2006 AMC 8. I particularly would like to express my gratitude to those teachers who personally paid the fees associated with the contest. eir dedication to the profession is certainly noteworthy and beyond the call of duty.

3

report of the director continued

III. sTaTIsTICal analysIs of resulTs

a fOUR YeaR COMPaRIsOn

Please note that the word “recorded” used throughout this section should be interpreted as indicating that the results are based on only the items that were mark-sensed by the students. Some of the students failed to mark items such as their grade or gender.

Table 1 - school & student registrations & averages

  • 2006
  • 2005
  • 2004
  • 2003
  • 2002

Registered schools

2,269

  • 2,629
  • 2,70ꢀ
  • 2,396
  • 2,493

Registered students

Schools sending in results Answer forms received

ꢀ69,650 ꢀ72,380 ꢀ84,270 ꢀ83,830 ꢀ88,530

  • 2,235
  • 2,504
  • 2,628
  • 2,737
  • 2,390

ꢀ47,2ꢀ0 ꢀ43,428 ꢀ54,590 ꢀ58,309 ꢀ57,778
Recorded female participants 68,ꢀ45 Recorded male participants Average Top Score Average of all participants Average of female participants Average of male participants Perfect papers
67,655 73,275
ꢀ7.5ꢀ 9.835
9.32
ꢀ0.34
79
73,647 78,265
ꢀ7.42
9.94
75,574 80,0ꢀ0
ꢀ8.32 ꢀ0.27
9.72
ꢀ0.79
ꢀ92
75,880

  • 78,825
  • 76,283

ꢀ8.08 ꢀ0.ꢀ4
9.6ꢀ
ꢀ0.64

293

ꢀꢀ.09 ꢀ0.47 ꢀꢀ.68
ꢀ26
9.47 ꢀ0.4
50

Table 2 - Participation & average score by Grade levels

Participation by Recorded Grade Levels

GRaDe

4 & 5

  • 2006
  • 2005
  • 2004
  • 2003
  • 2002
  • 2001

ꢀ,853

2,0ꢀ4

  • ꢀ,496
  • ꢀ,553
  • ꢀ,560 ꢀ,736

678

ꢀ6,639 48,479 78,038

ꢀ4,268 48,4ꢀ8 76,877

ꢀ5,248 ꢀ5,4ꢀ4 ꢀ4,707 ꢀ4,7ꢀ0 5ꢀ,679 53,ꢀ70 53,804 58,27ꢀ 84,399 86,542 85,35ꢀ 93,090

Average Score by Recorded Grade Levels

Grade

4 & 5

2006

8.2ꢀ

2005

8.05

2004

8.ꢀ5

2003

7.62

2002

7.76

2001

6.73

678

  • 8.5ꢀ
  • 8.ꢀ6

9.37
8.5ꢀ 9.55
8.58 9.77

  • 9.3ꢀ
  • 8.28

9.82
ꢀ0.75
ꢀ0.59 ꢀꢀ.79
9.49

  • ꢀ0.66
  • ꢀ0.52
  • ꢀ0.48
  • ꢀ0.93

ere were a total of 3,ꢀ72 school winners at 2,249 schools, and the following number of certificates were sent to schools: Gold – 9,238, Silver- ꢀꢀ,827, Bronze- ꢀ4,9ꢀ6. While the AMC 8 is primarily intramural, the national results make some comparison at that level viable, but the examination, which has no established norms, is comparable to neither classroom nor placement tests.

IV. sChool Plaque/namePlaTes

e AMC 8 School Winner Plaque is a beautiful 9”xꢀ2” plaque with space for ꢀ2 engraved name plates with the names of your year-by-year school winners. Teachers who gave the AMC 8 (then known as the AJHSME) in the ꢀ980’s and ꢀ990’s will remember the AMC provided an initial plaque and a school

4

report of the director continued

winner plate each year to schools participating in the AMC 8. Comments from teachers told us you missed having this recognition program for your math scholars. In your AMC 8 report from December 2006 you will find the web address that will allow you to order directly from our supplier. e AMC pays for the initial plaque and engraved name plate in the first year and the annual engraved plates in subsequent years.

V. 2006 sTaTe ToP sTudenT Plaque

This year we are presenting 174 Top Student Plaque awards. This plaque is awarded to the top scoring student in each state, all of the U.S. Territories and schools abroad, and Canada. This student award program that began in 1991 will continue to be presented on an annual basis. You will note, this Summary contains photos of some of last year’s awardees.

VI. area of ParTICIPaTIon

Participation in the AMC 8 extends beyond the USA. Registrations were received from 5ꢀ APO/FPO Schools (Junior High or Middle Schools associated with U.S. military bases abroad), American schools associated with U.S. Embassies, U.S. Territories (Guam, Puerto Rico and the Virgin Islands), Canadian, and foreign schools (including a group of ꢀ67 schools fromTaiwan, 39 Canadian schools, 26 from Singapore and 40 other international schools), for a total of ꢀ4,557 students. Recognition is hereby given to those teachers abroad who have provided on opportunity for their students to participate in the AMC 8. A complete list of participating schools from foreign countries and U.S. territories is found in this summary. We are grateful to the Department of Defense for providing the financial support which enabled the students in its dependent schools to participate in the AMC 8.

VII. daTe and formaT of The 2007 amC 8

e date of the 2007 AMC 8 is Tuesday, November ꢀ3th. e exam will again be a 40 minute, twenty-five question, multiple-choice contest. e scoring procedure will be the same as in 2006: one point will be given for each correct answer. Registration materials will be mailed to schools during the months of May, September, and October. If you wish to register now, a 2007 Registration Form has been included in this Summary on Page ꢀ69. We hope that your school will participate again in the 2007 AMC 8.

Completion of your registration as early as possible is always appreciated. Contest materials will be mailed in October and early November, so there is ample time for planning and completion of pre-contest activities. We would be most grateful if you would encourage neighboring schools to participate in the AMC 8.

e National Association of Secondary School Principals will continue to place the AMC 8 on its advisory list of contests and activities for 2007-2008.

5

report of the director continued

VIII. aCknowledGemenT

We are fortunate to have dedicated staff members who fulfill the responsibilities associated with the entire examination process in a timely and efficient manner. One can imagine all of the expected and unexpected duties and operations which must be completed during the examination year. AMC staff members include Donita Bowers, Marsha Conley, Rachelle Jensen, Linda McCormick, RuthAnn Royer, Sawsan Mohamed, Kay Michael and Elizabeth Price. Our

University student helpers are Gabe Smith and Nikki Bohl. I welcome this

opportunity to commend their excellent work. On November 25, 2006 AMC Registration Clerk Janet Sylvester died suddenly and unexpectedly. Janet had been employed at the AMC office since January 200ꢀ, and her good spirit, ready laugh, efficiency and readiness to do whatever was needed at the AMC office will be missed.

Janet Sylvester

IX. sTaTe dIreCTors, sPonsors, donors and

CommITTees

In this summary you will find a current list of the AMC 8 State Directors, sponsors, donors, members of the Committee on the American Mathematics Competitions (CAMC), and AMC 8 committee. In addition, the members of the American Mathematics Competitions (AMC) Panel are listed, and they are hereby recognized for submitting and reviewing potential problems for the examinations of the AMC. e work of the State Directors is significant

6

report of the director continued

to the success of the AMC 8. ey send follow-up mailings and attend state mathematics meetings to give talks about the examination in order to encourage more schools to participate. e Directors and Panelists receive no financial compensation for their work. e CAMC recognizes these individuals for their outstanding dedication to the profession of mathematics educators.

Special recognition is given for the work of the AMC 8 and CAMC committee members. ey devote considerable time, under the direction of Bonnie Leitch (AMC 8 Chair) and Elgin Johnston (CAMC Chair), reviewing examination policy, proposing problems, and developing the examination. e contributions of our sponsors and donors are hereby recognized. e significance of the examination is greatly enhanced by their continued support.

X. doCumenTary fIlm - usamo and Imo

MAA President-Elect Joe Gallian secured a grant from Larry Penn, who was on the US Math Olympiad Team in the late ꢀ970s, to fund a documentary film about the students in the USAMO, and the USA team at the 2006 International Math Olympiad. Larry Penn is now a partner in the investment firm Ellington Management Group. e film will be similar to the book “Count Down” and the film “Spellbound”. e director is George Csicsery. He produced an award winning film on the mathematician Paul Erdos and is working on a film about the American mathematician Julia Robinson. As of this writing, filming at the USAMO awards ceremony and the IMO is complete, and post-production has begun. We expect to see the finished film in late 2007.

XI. ConTrIbuTIons of The unIVersITy of nebraska

e AMC office has been located at the University of Nebraska in Lincoln since ꢀ970. For the past 36 years the University has made significant contributions in support of the American Mathematics Competitions including providing utilities, office space, and the use of university facilities.

XII. fInal CommenTs

I welcome and appreciate your comments regarding this Summary and the procedures associated with the AMC 8.

steven dunbar, director

The Mathematical Association of America American Mathematics Competitions
University of Nebraska - Lincoln, 1740 Vine Street
Lincoln, NE 68588—0658

7

ReGIOns Of THe aMC 8

Listed below are the eleven regions of the U.S.A. and Canada.

REGION

  • 0
  • Connecticut, Maine, Massachusetts, New Hampshire, Pennsylvania,

Rhode Island, Vermont

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    Abstracts for the MAA Undergraduate Poster Session San Diego, CA January 12, 2018 Organized by Eric Ruggieri College of the Holy Cross and Chasen Smith Georgia Southern University Dear Students, Advisors, Judges and Colleagues, If you look around today you will see over 400 posters (a record number!) and more than 650 student pre- senters, representing a wide array of mathematical topics and ideas. These posters showcase the vibrant research being conducted as part of summer programs and during the academic year at colleges and univer- sities from across the United States and beyond. It is so rewarding to see this session, which offers such a great opportunity for interaction between students and professional mathematicians, continue to grow. The judges you see here today are professional mathematicians from institutions around the world. They are advisors, colleagues, new Ph.D.s, and administrators. Many of the judges are acknowledged here in this booklet; however, a number of judges here today volunteered on site. Their support is vital to the success of the session and we thank them. We are supported financially by the National Science Foundation, Tudor Investments, and Two Sigma. We are also helped by the members of the Committee on Undergraduate Student Activities and Chapters (CUSAC) in some way or other. They are: Dora C. Ahmadi; Benjamin Galluzzo; TJ Hitchman; Cynthia Huffman; Aihua Li; Sara Louise Malec; May Mei; Stacy Ann Muir; Andy Niedermaier; Peri Shereen; An- gela Spalsbury; Violetta Vasilevska; Gerard A. Venema; and Jim Walsh. There are many details of the poster session that begin with putting out the advertisement in FOCUS, ensuring students have travel money, mak- ing sure the online submission system works properly, and organizing poster boards and tables in the room we are in today that are attributed to Gerard Venema (MAA Associate Secretary), Kenyatta Malloy (MAA), and Donna Salter (AMS).
  • A Calendar of Mathematical Dates January

    A Calendar of Mathematical Dates January

    A CALENDAR OF MATHEMATICAL DATES V. Frederick Rickey Department of Mathematical Sciences United States Military Academy West Point, NY 10996-1786 USA Email: fred-rickey @ usma.edu JANUARY 1 January 4713 B.C. This is Julian day 1 and begins at noon Greenwich or Universal Time (U.T.). It provides a convenient way to keep track of the number of days between events. Noon, January 1, 1984, begins Julian Day 2,445,336. For the use of the Chinese remainder theorem in determining this date, see American Journal of Physics, 49(1981), 658{661. 46 B.C. The first day of the first year of the Julian calendar. It remained in effect until October 4, 1582. The previous year, \the last year of confusion," was the longest year on record|it contained 445 days. [Encyclopedia Brittanica, 13th edition, vol. 4, p. 990] 1618 La Salle's expedition reached the present site of Peoria, Illinois, birthplace of the author of this calendar. 1800 Cauchy's father was elected Secretary of the Senate in France. The young Cauchy used a corner of his father's office in Luxembourg Palace for his own desk. LaGrange and Laplace frequently stopped in on business and so took an interest in the boys mathematical talent. One day, in the presence of numerous dignitaries, Lagrange pointed to the young Cauchy and said \You see that little young man? Well! He will supplant all of us in so far as we are mathematicians." [E. T. Bell, Men of Mathematics, p. 274] 1801 Giuseppe Piazzi (1746{1826) discovered the first asteroid, Ceres, but lost it in the sun 41 days later, after only a few observations.
  • 2021 May-June

    2021 May-June

    Newsletter VOLUME 51, NO. 3 • MAY–JUNE 2021 PRESIDENT’S REPORT I wrote my report for the previous newsletter in January after the attack on the US Capitol. This newsletter, I write my report in March after the murder of eight people, including six Asian-American women, in Atlanta. I find myself The purpose of the Association for Women in Mathematics is wondering when I will write a report with no acts of hatred fresh in my mind, but then I remember that acts like these are now common in the US. We react to each • to encourage women and girls to one as a unique horror, too easily forgetting the long string of horrors preceding it. study and to have active careers in the mathematical sciences, and In fact, in the time between the first and final drafts of this report, another shooting • to promote equal opportunity and has taken place, this time in Boulder, CO. Even worse, seven mass shootings have the equal treatment of women and taken place in the past seven days.1 Only two of these have received national girls in the mathematical sciences. attention. Meanwhile, it was only a few months ago in December that someone bombed a block in Nashville. We are no longer discussing that trauma. Many of these events of recent months and years have been fomented by internet communities that foster racism, sexism, and white male supremacy. As the work of Safiya Noble details beautifully, tech giants play a major role in the creation, growth, and support of these communities.
  • Emissary | Spring 2016

    Emissary | Spring 2016

    Spring 2017 EMISSARY M a t h ema t i cal Sc ien c es R e sea r c h Insti tute www.msri.org Analytic Number Theory Emmanuel Kowalski The basic problem in analytic number theory is often to compute how many integers, or primes, or other arithmetic objects (from L- functions to modular forms to elliptic curves to class groups to . ), satisfy certain condi- tions. Such questions have fascinated math- ematicians for centuries; for instance, one can ask: (a) How many prime numbers are there less than some large given number x? How many n 6 x are there such that the interval from n to n + 246 contains at least two prime numbers? (b) Fixing integers k > 2 and r > 1, for a large integer n, how many integers n1, k ..., nr are there such that n1 + + k ··· nr = n (Waring’s problem)? (c) How many integers in an interval x< 1/2 A Kloosterman path joins the partial sums of Kl (1;10007) over 1 x k when n 6 x + x are only divisible by 2 6 6 primes smaller than x1/20? or larger k varies from 0 to 10006. (The figure appears with axes on page 5.) 1/1000 than x ? or both? algebraic geometry, representation theory, lée Poussin. Similarly, representation theory (d) How many integers n 6 x are such that all — and others — find their use). And the took some of its inspiration from Dirichlet’s n4 + 2 is not divisible by the square of influence of such problems on mathematics study of primes in arithmetic progressions; any prime? often goes well beyond the original moti- much more recently, the ideas of higher- vation of solving them.
  • 01Case Part-I 1-76

    01Case Part-I 1-76

    1. From the Twentieth Century The women appearing in this chapter, inspirations for many currently active mathematicians, received their doctorates before 1960. Their stories look back through time, arranged in reverse chronological order of the women’s births. It is appropriate to begin with reminiscences of Olga Taussky Todd, namesake of the conference that spurred the compilation of this book. Her charmingly informal remarks are augmented by excerpts from a memorial article about her. Later in the book, two articles give more formal descrip- tions of her life and work. Julia Robinson’s mathematics led to her election as the first woman mathematician in the National Academy of Sciences (NAS); she was also the first woman elected president of the American Mathematical Society (AMS). In 1980 she was the second woman to present the prestigious AMS Colloquium Lectures. The first woman to do so, fifty-three years earlier in 1927, was Anna Johnson Pell Wheeler.1 Among the few women who earned a doctorate in mathematics in the first half of the twentieth century, Pell Wheeler was singular in receiving such recognition for her mathematics. She is likely best known for her active mentoring of other women. While chair of the mathematics department at Bryn Mawr College, she brought two refugees to her institution: Emmy Noether, then an established mathemati- cian, and also, to work with Noether, the young Taussky Todd—the women whose lives bracket this chapter. Noether is indisputably among the greatest mathematicians of the twentieth century. In recognition of the importance of her work, the Association for Women in Mathematics (AWM) has sponsored since 1980 the annual Noether Lecture series at the Joint Mathematics Meet- ings (JMM).