The Fifteenth Annual
Harvard-MIT
Mathematics Tournament
February 10th-February 11th, 2012
The Harvard-MIT Mathematics Tournament
290 Massachusetts Avenue • Cambridge, MA 02139
http://hmmt.mit.edu/ • hmmt-request@mit.edu
February 11, 2012
It is our great pleasure to welcome you to the 2012 Harvard-MIT Mathematics Tournament
(HMMT). After a highly successful 2011 HMMT, we are excited to see all of you back in Cambridge
and ready to compete.
We are deeply committed to running a successful tournament and promoting mathematical curiosity
in high school students. By bringing in nearly 900 competitors from across the nation and the world,
we believe we can be the incubator to make that happen - but you have to be the ignition! While you
are here, please do take some time to introduce yourself to the other competitors and meet people
from other teams, especially during the Friday night activities at MIT. When we were competitors,
one of our greatest joys was meeting others, and we hope that you will be able to share in that joy
as well.
All of our problems are written and compiled by Harvard and MIT undergraduates. We hope that
they are not only fascinating and challenging, but also engaging. Our staff of over 30 Harvard and
MIT undergraduates, and countless more contributors, have been working tirelessly over the past
year to make this competition a reality. In addition to the traditional team round and signature
Guts event, this year’s individual round will feature three separate tests in Algebra, Geometry, and
Combinatorics.
Should you have any questions, comments, or concerns during the tournament, please feel free to
ask any of the staff members of the tournament for help - we can be identified by our crimsoncolored T-shirts. Should you have any feedback about the tournament, please email us at hmmtfeedback@mit.edu. If you are interested in general updates and mathematical thoughts about
HMMT, please follow us on our Twitter @mathtournament and our Facebook at facebook.com/hmmt.info.
Announcements will be tweeted throughout the day.
Whether you are new to HMMT or a returning veteran, good luck and enjoy your time here! On
behalf of the entire staff, we look forward to welcoming you to the 2012 Harvard-MIT Mathematics
Tournament!
Sincerely,
William Chen, Harvard
Mark Velednitsky, MIT
Tournament Directors
Brian Hamrick
Travis Hance
George Xing
Problem Czars
HMMT 2012 Sponsors
The Harvard Mathematics Department
The MIT Mathematics Department
MIT Copytech
Jane Street Capital
The D. E. Shaw Group
Two Sigma
PROMYS
HMMT 2012 Staff
Tournament Directors
William Chen
Mark Velednitsky
Problem Czars
Brian Hamrick
Travis Hance
George Xing
Registration Director
Eli Ross
Finance Director
Stephanie Lin
Finance Associate
David Ke
Scripts Director
Daniel Li
Webmaster
Patrick Hulin
General Officer
Paul Handorff
Head Proctor
Olive Tang
Spokespeople
Charles Du
Ben Kraft
Carl Lian
Allen Park
Felix Sun
Prize Czar
Ina Chen
Head Graders
Sam Trabucco
Joy Zheng
Head Scorekeeper
Brian Hamrick
Assistant Scorekeepers
Lars Johnson
Gaurav Singh
Taylor Sutton
Hosting Coordinators
Felix Sun
Jonathan Li
Mini-events Director
Andrew Liu
Guts Director
Anthony Liu
Food Czars
Grace Gee
Michelle Yang
Sign Czars
Angela Li
Joel Schneider
Runners
Nevin Daniel
Marco Gentili
Jamin Liu
Photographer
Sherry Wu
Problem Writers
Justin Brereton
Casey Fu
Carl Lian
Jonathan Schneider
Jeffrey Wu
Sherry Wu
Joy Zheng
...and all of our day-of
volunteers!
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School
AAST
Accord Institute - IDEA MATH
Acton-Boxborough High School
Advanced Math and Science Academy
Albany Area Math Circle
American School of Brasilia - EAB
Archbishop Molloy High School
Baton Rouge Magnet High School
Beijing STFX
Bejing University High School
Belmont High School
Bishop Brady High School
BNL Gifted Math Program
Canyon Crest Academy
Choate Rosemary Hall
Collegiate School
Diamond Study Centre
Dougherty Valley High School
Fairfax Math Circle
Fairfield High Preparatory School
Florida
Georgetown Day School
Greensburg Salem High School
Groton School
Gunn
Hamden Hall Country Day School
Harmony School of Excellence
HCSSiM 2011 Alumni
Hotchkiss School
Hunter College High School
IDEA Math
Illinois Math and Science Academy
Individuals
Interlake High School
Ithaca Math Circle
Team
Bergen Academies AVPA
Bergen Academies AEDT
Bergen Academies ABF
Accord A* IDEA MATH Air
Accord A* IDEA MATH Fire
Accord A* Earth
WeMade USAMO
AMSA Charter
Steinmetz
EAB Bulls
Stanford OHS
Baton Rouge Math Team
Beijing STFX A1
Beijing STFX A2
BUHS 1
Robert’); DROP TABLE Students;–
Bishop Brady High School
BNL Octachorons
BNL Tesseracts
Canyon Crest Academy SoCalCoolUs
Choate A
Choate B
Collegiate School
Diamond Study Centre
DVHS Math Team
Fairfax Math Circle 1
Fairfax Math Circle 2
Fairfield High 1
Fairfield High 2
Florida A
Florida B
Florida C
Georgetown Day School 1
Georgetown Day School 2
Team Fractal
Groton School
Gunn
Hamden Hall
HSE 1
Derivatives of Abelian Groups
Hotchkiss
More Cowbell
These Go To Eleven
Shanghai Foreign Language School
United Idea
IMSA
Individuals 1
Individuals 2
Individuals 3
WashingtonGold
Ithaca Math Circle
Team Round
Maxwell–Dworkin 323
Maxwell–Dworkin 221
Sever 101
Science Center 221
Maxwell–Dworkin 221
Emerson 210
Maxwell–Dworkin 319
Maxwell–Dworkin G125
Science Center 309
Sever 107
Sever 213
Sever 113
CGIS Knafel K107
Sever 201
Science Center 304
Maxwell–Dworkin 119
Pierce 100F
Sever 213
Sever 104
Maxwell–Dworkin 119
Science Center 309
Emerson 210
Maxwell–Dworkin G125
Sever 102
Maxwell–Dworkin G125
Sever 104
Sever 110
Sever 109
Sever 206
Sever 212
Science Center 309
Sever 106
Sever 213
Sever 110
Maxwell–Dworkin 119
Sever 205
Sever 105
Pierce 209
Pierce 209
Sever 215
Pierce 100F
Sever 102
Emerson 210
Sever 212
Maxwell–Dworkin G125
Sever 206
Sever 202
Sever 202
Sever 202
Sever 208
Sever 206
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School
Kent School
Lehigh Valley
Team
Lions
Lehigh Valley Fire
Lehigh Valley Ice
Lehigh Valley Lightning
Lexington High School
Quasi Houbenators
Loomis Chaffee
MTC A
BNL Hypercubes
MISTerminators
Montgomery Blair A
Montgomery Blair B
Morgantown 1
Nashoba 1
101010 (Base 2)
Murph and the Magictones
Party Posse
JPS Hawks
Team Round
Sever 111
CGIS Knafel K105
Pierce 209
Maxwell–Dworkin G135
Sever 207
Sever 113
Maxwell–Dworkin G135
Sever 112
Maxwell–Dworkin 119
Sever 111
Pierce 114
Sever 110
Sever 210
Sever 112
Sever 203
Pierce 320
Sever 113
CGIS Knafel K109
West Windsor-Plainsboro HS South
Westborough High School
Western Mass ARML
Newton South Sines
Newton South Cosines
North Carolina
North Houston
Hoggers
Phillips Academy 1
Phillips Academy 2
PEARL
Pottsville
Little Tigers
RUHS 1
RUHS 2
World Domination
Sabio Math
South Brunswick Duckies
St. Mark’s School of Texas A
The Lawrenceville School
Thomas Jefferson A
Thomas Jefferson B
Thomas Jefferson C
Twin Cities West
Twin Cities East
UCC
Ward Melville Team A
Wasatch Academy A1
Wasatch Academy B1
Knights A
Knights B
WW-P Pirates
Westborough HS
Western Mass ARML All Stars
Sever 211
Sever 101
Maxwell–Dworkin 223
Emerson 210
Sever 203
Sever 211
Sever 113
CGIS Knafel K108
Emerson 105
Sever 210
Science Center 222
Sever 209
Sever 103
Sever 210
Emerson 105
Sever 204
Emerson 105
CGIS Knafel K106
Sever 209
Sever 203
Sever 102
Sever 106
Sever 208
Emerson 105
Sever 106
Sever 103
Sever 215
Emerson 210
Sever 109
Emerson 105
Sever 207
Western PA
Western PA
Sever 201
Lexington High School
Livingston High School
Loomis Chaffee School
Math Training Center
BNL Gifted Math Program
MIST Academy
Montgomery Blair High School
Morgantown High School
Nashoba Regional High School
Needham High School
New York City Math Team
JP Stevens High School
Newton South High School
North Carolina
North Houston Acad of Sci and Math
Northfield Mount Hermon School
Phillips Academy
Phillips Exeter Academy
Pottsville Area High School
Princeton High School
Renmin University High School
Sabio Academy
South Brunswick High School
St. Mark’s School of Texas
The Lawrenceville School
Thomas Jefferson HS for Sci and Tech
Twin Cities All-Stars
Upper Canada College
Ward Melville High School
Wasatch Academy
West Windsor-Plainsboro HS North
Schedule of Events
Time
All Day
8:00–8:45am
8:00–8:45am
9:00–12:00pm
10:00–10:30am
12:30–1:30pm
1:30–2:30pm
3:00–4:30pm
5:00–5:30pm
Event
Coach/Parent Lounge
Lost & Found, Protests
Registration
Breakfast
Individual Rounds
Proctor Meeting
Team Round
Lunch
Guts Round
Awards Ceremony
Awards Overflow
Location
Holden Chapel
Sci Center 109
Science Center Lobby
Science Center Lobby
Sci Center A (Teams 1–10)
Sci Center B (Teams 11–43)
Sci Center C (Teams 44–66)
Sci Center D (Teams 67–82)
Emerson 105 (Teams 83–103)
Emerson 210
Various
Harvard Square
Sci Center B (Teams 1–48)
Sci Center C (Teams 49–82)
Sci Center D (Teams 83–103)
Sci Center B
Sci Center C
Rules Highlights
http://web.mit.edu/hmmt/www/rules.shtml
Calculators and Computational Aids
You may not use books, notes, calculators, pocket organizers, slide-rules, abaci, or any other computational aids. Similarly, you may not use graph paper, rulers, protractors, compasses, architectural tools,
or any other drawing aids. Translators are also prohibited. Translation sheets will be available upon
request.
Grading Policy
• Answers must be simplified and exact unless otherwise specified.
• Rational numbers should be given in lowest terms.
• An nth root should be simplified so that the radicand contains no fractions and is not divisible by the
nth power of any prime.
• Denominators do not need to be rationalized.
• Correct mathematical notation must be used.
• No partial credit will be given unless otherwise specified.
Protests
If a student believes that an answer given on the answer key for a morning event is incorrect, an appeal
must be submitted in Science Center 109 by 2:00 pm.
Decisions of the coordinators of the Harvard-MIT Mathematics Tournament are final.