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21st Harvard-MIT Mathematics Tournament

HMMT November Saturday, November 11, 2017 November 11, 2017

Hello HMMT community,

elcome to the 2017 Harvard-MIT November Math Tournament, W our 21st year running! After months of writing problems, organizing registration, and developing software, we are thrilled to see all of you in Cambridge. Whether you’re joining us from Shanghai, Tegucigalpa, or Boston, we hope you’re as excited as we are! We at HMMT seek to share our passion for mathematical problem- solving. While math itself remains a core piece in our mission, we see year after year that people are just are as important. Many of us helping to organize HMMT can trace our friendships to events like these, long after we have graduated. While we won’t speculate why these events engender such lasting bonds, we encourage you to take advantage of this amazing opportunity. Over breakfast, between rounds, and even during our cere- monies, introduce yourself to students from other teams. Say hello to some of our sponsors in the Science Center lobby. And most importantly, thank your coaches and parents. As scholars, you’re here with over 800 of the brightest, most mathematically-gifted students in the world. Who knows what you will find? As usual, all of our problems have been written and compiled by Harvard and MIT undergraduates. No matter how many of them you solve, we hope you find them both challenging and engaging. To the coaches, teachers, and parent chaperones: thank you for your dedication to helping your students compete. We have a talk by Po-Shen Loh that will be specifically aimed at parents, coaches, and proctors, and we hope you find it engaging. There will be a guts round viewing room available as well. To help navigate the campus, we have included a map inside the pro- gram and posted signs to direct you around Harvard. Moreover, all teams’ room assignments can be found inside the program. Should you have any questions, comments, or concerns, please don’t hesitate to ask an HMMT staff member (dressed in pink HMMT shirts) for help. Whether this is your first or fourth HMMT, we hope that you enjoy the tournament we’ve put together. Good luck today, and above all, have fun!

Regards,

Andrew He Ashwin Sah Nathanael Cho Michael Tang Tournament Directors Yannick Yao November Problem Czars Teams and Room Assignments

Building Codes: YC Aud. = Yenching Auditorium, SC = Science Center, Sv = Sever

Nickname Organization Indiv. Team Guts Awards ACM ACM SC 309 SC 309 SC D SC C Stevenson Adlai E. Stevenson SC 310 SC 310 SC E SC C High School AAMC Cards Albany Area Math SC C SC C SC C SC C Circle AHS Albemarle High YC Aud. YC Aud. SC B SC B School Almaty 1 Almaty SC E Sv 214 SC D SC C ASFG american school YC Aud. YC Aud. SC B SC B foundation of guadalajara AMSA1 AMSA Charter SC D SC D SC C SC C School AMSA2 AMSA Charter SC D Sv 112 SC C SC C School Warriors Andover High School SC E SC E SC D SC B Austin Math Austin Math Circle SC B10 SC B10 SC D SC B Beijing BDF Beijing Math Circle YC Aud. Sv 209 SC B SC B Team De Belmont High School SC 309 SC 309 SC D SC C BCA 1 Bergen County SC B SC B SC A SC A Academies BCA 2 Bergen County SC B Sv 103 SC A SC A Academies BCA 3 Bergen County SC B Sv 106 SC A SC A Academies Bucky Buckingham Browne SC C SC C SC B SC B & Dayz Buckingham Browne SC C SC B SC B SC B & Nichols School CATS YC Aud. Sv 210 SC B SC B Seq Chicago Chicagoland SC B Sv 202 SC A SC A Corollaries Comm A Commonwealth SC C SC C SC C SC C School Comm B Commonwealth SC C Sv 111 SC C SC C School CA1 SC D SC D SC B SC B CA2 Concord Academy SC D Sv 202 SC B SC B Building Codes: YC Aud. = Yenching Auditorium, SC = Science Center, Sv = Sever

Nickname Organization Indiv. Team Guts Awards Chen One Concord Carlisle SC D SC D SC C SC C High School Smurfs Concord Carlisle SC D Sv 203 SC C SC C High School CASLV Falcons Coral Academy of SC B10 SC B10 SC E SC B A Science Las Vegas GMinor Deerfield Academy SC D Sv 201 SC B SC B Green Deerfield Academy SC D SC D SC B SC B EB Circle East Brunswick High SC C SC C SC C SC C School EB Square East Brunswick High SC C Sv 109 SC C SC C School Jesters Emma Willard YC Aud. YC Aud. SC B SC B School FMC Fairfax Math Circle SC 222 SC 222 SC E SC C GVMathClub Garnet Valley High SC B SC B SC B SC B School Gauss Math Gauss Academy of SC B Sv 107 SC A SC A Mathematical Education Gauss Tiger Gauss Math Circle SC B Sv 206 SC B SC B MKS High Technology SC A Sv 213 SC D SC B High School ICAE1 ICAE SC C SC C SC B SC B ICAE2 ICAE SC C Sv 110 SC B SC B IM-1 Ideamath SC D SC D SC B SC B IM-2 Ideamath SC D Sv 203 SC B SC B Individual 1 Individuals SC A SC A SC C SC C Individual 2 Individuals SC A SC A SC C SC C Individual 3 Individuals SC A SC A SC C SC C Individual 4 Individuals SC A SC A SC C SC C KentCT YC Aud. Sv 211 SC B SC B Kents YC Aud. Sv 210 SC C SC C KGV King George V SC E Sv 214 SC D SC B School LVMC-A Las Vegas Math SC B Sv 207 SC C SC C Circle LVMC-A2 Las Vegas Math SC B SC B SC C SC C Circle Building Codes: YC Aud. = Yenching Auditorium, SC = Science Center, Sv = Sever

Nickname Organization Indiv. Team Guts Awards LCA1 Lexington Christian SC 221 SC 221 SC E SC C Academy Lex Delta Lexington High SC B Sv 111 SC A SC A School Lex Epsilon Lexington High SC B SC B SC A SC A School Lex Gamma Lexington High SC B Sv 102 SC A SC A School Big L Club Livingston High SC C Sv 110 SC D SC A School Math Team Little L Club Livingston High SC C SC C SC D SC A School Math Team LAMC A Louisiana Math SC C SC C SC A SC A Circle LAMC B Louisiana Math SC C SC B SC A SC A Circle LAMC C Louisiana Math SC C Sv 107 SC A SC A Circle ML Martin Luther High SC B10 SC B10 SC D SC C School Golden Eagles Middlesex County YC Aud. Sv 208 SC C SC C Academy MWM Midwest Math Circle SC E Sv 214 SC D SC C (H)(MMT) SC C Sv 106 SC C SC C H(a)MMT Milton Academy SC C SC C SC C SC C H(y)MMT Milton Academy SC C SC B SC C SC C MBHS 1 Montgomery Blair SC C SC C SC A SC A High School MBHS Two Montgomery Blair SC C SC B SC A SC A High School Moses Brown A SC D Sv 206 SC B SC B Moses Brown B Moses Brown School SC D Sv 207 SC B SC B T1 Mystic Valley SC 222 SC 222 SC E SC B Regional Charter School (MVRCS) NNHS-C Naperville North SC B SC B SC C SC C High School NNHS+C Naperville North SC B Sv 110 SC C SC C High School Algebros Nashoba Regional SC 309 SC 309 SC D SC B High School Building Codes: YC Aud. = Yenching Auditorium, SC = Science Center, Sv = Sever

Nickname Organization Indiv. Team Guts Awards NNHS Tigers Newton North High SC A Sv 213 SC D SC C School Fermat Newton south SC B Sv 208 SC A SC A NHSMT 1 NH State Math SC C SC C SC B SC B Team NHSMT 2 NH State Math SC C SC B SC B SC B Team NHSMT 3 NH State Math SC C Sv 106 SC B SC B Team North Georgia A North Georgia A SC B Sv 209 SC B SC B Jeju North London YC Aud. Sv 208 SC B SC B Collegiate School Jeju NHS A Northview High SC 309 SC 309 SC D SC B School OH Ohio ARML SC B Sv 112 SC B SC B Andover A1 SC B Sv 106 SC C SC C Andover A2 Phillips Academy SC B SC B SC C SC C Andover A3 Phillips Academy SC B Sv 102 SC C SC C PEA IdeaMath Phillips Exeter SC B Sv 103 SC B SC B Ci Academy PEA IdeaMath Phillips Exeter SC B Sv 101 SC B SC B Un Academy PEAK Phillips Exeter SC B Sv 102 SC B SC B Academy PEAR Phillips Exeter SC B SC B SC B SC B Academy PEAT Phillips Exeter SC B Sv 104 SC B SC B Academy PVMT 1 Pleasant Valley High SC C SC C SC D SC A School PRISMS Eyas Princeton SC B Sv 105 SC B SC B International School of Math and Science PRISMS Falcons Princeton SC B Sv 104 SC B SC B International School of Math and Science Princeton Lion Princeton Math YC Aud. YC Aud. SC C SC C Circle Pui Ching PuiChing Middle SC B Sv 103 SC A SC A School Macau Building Codes: YC Aud. = Yenching Auditorium, SC = Science Center, Sv = Sever

Nickname Organization Indiv. Team Guts Awards RVMT - A Raritan Valley Math SC B Sv 109 SC B SC B Team Raccoon A RIA Global SC E Sv 214 SC D SC B Education Ridge Lion Ridge Math Circle YC Aud. YC Aud. SC C SC C RSM RSM SC D Sv 206 SC D SC A SagerHmmt Sager Education YC Aud. Sv 210 SC C SC C CS Shanghai American SC A SC A SC D SC C School SCS Sierra Canyon School YC Aud. Sv 211 SC B SC B Cardinal SC D SC D SC B SC B SKS Black South Kent School SC D Sv 202 SC B SC B SMWL St. Mark’s School SC B Sv 204 SC C SC C SPS-A St. Paul’s School SC E SC E SC D SC C TAS Taipei Math Circle YC Aud. YC Aud. SC B SC B Beijing Star TAL SC A SC A SC B SC B TAMS Texas Academy of SC A Sv 212 SC D SC C Mathematics and Science Brearley The Brearley School SC A Sv 213 SC D SC B Bromfield The Bromfield SC E SC E SC D SC B School RLS The Roxbury Latin SC 309 SC 309 SC D SC B School SBS The Stony Brook SC 310 SC 310 SC E SC B School TGG The SC E Sv 215 SC D SC B WNSX The Williston YC Aud. YC Aud. SC C SC C Northampton School Saltyfish Thinktown SC B10 SC B10 SC E SC B Institution TT TT Math School YC Aud. Sv 210 SC B SC B UTS A University of SC B Sv 203 SC B SC B Toronto Schools UTS B University of SC B SC B SC B SC B Toronto Schools Chipmunks Walton High School SC C SC C SC B SC B Wu-Shu Walton High School SC C SC B SC B SC B Utah Wasatch Academy SC D Sv 204 SC C SC C Building Codes: YC Aud. = Yenching Auditorium, SC = Science Center, Sv = Sever

Nickname Organization Indiv. Team Guts Awards Wasatka Wasatch Academy SC D Sv 205 SC C SC C Rams Watkinson School YC Aud. Sv 207 SC B SC B WV West Virginia SC 222 SC 222 SC E SC B WAMC Westchester Area SC B Sv 205 SC A SC A Math Circle WMA Mars Western Mass SC B Sv 103 SC C SC C ARML WMA Stars Western Mass SC B Sv 201 SC C SC C ARML WMA Suns Western Mass SC B SC B SC C SC C ARML WMA Venus Western Mass SC B Sv 105 SC C SC C ARML Western PA Western YC Aud. YC Aud. SC B SC B Pennsylvania Ghost Math Westford Academy SC A Sv 212 SC B SC B WestyBlack Westminster School SC C SC C SC C SC C WestyGold Westminster School SC C SC B SC C SC C Weston A Weston High School SC 221 SC 221 SC E SC B Wheeler1 SC D SC D SC B SC B Wheeler2 Wheeler School SC D Sv 203 SC B SC B WEHS-A Williamsville East SC E Sv 215 SC D SC B High School (GMP)’ Williamsville North SC E SC E SC D SC B WHS 1 Winchester High SC A Sv 213 SC B SC B School WAnabees YC Aud. YC Aud. SC B SC B Yu’s A Team Yu’s Elite Education SC B Sv 101 SC C SC C Yu’s B Team Yu’s Elite Education SC B SC B SC C SC C Yu’s C Team Yu’s Elite Education SC B Sv 102 SC C SC C Student Schedule

Friday, November 10, 2017 (MIT) Time Event Location 7:00PM – 8:00PM Early Registration Building E51 7:00PM – 7:45PM Pizza and Ice Cream Building E51 7:45PM – 8:00PM Mini-Events Preview Building E51 8:00PM – 9:00PM Mini-Events Building E51

Saturday, November 11, 2017 (Harvard) Time Event Location 8:00AM – 9:00AM Breakfast Science Center Lobby 8:00AM – 9:00AM Registration Science Center 112 9:00AM – 10:15AM Individual Round Varies 10:30AM – 11:45AM Theme Round Varies 12:00PM – 1:00PM Team Round Varies 1:00PM – 2:30PM Lunch On Own* 2:30PM – 4:15PM Guts Round Science Center (Varies) 4:20PM – 4:40PM Integration Bee Finals Science Center B 5:00PM – 5:45PM Awards Ceremony Science Center (Varies)

* Teams that pre-ordered pizzas will be able to pick them up in the Science Center Lobby during lunchtime. Coaches, Proctors, and Parents Schedule

Friday, November 10, 2017 (MIT) Time Event Location 7:00PM – 8:00PM Early Registration Building E51 7:00PM – 7:45PM Pizza and Ice Cream Building E51 7:45PM – 8:00PM Mini-Events Preview Building E51 8:00PM – 9:00PM Mini-Events Building E51

Saturday, November 11, 2017 (Harvard) Time Event Location 8:00AM – 9:00AM Breakfast Science Center Lobby 8:00AM – 9:00AM Registration Science Center 112 9:15AM – 10:30AM Talk by Po-Shen Loh Jefferson 250 11:15AM – 11:45AM Proctor Meeting Jefferson 250 12:00PM – 1:00PM Team Round Proctoring Varies 1:00PM – 2:30PM Lunch On Own* 2:30PM – 4:15PM Guts Round Viewing Jefferson 250 4:20PM – 4:40PM Integration Bee Finals Science Center B 4:45PM – 5:30PM Awards Ceremony Science Center

A Help Center is located in Science Center 104.

* Teams that pre-ordered pizzas will be able to pick them up in the Science Center Lobby during lunchtime. 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. Translators are 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 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 is incorrect, they must go to the designated Appeals Room and submit an appeal in writing. The deadline to do so will be announced on the day of the contest.

Decisions of the coordinators of the Harvard-MIT Math Tournament Novem- ber are final. Local Information

Food

• Clover ($) Vegetarian sandwiches and plates. 1326 Mass Ave. Cambridge, MA 02138

• Shake Shack ($) American fast casual food, famous 92 Winthrop St burgers and shakes. Cambridge, MA 02138

• Felipes ($) Made-to-order fresh Mexican food 21 Brattle St with aunthetic recipes Cambridge, MA 02138

• Chipotle ($) Mexican fast casual food, including 1 Brattle St custom-made tacos and burritos. Cambridge, MA 02138

• El Jefes Taqueria ($) Assembled-to-order Mexican eats 83 Mt Auburn St with lots of toppings and sauces. Cambridge, MA 02138

• Subway ($) Build-your-own sandwiches with 36 John F. Kennedy St many healthy options. Cambridge, MA 02138

Emergency

In the event of fire or life-threatening emergency, call (617) 495-1212, not 911. This number will reach the Harvard Police Department; they will respond faster than 911. Harvard Campus Map (http://map.harvard.edu/) HMMT Officers and Staff

Officer Team

Tournament Directors: Andrew He, Nathanael Cho Problem Czars (November): Ashwin Sah, Michael Tang, Yannick Yao Problem Czars (February): Allen Liu, Henrik Boecken, Kevin Sun Finance Directors: Ellen Li, Emily Cheng Registration Directors: Lillian Zhang, Zachary Diamandis Schedule Coordinators: Cynthia Liu, Grace Whitney Scripts Director: Yang Liu Scripts Team: Alexander Wei, Ashley Kim, Roger Jin Webmaster: Grace Zhang Outreach Chairs: Ahaan Rungta, Brice Huang Social Chair: Celine Liang General Officer: Emery Liu Special Projects Directors: Jakob Weisblat, Michelle Chen Historian: Evan Chen

Staff Members

Assistant Historian: David Liang Brochure Coordinator: Sarah Wu Building Coordinators: Andres Fabrega, Cindy Yang, Francisco Proskauer, Jane Ahn, Luke Melas-Kyriazi, Natalia Hajlasz Driver: William Qian Finance Associates: Anurag Golla, Isabella Kang, Kim Nguyen, Rebecca Shin Food Czars: Ting Li, Xiaolu Guo Guts Round Directors: Daniel Qu, Karen Chen, Sophia Xia Head Graders: Aaditya Singh, Mark Kong, Nelson Niu Head Proctor: Matthew Hambacher Logistics Coordinators: Aditi Chitkara, Sambuddha Chattopadhyay, William Ho Problem Committee: Alec Sun, Amanda Wang, Angela Deng, Caleb He, Calvin Yost-Wolff, Chad Qian, Christopher Shao, Christopher Xu, Cory Hixson, Daniel Chiu, David Xiang, Dhamanpreet Kaur, Dhruv Rohatgi, Fan Zhou, Faraz Mas- roor, Farrell Wu, Gabriel Mintzer, Jason Lu, John Wu, Johnny Gao, Junyao Peng, Kevin Chen, Kevin Yang, Mehtaab Sawhney, Mendel Keller, Michael Ku- ral,Nathan Weckwerth, Nikhil Reddy, Qi Qi, Quilee Simeon, Rachel Zhang, Sanath Devalapurkar, Saranesh Prembabu, Serina Hu, Sujay Kazi, Varkey Alumootil Photographers: Abraham Park, Joyce Tian, Steven Hao, Vivian Huang Prize Czars: Anton Cao, Baula Xu, Byron Xu Sign Czars: Ben Nguyen, Christina Chen, Kenz Kallal Spokespersons: Cathy Wang, Hadrian Merced, Justin Yu, Michael Xie, Paolo Adajar, Rogers Epstein, Thomas Xiong, Vaughan McDonald Sponsors

Special thanks to our sponsors: 21st Harvard-MIT Mathematics Tournament November Tournament 2017