23rd Harvard-MITMathematics Tournament

HMMT November TournamentSaturday, November 9, 2019

November 9, 2019

Hello HMMT 2019 November Contestants!

T he HMMT organizing team is delighted to have you join us here in Cam-bridge! We have all been committed to building this November’s competition

into an experience that we hope you will recall fondly for many years to come.From crafting the most intriguing of problems to setting up the spaces for optimaltest taking, our team of over 100 MIT and Harvard undergraduates presents toyou the 23rd season of HMMT.

These couple of days every year are truly special for us, and we hope that youall will find the same statement holds for you. We are honored by how eager manyin the high school math community are to take part in HMMT. As we celebratethis wonderful discipline, never forget that even more importantly, we gather hereto showcase and laud the achievements of all you fantastic mathletes.

Those of you who are seasoned veterans of HMMT may recall that, for years,our tournament directors have equated the importance of the human element ofHMMT to that of math. This year, we take this statement further; the humanelement of HMMT is crucial to our competition. For all its purity and intrinsicbeauty awaiting discovery by bright minds such as your own, math cannot holdthe sole, dominant place in our minds and being. Take the time to speak to theother incredibly bright students around you, but more importantly, listen. Wehope that you will find unparalleled appreciation and wonder by familiarizingyourself with your exceptional peers.

To the coaches, parents, and chaperones: we have seen how much you careabout granting your students opportunities to pursue their interests in math,and we cannot express enough how appreciative we are for your support of thenext generation of mathletes. Through the toils of registration, the anxiety ofthe lottery, and the logistical taxes of organizing the trip here, your commitmentshines. While today is primarily about the students, we hope that you will enjoyPo-Shen Loh’s talk and cheer on your students during the guts round.

As you navigate Harvard College’s campus, we hope you will find the mapsin this brochure helpful. Please check your team’s room assignments carefullyand notify any HMMT member (clad in turquoise blue staff shirts or red officerjackets) if you require any assistance. We understand that the campus can behard to navigate at first, and we’re here to help!

Once again, we are excited to have you join us for today’s tournament, and wewish you all the best in the competition!

Regards,

Jeffery YuWilliam HoTournament Directors

Evan ChenWilliam QianChief Advisors

Carl SchildkrautKrit BoonsirisethMilan HaimanNovember Problem Czars

Student Schedule

Friday, November 8, 2019 (MIT)

6:30PM 8:00PM Early Registration Building W20 La Sala

6:30PM 7:15PM Pizza and Ice Cream Building W20 La Sala

7:15PM 7:30PM Mini-Events Preview Building W20 La Sala

7:30PM 9:00PM Mini-Events Buildings 1, 3, 5

Saturday, November 9, 2019 (Harvard)

8:00AM 9:00AM Breakfast Science Center Arcades

8:00AM 9:00AM Registration* Science Center B, C

8:30AM 9:00AM Individual CompetitorMeeting Science Center 109

9:00AM 10:15AM General Round varies

10:30AM 11:45AM Theme Round varies

12:15PM 1:30PM Team Round varies

1:30PM 2:45PM Lunch

Science Center ArcadesTicknor LoungeBauer NaitoCambridge Queen’s Head

2:45PM 4:30PM Guts Round varies

4:40PM 5:25PM Integration Bee Finals Science Center B

5:35PM 6:30PM Awards Ceremony Science Center B, C

* Registration and awards are in the same room.

Coaches, Proctors, and Parents Schedule

Friday, November 8, 2019 (MIT)

6:30PM 8:00PM Early Registration Building W20 La Sala

6:30PM 7:15PM Pizza and Ice Cream Building W20 La Sala

7:15PM 7:30PM Mini-Events Preview Building W20 La Sala

7:30PM 9:00PM Mini-Events Buildings 1, 3, 5

Saturday, November 9, 2019 (Harvard)

8:00AM 9:00AM Breakfast Science Center

8:00AM 9:00AM Registration* Science Center

9:30AM 10:00AM Proctor Meeting Geological Lecture Hall

10:15AM 11:45AM Talk by Po-Shen Loh Geological Lecture Hall

12:15PM 1:30PM Team Round varies

1:30PM 2:45PM Lunch

Science Center ArcadesTicknor LoungeBauer NaitoCambridge Queen’s Head

2:45PM 4:30PM Guts Round Viewing Geological Lecture Hall

4:40PM 5:25PM Integration Bee Finals Science Center B

5:35PM 6:30PM Awards Ceremony Science Center B, C

* Registration and awards are in the same room.

A help center is located in Science Center 112.

Teams and Room AssignmentsNote: Registration and awards will take place in the same room.

SC = Science Center, SH = Sever, BN = Bauer Naito, CQH = Cambridge Queen’s Head

Team Name Organization Indiv. Team Lunch Guts Awards

(Mostly Not) Naperville North Naperville North High School SC D SH 105 SC SC D SC B

4Schoolers Mathletics 4Schoolers SH 113 SH 207 Ticknor SH 113 SC C

A** Nanjing Foreign LanguageSchool SC D SC D SC SC B SC B

AB Blue Turtle ABRHS SC B SC 309 BN SC B SC B

AB Green Frog ABRHS SC B SC B10 BN SC B SC B

Academy Eagles 1 Academy Eagles SH 113 SH 113 Ticknor SC D SC C

ACM ACM SC B SC 304 BN SC B SC B

Ada Lovelace Williston Northampton School SC B SC B SC SC C SC C

Albany Area Math CircleCardinals

Albany Area Math Circle SC C SC C BN SC C SC C

AlphaStar Academy LIGHTNING AlphaStar Academy SC B SC B BN SC B SC B

AlphaStar Academy THUNDER AlphaStar Academy SC B SC 104 BN SC B SC B

Amsa Omega Algonquin SH 113 SH 202 Ticknor SH 113 SC B

Belmont Hill School Belmont Hill School SC D SC D BN SC B SC B

BHS Math Team: Imaginaries Brookline High School SC D SH 211 Ticknor SC D SC C

Big L Club Livingston High School SC B SC B SC SC C SC C

BISV Bobcats B Basis Independent Silicon Valley SC C SH 213 Ticknor SC B SC B

Boston Latin School Boston Latin School SC D SC D SC SC B SC B

Bromfield School Bromfield School SH 113 SH 113 CQH SC D SC B

CCHS Math Team Concord Carlisle Math Team SH 113 SH 113 Ticknor SC D SC C

Team Name Organization Indiv. Team Lunch Guts Awards

Cheshire Academy Team A Cheshire Academy SH 113 SH 202 Ticknor SH 113 SC C

Collegiate School Collegiate School SC C SH 103 CQH SC C SC C

DA Math Team Deerfield Academy SH 113 SH 206 Ticknor SH 113 SC B

Daisy The Hockaday School SC C SC C SC SC B SC B

Doormat HPRPL SC D SH 111 CQH SC D SC C

DTSAO Beijing National Day School SC C SC C BN SC C SC C

East Brunswick Math Square East Brunswick Math Circle SC B SC A SC SC C SC C

East Brunswick Math Triangle East Brunswick Math Circle SC B SC B SC SC C SC C

Einsteinian Monkeys Jim Thorpe Mathematics Club SC D SH 112 CQH SC D SC B

Emma Willard School Emma Willard School SC C SC C SC SC C SC C

Enormous L Club Livingston High School SC B SC E SC SC C SC C

ESBNUExperimental High SchoolAttached To Beijing NormalUniversity

SC C SH 214 CQH SC C SC C

Exeter Kids in Khakis Exeter Math Club SC B SC 222 SC SC B SC B

Exeter Red Exeter Math Club SC B SC B SC SC B SC B

fighting Tsinghua international school SC C SH 214 SC SC D SC C

First State Math A First State Math SC D SC D SC SC D SC C

G-Y United Tiger Gauss Academy of MathematicalEducation SC B SC E SC SC C SC C

G-Y United Lion Gauss Academy of MathematicalEducation SC B SC A SC SC C SC C

Garnet Valley Math Team Garnet Valley High School SC C SC C SC SC C SC C

GRAVITON [A] Mengersponge SH 113 SH 113 CQH SC D SC B

ICAE1 Indus Center for AcademicExcellence SC B SC B SC SC B SC B

Team Name Organization Indiv. Team Lunch Guts Awards

ICAE2 Indus Center for AcademicExcellence SC B SC E SC SC B SC B

IDEA MATH E IDEA MATH SC B SC B10 SC SC C SC C

Individuals 1 Individuals SC B SH 203 CQH SC C SC C

Individuals 2 Individuals SC C SH 203 CQH SC C SC C

Individuals 3 Individuals SC D SH 203 CQH SC C SC C

Jericho A Jericho High SC B SC B SC SC B SC B

Katherine Johnson Williston Northampton School SC B SC 310 SC SC C SC C

Kentucky A Origami for Education SC C SC C SC SC C SC C

Lawrenceville Math Club 1 The Lawrenceville School SC B SC B SC SC C SC C

Lawrenceville Math Club 2 The Lawrenceville School SC B SH 102 SC SC C SC C

Lexington Delta Lexington High School SC C SH 213 CQH SC C SC C

Long Island Super Long Island Super SC B SC 309 SC SC B SC B

MacDu�e The MacDu�e School SC D SH 104 SC SC D SC C

Maple TTMath School SC C SC C SC SC B SC B

MBHS A Montgomery Blair High School SC D SH 109 SC SC D SC C

Midwest Math Circle A Midwest Math Circle SC B SC 105 SC SC C SC C

Milton Mustangs Milton Academy SH 113 SH 113 SC SC D SC C

Mozart Bellevue Math Team SC D SH 210 CQH SC B SC B

Munster High School Munster High School SC D SH 107 SC SC B SC B

MVRCS Team 1A Mystic Valley Regional CharterSchool SH 113 SH 205 Ticknor SH 113 SC B

New Britain High Math Team Consolidated School District ofNew Britain SC D SH 209 CQH SC D SC C

Team Name Organization Indiv. Team Lunch Guts Awards

NewstarTHE HIGH SCHOOL AFFILIATEDTO BEIJING NORMALUNIVERSITY

SC C SH 214 SC SC B SC B

Newton North High SchoolTigers Newton North High School SC C SH 103 SC SC B SC B

North Georgia A North Georgia A SC B SC 309 SC SC C SC C

Northfield Mount HermonMath Club Northfield Mount Hermon SH 113 SH 201 Ticknor SH 113 SC B

Northview High School Northview High School SC C SC C SC SC B SC B

Open mind Nanjing Shanghai Foreign LanguageSchool SC C SH 110 SC SC D SC C

PEA Ladies in Lavender Exeter Math Club SC B SC 221 SC SC B SC B

Phillips Academy A1 Phillips Academy SC C SC C SC SC B SC B

Phillips Academy A3 Phillips Academy SC C SC B SC SC B SC B

Plano Math Circle Plano Math Circle SH 113 SH 113 SC SC D SC B

PRISMS Falcons Princeton International School ofMath and Science SC B SC B SC SC C SC C

PRISMS Young Falcons Princeton International School ofMath and Science SC B SC 309A SC SC C SC C

PSAM A Pinellas Student Association ofMathematics SC B SC B SC SC B SC B

PSAM B Pinellas Student Association ofMathematics SC B SC A SC SC B SC B

Random Math Team A Random Math SC D SC D SC SC D SC C

Ransom Everglades Mu AlphaTheta Math Team 1

Ransom Everglades SC C SH 102 SC SC B SC B

Raritan Valley Math Team -Newton

Raritan Valley Math Group SH 113 SH 113 SC SC D SC C

Russian School of MathematicsTeam A Russian School of Mathematics SC B SC B SC SC B SC B

Russian School of MathematicsTeam B Russian School of Mathematics SC B SC E SC SC B SC B

Shenzhen The A�liated High School ofSCNU SC C SH 106 SC SC B SC B

Shenzhong Flametree Shenzhen Middle School SC D SC D SC SC D SC C

Team Name Organization Indiv. Team Lunch Guts Awards

St. Mark’s Number Lions St. Mark’s School SC D SH 210 SC SC D SC C

St. Paul’s School St. Paul’s School SC C SH 103 SC SC C SC C

Stony Brook School Math Team Stony Brook School SH 113 SH 206 Ticknor SH 113 SC C

Tabor Academy Tabor Academy SC D SH 101 SC SC B SC B

Taipei Math Team A Taipei Math Team SH 113 SH 204 Ticknor SH 113 SC C

Team Je� Kent School SC C SH 106 CQH SC D SC C

TEHSATBNU TEHSATBNU SC C SC B SC SC C SC C

Texas Academy of Math andScience

Texas Academy of Math andScience SC B SC A SC SC B SC B

Texas Thunder Austin Math Circle SC B SC 309A BN SC B SC B

Texas Tornado Austin Math Circle SC B SC B BN SC B SC B

Texas Typhoon Austin Math Circle SC B SC 222 BN SC B SC B

The Galois Group The Taft School SC C SC C SC SC B SC B

THINKTOWN 1 ASDAN China SC B SC B BN SC C SC C

THINKTOWN 2 ASDAN China SC B SC A BN SC C SC C

Thousand Oaks Techies Thousand Oaks Techies SC D SC D SC SC D SC C

TLPS SEP 5 Transitions Lab PreparatorySchool SC C SC C SC SC B SC B

Trinity Math Team Trinity School SC D SC D SC SC D SC C

University of Toronto Schools B University of Toronto Schools SC B SC B SC SC B SC B

University of Toronto Schools C University of Toronto Schools SC B SC A SC SC B SC B

University of Toronto Schools D University of Toronto Schools SC B SC E SC SC B SC B

Vancouver Volcanoes V Square Math SC C SC C SC SC C SC C

Team Name Organization Indiv. Team Lunch Guts Awards

Walton Math Team Walton High School SH 113 SH 113 SC SC D SC B

Ward Melville Math Team A Ward Melville Senior High School SH 113 SH 202 Ticknor SH 113 SC B

Westchester Area Math Circle Westchester Area Math Circle SC D SH 208 SC SC B SC B

Western Mass ARML Stars Western Mass ARML SC B SC B SC SC B SC B

Western Mass ARML Suns Western Mass ARML SC B SH 102 SC SC B SC B

Weston High School Alpha Weston High School SC C SC C SC SC C SC C

Weston High School Theta Weston High School SC C SC B SC SC C SC C

Wilbraham & Monson Academy Wilbraham & Monson Academy SC D SC D SC SC B SC B

Wildcats A The Wheatley School SC D SC D SC SC B SC B

Winchester 1 Winchester High School SC C SH 213 SC SC B SC B

WSHS 1 White Station High School SC C SC C SC SC B SC B

Y-G Alligator Yu’s Elite Education SC B SC B SC SC C SC C

Yu’s Bu�alo Yu’s Elite Education SC B SC 221 SC SC C SC C

Zhaoyang RDFZ Chaoyang Branch School SC C SH 110 SC SC B SC B

Test Information

Rules

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. Translatorsare also prohibited.

Guidelines for Acceptable Answers

• 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 fractionsand 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 is incorrect, they mustgo to the designated Appeals Room and submit an appeal in writing. The deadlineto do so will be announced on the day of the contest.

Decisions of the November HMMT coordinators are final.

Rules of Conduct and Expectations for HMMTAttendees

Introduction

HMMT is organized by undergraduate students at Harvard and MIT under theeponymous student organization at MIT and the Harvard Undergraduate Math-ematics Association at Harvard. While we receive support from both universities,HMMT is a distinct entity from Harvard University and the Massachusetts Instituteof Technology. When you attend our competition, you are a HMMT-sponsoredvisitor to the campuses of Harvard and MIT. As such, we mandate that HMMT at-tendees abide by the following rules of conduct. We strongly urge the coach orresponsible adult of each organization to carefully read this document and ensurethat students are also made familiar with it. Repeated or severe failure to complywith any subset of these rules will result in disqualifications and revocation of yourorganization’s participation in HMMT.

HMMT Security and Sta�ng

HMMT will be overseen by multiple groups. You must abide by direction from eachof these entities:

• HMMT organizers (volunteers, sta�, and o�cers): These are the studentswho run HMMT. HMMT organizers are instructed to correct any violations ofthis code of conduct.

• Harvard Campus Services: Harvard Campus Services Custodial Sta� willbe on the premises throughout the day. If you are unable to comply withCustodial Sta� directives, you should inform a HMMT o�cer immediatelyfor resolution.

• HMMT executive o�cers (Tournament Directors and Chief Advisors): TheTournament Directors (William Ho and Je�ery Yu) and Chief Advisors (EvanChen and William Qian) have multiple privileges, including the power toimmediately suspend degenerate teams.

• Securitas: Securitas is a contracting security company used by Harvard Uni-versity. Securitas guards will be patrolling the Science Center. If Securitasasks you to do something you perceive as unreasonable, you should inform aHMMT o�cer immediately for mediation. Securitas has the right to dismissyou from the premises.

• Harvard University Police Department: A HUPD detail will be present dur-ing the morning of HMMT while registration fees are being paid. You mustobey HUPD. If mediation is required, you should inform a HMMT o�cer im-mediately.

Conduct

While you are an attendee of HMMT, you must abide by these rules:

• Be respectful of people. This includes attendees from other organizations,members of the HMMT organizing team, and sta� of the universities. Rudebehavior, inappropriate language, and threats against another person of anynature, whether verbal or physical, executed or suggested, will not be toler-ated in the slightest.

• Dishonesty accompanied by ill intent (including, but not limited to, lying,cheating, doping, and scamming) is forbidden.

• Be respectful of the spaces. While attending the events of HMMT, you area visitor to the private property of Harvard University or MIT. Do not defacethe rooms.

• Do NOT eat or drink in classrooms unless they are designated for eating.Your organization’s eating location should be clearly denoted in the brochure.Do not dispose of your food scraps in classroom waste containers. Eat onlywithin your designated food location. You will be issued wristbands basedon your food service location. If your team has ordered food, it will be deliv-ered to your location by HMMT sta�. Securitas and HMMT sta� are instructedto enforce food assignments.

• Do not transfer furniture between rooms. If you require more furniture, alerta HMMT o�cer, and arrangements will be made for you by Harvard CampusServices.

• Do not linger around doors and stairs. Do not obstruct hallways. You shouldwait in open locations or your assigned rooms to prevent disruptive tra�ccongestion.

• Be responsible for your organization’s members. If another person from yourorganization is acting improperly, you should try to be a corrective force. Ifthis does not work, alert a HMMT organizer. Penalties may be leveragedagainst an entire organization based on a single person’s actions.

Consequences

The HMMT Tournament Directors reserve the right to disqualify any team at anytime for any reason. In severe situations of misconduct, you may be escorted o� thepremises. Your test results will be nullified, and you will be disqualified. Permanentbans will be imposed upon organizations that severely infringe upon the code ofconduct.

Local Information

Food

• Clover ($)Vegetarian sandwiches andplates.

1326 Mass Ave.Cambridge, MA 02138

• Shake Shack ($)American fast casual food,famous burgers and shakes.

92 Winthrop St.Cambridge, MA 02138

• Felipe’s ($)Made-to-order fresh Mexicanfood with aunthetic recipes.

21 Brattle St.Cambridge, MA 02138

• Chipotle ($)Mexican fast casual food,including custom-made tacosand burritos.

1 Brattle St.Cambridge, MA 02138

• El Jefe’s Taqueria ($)Assembled-to-order Mexicaneats with lots of toppings andsauces.

83 Mt. Auburn St.Cambridge, MA 02138

• Subway ($)Build-your-own sandwicheswith many healthy options.

36 John F. Kennedy St.Cambridge, MA 02138

EmergencyIn the event of fire or life-threatening emergency, call (617) 495-1212, not 911. Thisnumber will reach the Harvard University Police Department; they will respondfaster than 911.

Harassment PolicyHMMT strictly does not tolerate harassment or violence of any form to our competi-tors and sta�, regardless of the identity of the o�ender. If you believe that you oryour competitors have been harassed, please immediately call the Harvard Univer-sity Police Department at (617) 495-1212. If you do not feel comfortable contactingpolice yourself, please visit the help center in Science Center 112 and request helpwith your incident, and we will make every e�ort to help you decide on an appro-priate response and recourse.

SponsorsSpecial thanks to our sponsors:

HMMT O�cers and Sta�

O�cer TeamTournament Directors

• Je�ery Yu• William Ho

Chief Advisors• Evan Chen• William Qian

Education Chair• Agustin Garcia

Finance Directors• Aileen Ma• Christine Cai

Friday Night Events Chair• Andrew Lin

Inventory Czar• Katherine Xiong

Membership Coordinators• Cindy Yang• Junu Lee

Problem Czars (November)• Carl Schildkraut• Krit Boonsiriseth• Milan Haiman

Problem Czars (February)• Andrew Gu• James Lin• Michael Ren

Publicity Chair• Jason Chen

Registration Directors• Jenny Choi• Richard Xu

Schedule Coordinators• Lauren Oh• Richard Allen

Social Chairs• Assel Ismoldayeva• Celine Liang

Software Co-Directors• Dennis Giese• Gabriel Mintzer• Mark Kong• Yannick Yao

Tournament Director Emeritus• Emery Liu• Lillian Zhang

Sta� MembersAssistants to the Tournament Director

• Aaditya Singh• Ted Pyne

Brochure Coordinator• Ivy Mao

Chief Proctors• Andrew Wu• Richard Luo• Carolyn Ge• Haneul Shin• Jason Lee• Tristan Shin

Education Sta�• Melinda Sun• Merrick Cai• Vincent Pan

Finance Associates• Allen Chen• Lucy He• Lucy Jiao

Food Czars• Clara Shi• Hee Jae Hong

Friday Night Assistants• Emma Qin• Vincent Huang

Language/Online Coordinator• Kristy Chang

Membership Deputies• Benji Kan• Matthew Ho

Photographers• Alison Chen• Andy Zhu• Michael Diao

Prize Czar• Alison Fang

Problem Captains• Dain Kim• Daniel Zhu• Joshua Lee• Ryan Kim• Sheldon Tan• Shengtong Zhang• Sujay Kazi• Wanlin Li• Zachary Chroman

Problem Editors• Ahmed Zawad Chowdhury• Alan Peng• Alec Sun• Anders Olsen• Anqi Li• Catherine Wu• Chad Qian• Dylan Liu

Problem Editors (continued)• Freddie Zhao• Guanpeng Xu• Hahn Lheem• Handong Wang• Joseph Heerens• Katherine Cheng• Kevin Ren• Nurullah Giray Kuru• Sean Anderson Ty• Siva Muthupalaniappan• Siwakorn Fuangkawinsombut• Victor Luo• Vincent Bian

Registration Associates• Catherine Mei• Priya Ganesh• Sanjana Ramrajvel

Room Czars• Joshua Pan• Brian Tobin• Austin Li• Allen Wang• Amber Li• Hanna Yang

Software Team• Aaron Sun• Sophia Zhi• Wendy Wu

T-shirt Designer• Raphi Kang

Map also at http://map.harvard.edu

23rd Harvard-MIT Mathematics TournamentNovember 2019