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1011 volume 1_mathcounts_school_handbook_without_solutions 1011 volume 1_mathcounts_school_handbook_without_solutions Document Transcript

  • 2010–2011 School Handbook: Volume I Contains more than 100 creative math problems that meet NCTM standards for grades 6-8. For questions about your local MATHCOUNTS program, please contact your chapter (local) coordinator. Coordinator contact information is available through the Find a Coordinator option of the Competition Program link on www.mathcounts.org. © 2010 MATHCOUNTS Foundation 1420 King Street, Alexandria, VA 22314 703-299-9006 info@mathcounts.org www.mathcounts.orgUnauthorized reproduction of the contents of this publication is a violation of applicable laws. Materials may be duplicated for use by U.S. schools.MATHCOUNTS® and Mathlete® are registered trademarks of the MATHCOUNTS Foundation.
  • AcknowledgmentsThe 2009–2010 MATHCOUNTS Question Writing Committee developed the questions for the2010–2011 MATHCOUNTS School Handbook and competitions: • Chair: Sam Baethge, San Antonio, TX • Tom Butts, University of Texas at Dallas, Richardson, TX • Barbara Currier, Greenhill School, Addison, TX • Joyce Glatzer, West New York Public Schools, West New York, NJ • Tina Pateracki, Bluffton, SC • Tom Price, Norris Middle School, Firth, NE • Tricia Rothenberg, Georgetown, TXNational Judges review competition materials, develop Masters Round questions and serve as arbiters at theNational Competition: • Richard Case, Computer Consultant, Greenwich, CT • Flavia Colonna, George Mason University, Fairfax, VA • Peter Kohn, James Madison University, Harrisonburg, VA • Carter Lyons, James Madison University, Harrisonburg, VA • Monica Neagoy, Mathematics Consultant, Washington, DC • Harold Reiter, University of North Carolina-Charlotte, Charlotte, NC • Dave Sundin (STE 84), Statistics and Logistics Consultant, San Mateo, CANational Reviewers proofread and edit the problems in the MATHCOUNTS School Handbook and/or competitions:William Aldridge, Springfield, VA Chris Jeuell, Kirkland, WASusanna Brosonski, Orlando, FL Doug Keegan (STE 91, NAT 92), Austin, TXDan Cory (NAT 84, 85), Seattle, WA Stanley Levinson, P.E., Lynchburg, VACraig Countryman, San Diego, CA Howard Ludwig, Ocoee, FLRoslyn Denny, Valencia, CA Randy Rogers, West Des Moines, IAEdward Early (STE 92), Austin, TX Frank Salinas, Houston, TXNancy English, Glendale, MO Craig Volden (NAT 84), Columbus, OHBarry Friedman (NAT 86), Westfield, NJ Chaohua Wang, Bloomington, ILDennis Hass, Newport News, VA Becky Weiss, Vienna, VAHelga Huntley (STE 91), Newark, DE Deborah Wells, Rockville, MDAUVSI Foundation Writers/Editors: Wendy Amai, Sandia National Laboratories; Dr. Kenneth Berry, Assistant Directorof the Science and Engineering Education Center, University of Texas at Dallas; Angela Carr, Web Services Manager,AUVSI; Dr. Deborah A. Furey, DARPA Program Manager; Lisa Marron, Sandia National Laboratories and Dr. DaveNovick, Sandia National LaboratoriesTAF Writers/Editors: Charles Cicci, ACAS, MAAA; Kevin Cormier, FCAS, MAAA; Jeremy Fogg; Mubeen Hussain;Pete Rossi, FSA, FCA, CERA, MAAA; Jeremy Scharnick, FCAS, MAAA; John Stokesbury, FSA, FCA, EA, MAAA andPatricia Teufel, FCAS, MAAA.CEA Writers/Editors: Bill Belt; Deepak Joseph; Brian Markwalter, P.E. and Dave WilsonThe Solutions were written by Kent Findell, Diamond Middle School, Lexington, MA.The Problems and Solutions for the MATHCOUNTS School Handbook and competitions are reviewed by Brian Edwards(STE 99, NAT 00), Project Coordinator, MATHCOUNTS Foundation; Mady Bauer, Bethel Park, PA; Jane Lataille, LosAlamos, NM.MathType software for handbook development contributed by Design Science Inc., www.dessci.com, Long Beach, CA. Editor and Contributing Author: Jessica Liebsch, Manager of Education MATHCOUNTS Foundation Content Editor and Contributing Author: Kristen Chandler, Deputy Director & Program Director MATHCOUNTS Foundation Introduction and Program Information: Archana Mehta, Director of Marketing MATHCOUNTS Foundation Executive Director: Louis DiGioia MATHCOUNTS Foundation Honorary Chair: William H. Swanson Chairman and CEO, Raytheon Company
  • TABLE OF CONTENTS Count Me In! Critical 2010–2011 Dates ........................................................................ 4 A contribution to the MATHCOUNTS Foundation Introduction ............................................................................................. 5 will help us continue to make this worthwhile program New This Year.......................................................................................... 6 available to middle school students nationwide. Eligibility Update for Small Schools, Homeschools and Virtual Schools. ............................ 6 The MATHCOUNTS Foundation Updated Website.................. . .................................... 6 will use your contribution for Step-By-Step Solutions for OPLET........................... 7 programwide support to give thousands of students the What About Math?..................................................... 7 opportunity to participate. New Silver Level Status Packet ................................ 7 Referral Program ....................................................... 7 MATHCOUNTS and Social Media. .......................... 7 To become a supporter of MATHCOUNTS, send The MATHCOUNTS OPLET .............................................................. 8 your contribution to: Building a Competition and/or Club Program .................................... 9 MATHCOUNTS Foundation Recruiting Mathletes® ............................................... 9 1420 King Street Maintaining a Strong Program .................................. 9 Alexandria, VA 22314-2794 MATHCOUNTS Competition Program ............................................. 10 Or give online at: Preparation Materials............................................... 10 www.mathcounts.org/donate Coaching Students ................................................... 11 Official Rules and Procedures ................................ 13 Registration......................................................... 13 Other ways to give: • Ask your employer about Eligible Participants............................................ 14 matching gifts. Your donation Levels of Competition ........................................ 15 could double. Competition Components ................................... 16 • Remember MATHCOUNTS Additional Rules ................................................. 17 in your United Way and Scoring ............................................................... 18 Combined Federal Campaign at work. Results Distribution ............................................ 18 • Leave a legacy. Include Forms of Answers ............................................... 19 MATHCOUNTS in your will. Vocabulary and Formulas ................................... 20 For more information regarding MATHCOUNTS Club Program .......................................................... 23 contributions, call the director Club Materials ......................................................... 23 of development at 703-299-9006, Rules, Procedures and Deadlines ............................ 23 ext. 103 or e-mail Getting Started .................................................... 23 info@mathcounts.org. Attaining Silver Level Status .............................. 24 The MATHCOUNTS Foundation is a 501(c)3 Attaining Gold Level Status ............................... 24 organization. Your gift is fully tax deductible. Frequently Asked Questions .................................... 25 Handbook Problems Warm-Ups and Workouts......................................... 27 Stretch ...................................................................... 36 What About Math? ................................................................................. 37 The National Association of Secondary School Introduction ............................................................. 37Principals has placed this program on the NASSP Problems .................................................................. 38Advisory List of National Contests and Activities for 2010–2011. Answers to Handbook and What About Math? Problems ................. 47 Solutions to Handbook Problems ........................................................ 51 Solutions to What About Math? Problems........................................... 572011 MATHCOUNTSNational Competition Sponsor Problem Index ....................................................................................... 61 Registration Form ................................................................................. 63The MATHCOUNTS Foundation makes its products and services available on a nondiscriminatory basis. MATHCOUNTS does not discriminateon the basis of race, religion, color, creed, gender, physical disability or ethnic origin.
  • CrITICAL 2010-2011 DATES Immediately For easy reference, write your local coordinator’s address and phone number here. Contact information for coordinators is available through the Find a Coordinator option of the Competition Program link on www.mathcounts.org or from the national office. September- Send in your school’s Registration Form to receive Volume II of the MATHCOUNTS School Dec. 10 Handbook, the Club in a Box Resource Kit and/or your copy of the 2010-2011 School Competition Kit. Items will ship shortly after receipt of your form, with the mailing of the School Competition Kit following this schedule: Registration Forms postmarked by Oct. 1: Kits mailed in early November. Kits continue mailing every two weeks. Registration Forms postmarked by Dec. 10 deadline: Kits mailed in early January. Mail or fax the MATHCOUNTS Registration Form (with payment if participating in the competition) to: MATHCOUNTS Registration, P.O. Box 441, Annapolis Junction, MD 20701 Fax: 301-206-9789 (Please fax or mail, but do not do both.) Questions? Call 301-498-6141 or confirm your registration via the Registered Schools database and/or 2010-2011 Bronze Level Schools list at www.mathcounts.org in the Registered Schools or Club Program sections, respectively. Oct. 26-28 American Math Challenge, brought to you by MATHCOUNTS and powered by Mathletics® Dec. 10 Competition registration Deadline (postmark) In some circumstances, late registrations might be accepted, at the discretion of MATHCOUNTS and the local coordinator. Register on time to ensure participation by your students. Early January If you have not been contacted with details about your upcoming competition, call your local or state coordinator! If you have not received your School Competition Kit by the end of January, contact MATHCOUNTS at 703-299-9006. Feb. 1-28 Chapter Competitions March 1-27 State Competitions March 4 Deadline for MATHCOUNTS Club Programs to reach Silver Level. Applications for Silver Level Status must be received by MATHCOUNTS as of this date for entry to the Silver drawing. March 25 Deadline for MATHCOUNTS Club Programs to reach Gold Level. Applications for Gold Level Status and Ultimate Math Challenges must be received by MATHCOUNTS as of this date for entry to the Gold drawing. April 15 Applications for Silver and Gold Levels will be accepted through April 15 for plaques and banners. (No entries to drawings will be accepted after March 4 and 25, respectively.) May 5-8 2011 Raytheon MATHCOUNTS National Competition in Washington, D.C. Interested in more coaching materials or MATHCOUNTS items? Purchase items from the MATHCOUNTS store at www.mathcounts.org or contact Sports Awards at 800-621-5803. Selected items also are available at www.artofproblemsolving.com.4 MATHCOUNTS 2010-2011
  • INTrODUCTIONThe mission of MATHCOUNTS is to inspire excellence, confidence and curiosity in U.S. middle schoolstudents through fun and challenging math problems. Currently in its 28th year, MATHCOUNTS meetsits mission by providing two separate but complementary programs for middle school students: theMATHCOUNTS Competition Program and the MATHCOUNTS Club Program. The MATHCOUNTS Competition Program is designed to excite and challenge middle school students. With four levels of competition-school, chapter (local), state and national-the Competition Program provides students with the incentive to prepare throughout the school year to represent their schools at these MATHCOUNTS-hosted* events. MATHCOUNTS provides the preparation materials and the competition materials, and with the leadership of the National Society of Professional Engineers, more than 500 Chapter Competitions, 56 State Competitions and one NationalCompetition are hosted each year. These competitions provide students with the opportunity to gohead-to-head against their peers from other schools, cities and states; to earn great prizes individuallyand as members of their school team; and to progress to the 2011 Raytheon MATHCOUNTS NationalCompetition in Washington, D.C. There is a registration fee for students to participate in the CompetitionProgram, and participation past the School Competition level is limited to the top 10 students per school.A more detailed explanation of the Competition Program follows on pages 10 through 21.The MATHCOUNTS Club Program (MCP) is designed to increase enthusiasm for math by encouraging the formation within schools of math clubs that conduct meetings that are fun and include a variety of math activities. The resources provided through the MCP are also a great supplement for classroom teaching. The activities provided for the MCP foster a positive social atmosphere, with a focus on students working together as a club to earn recognition and rewards in the MATHCOUNTS Club Program. Some rewards are participation based, and others are achievement based, butall rewards require a minimum number of club members (based on school size). Therefore, there is anemphasis on building a strong club and encouraging more than just the top math students within a schoolto join. There is no cost to sign up for the MATHCOUNTS Club Program, but a Registration Form mustbe submitted to receive the free club materials. (A school that registers for the Competition Program isautomatically signed up for the MATHCOUNTS Club Program.) A more detailed explanation of theMATHCOUNTS Club Program follows on pages 23 through 26.*While MATHCOUNTS provides the actual School Competition Booklet with the questions, answers and procedures necessary to run the SchoolCompetition, the administration of the School Competition is up to the MATHCOUNTS coach in the school. The School Competition is notrequired; selection of team and individual competitors for the Chapter Competition is entirely at the discretion of the school coach and need notbe based solely on School Competition scores.MATHCOUNTS 2010–2011 5
  • NEw THIS YEArELIgIBILITY UpDATE FOr SMALL SCHOOLS, HOMESCHOOLS AND vIrTUAL SCHOOLSMATHCOUNTS has made a few updates to the eligibility rules for schools defined as small schools,homeschools and virtual schools.Small Schools: MATHCOUNTS will no longer distinguish between the sizes of schools for CompetitionProgram registration and competition purposes. Every “brick-and-mortar” school will have the sameregistration allowance of up to one team of four students and/or up to six individuals. A school’sparticipants may not combine with any other school’s participants to form a team when registering orcompeting.Homeschools: The definition of homeschool and the requirements for participation (including completedaffidavits) are outlined on page 14, and these have not changed since last year. However, note that allhomeschool students now must register and compete as individuals. Homeschool students may nolonger combine with other homeschool students-even those in the same homeschool or homeschoolgroup/center-to form teams when registering or competing. There is no limit to the number of studentsfrom a homeschool or homeschool group/center who can register as individuals. However, students mustall register as individuals under their own homeschool’s name.Virtual Schools: Any virtual school interested in registering students must contact the MATHCOUNTSnational office at 703-299-9006 before December 10, 2010 for registration details. Any studentregistering as a virtual school student must register as an individual and compete in the MATHCOUNTSChapter Competition assigned according to the student’s home address. Additionally, virtual schoolcoaches must complete an affidavit verifying that the students from the virtual school are in the sixth,seventh or eighth grade and that the virtual school complies with applicable state laws. Completedaffidavits must be submitted to the local coordinator prior to competition. UpDATED wEBSITEOver the past year, MATHCOUNTS collected a significant amount of feedback regarding our website,and this summer quite a few additions were made to make the site more user-friendly and easier tonavigate. A few highlights are listed below.1. New Home Page - Upon visiting www.mathcounts.org, you will find a newly designed home page withnew menus and more targeted links that will make it easier for visitors to find information. The news feedhas been expanded to allow visitors to view more information at one glance. Similar to last year, the newssection will continue to focus on MATHCOUNTS events, MATHCOUNTS press notices and otherinteresting math news.2. Customized Landing Page - If you log in to mathcounts.org, you will find a brand new landing page.MATHCOUNTS has customized your landing page for you. As a coach, upon log-in, you will find a linkto the MCP Members Only page, a link to the coaches’ forum and a news feed with informationspecifically for coaches. You also will find other interesting information such as the Coach of the Weekprofile and grant and scholarship information for teachers.3. New Online Game - Students like to play games, and MATHCOUNTS recently worked withThinkFun® to provide a version of their popular Rush Hour logic game! The Math Arcade now features anew interactive game that can be played within the page. Students will have opportunities to learn criticalmath concepts while having fun and competing against their peers. In addition to Rush Hour, other onlinegames are accessible through a list of links.6 MATHCOUNTS 2010-2011
  • STEp-BY-STEp SOLUTIONS FOr OpLETThe MATHCOUNTS OPLET (Online Problem Library and Extraction Tool) has a new enhancement.MATHCOUNTS is pleased to announce that step-by-step solutions are now available for more than 4200problems in OPLET! Many of our OPLET subscribers requested that we provide step-by-step solutions inOPLET in addition to providing the answers to the problems. We are very excited to be able to offer thisnew feature. If you are not currently a subscriber, see page 8 to learn more about what OPLET can offer. wHAT ABOUT MATH?MATHCOUNTS once again has dedicated a special section of this handbook to What About Math?, aseries of problems designed to help bridge the gap between the knowledge that students gain fromMATHCOUNTS and the important applications of this knowledge in their lives. This year’s What AboutMath? section begins on page 37 and was written for us by the Association for Unmanned VehicleSystems International (AUVSI), The Actuarial Foundation (TAF) and the Consumer ElectronicsAssociation (CEA). The problems were written by professionals in these fields and focus on theapplications of math in these fields’ career paths. NEw SILvEr LEvEL STATUS pACkETIn order to make achieving Silver Level Status a more clear and attainable goal for all schools, we havemade it easier for you to access all the Silver Level Challenges required for your math club. This year inyour Club in a Box Resource Kit, you will find a new Silver Level Status Packet. The Silver Level StatusPacket contains five Silver Level Challenges, the number needed to fulfill the requirement for SilverLevel Status. In addition, a sixth (optional) Silver Level Challenge will be posted on the MCP MembersOnly page for your club. Solutions to Silver Level Challenges will be posted monthly. rEFErrAL prOgrAMOn a recent survey, over 90% of MATHCOUNTS coaches said they would refer MATHCOUNTS toa colleague. Your referral can be rewarded by having a portion of your registration fees reimbursed!MATHCOUNTS is continuing its referral and registration incentives for the Competition Program.Simply refer a middle school that is not currently participating in MATHCOUNTS*. If the schoolyou refer registers a team of four students for the MATHCOUNTS Competition Program this school year,your school and the referred school each will receive a refund of $10. With enough successful referrals,your school could be reimbursed for its entire registration fee! The new Registration Form on page 63 hasa section for you to indicate the school(s) you wish to refer. Incentive for MATHCOUNTS Club Program SchoolsSchools that previously have participated only in the MATHCOUNTS Club Program (and not in theCompetition Program) may also take advantage of the incentive program. If you choose to register a teamof four for the MATHCOUNTS Competition Program this school year, you will receive a refund of $10.Additional details may be found on the Registration Form on page 63. MATHCOUNTS AND SOCIAL MEDIADid you know that MATHCOUNTS is now on Facebook, Twitter and YouTube? Stay updated onimportant MATHCOUNTS news and math happenings through posts and tweets. Watch highlights fromthe National Competition, a promotional video about the Club Program and all the newestMATHCOUNTS Minis online!Experience MATHCOUNTS on social media at the following websites: www.facebook.com/mathcounts www.twitter.com/mathcounts www.youtube.com/mathcountsfdn*To qualify as a referred school, a school may not have participated in the MATHCOUNTS Competition Program for theprevious two school years.MATHCOUNTS 2010–2011 7
  • THE MATHCOUNTS OpLET (Online problem Library and Extraction Tool) . . . a database of thousands of MATHCOUNTS problems AND step-by-step solutions, giving you the ability to generate worksheets, flash cards and Problems of the DayThrough www.mathcounts.org, MATHCOUNTS is offering the MATHCOUNTS OPLET-a database of10,000 problems and over 4,000 step-by-step solutions, with the ability to create personalized worksheets,flash cards and Problems of the Day. After purchasing a 12-month subscription to this online resource, theuser will have access to MATHCOUNTS School Handbook and MATHCOUNTS competition problemsfrom the past 10 years and the ability to extract the problems and solutions in personalized formats. (Eachformat is presented in a pdf file to be printed.) purchase OpLET before October 15, 2010 and SAvE $25 off your subscription fee. If you register a team of 4 and up to six individual students for the MATHCOUNTS Competition Program, you could save a total of $75!* If you are renewing OPLET, MATHCOUNTS will extend the $25 discount to you through this program year. www.mathcounts.org/OpLETOnce the subscription is purchased, the user can access the MATHCOUNTS OPLET each time he or shegoes to www.mathcounts.org/oplet and logs in. Once on the MATHCOUNTS OPLET page, the user cantailor the output to his or her needs by answering a few questions. These are some options that can bepersonalized: ♦ Format of the output: Worksheet, Flash Cards or Problems of the Day ♦ Number of questions to include ♦ Solutions (whether to include or not for selected problems) ♦ Math concept: Arithmetic, Algebra, Geometry, Counting and Probability, Number Theory, Other or a Random Sampling ♦ MATHCOUNTS usage: Problems without calculator usage (Sprint Round/Warm-Up), Problems with calculator usage (Target Round/Workout/Stretch), Team problems with calculator usage (Team Round), Quick problems without calculator usage (Countdown Round) or a Random Sampling ♦ Difficulty level: Easy, Easy/Medium, Medium, Medium/Difficult, Difficult or a Random Sampling ♦ Year range from which problems were originally used in MATHCOUNTS materials: Problems are grouped in five-year blocks in the system.Once these criteria have been selected, the user either (1) can opt to have the computer select theproblems at random from an appropriate pool of problems or (2) can manually select the problems fromthis appropriate pool of problems.How does a person gain access to this incredible resource as soon as possible?A 12-month subscription to the MATHCOUNTS OPLET can be purchased at www.mathcounts.org. Thecost of a subscription is $275; however, schools registering students in the MATHCOUNTS CompetitionProgram will receive a $5 discount per registered student.If you would like to get a sneak peek at this invaluable resource before making your purchase, you cancheck out screen shots of the MATHCOUNTS OPLET at www.mathcounts.org/oplet. You will see theease with which you can create countless materials for your Mathletes, club members and classroomstudents. Imagine the time that can be saved in preparing for club meetings, practice sessions or classroom teaching!*The $75 savings is calculated using the special $25 offer plus an additional $5 discount per student registered for theMATHCOUNTS Competition Program.8 MATHCOUNTS 2010-2011
  • BUILDINg A COMpETITION AND/Or CLUB prOgrAM rECrUITINg MATHLETES®Ideally, the materials in this handbook will be incorporated into the regular classroom curriculum sothat all students learn problem-solving techniques and develop critical thinking skills. When a school’sMATHCOUNTS Competition Program or Club Program is limited to extracurricular sessions, allinterested students should be invited to participate regardless of their academic standing. Because thegreatest benefits of the MATHCOUNTS programs are realized at the school level, the more Mathletesinvolved, the better. Students should view their experience with MATHCOUNTS as fun, as well aschallenging, so let them know from the very first meeting that the goal is to have a good time whilelearning.Here are some suggestions from successful competition and club coaches on how to stimulate interest atthe beginning of the school year:• Build a display case showing MATHCOUNTS shirts and posters. Include trophies and photos from previous years’ club sessions or competitions.• Post intriguing math questions (involving specific school activities and situations) in hallways, the library and the cafeteria. Refer students to the first meeting for answers. Use the MATHCOUNTS poster sent with this handbook!• Make a presentation at the first pep rally or student assembly.• Approach students through other extracurricular clubs (e.g., honor society, science club, chess club).• Inform parents of the benefits of MATHCOUNTS participation via the school newsletter or PTA.• Create a MATHCOUNTS display for Back-to-School Night.• Have former Mathletes speak to students about the rewards of the program.• Incorporate the Problem of the Week from the MATHCOUNTS website (www.mathcounts.org/potw) into the weekly class schedule. MAINTAININg A STrONg prOgrAMKeep the school program strong by soliciting local support and focusing attention on the rewards ofMATHCOUNTS. Publicize success stories. Let the rest of the student body see how much fun Mathleteshave. Remember, the more this year’s students get from the experience, the easier recruiting will be nextyear. Here are some suggestions:• Publicize MATHCOUNTS meetings and events in the school newspaper and local media.• Inform parents of meetings and events through the PTA, open houses and the school newsletter.• Schedule a special pep rally for the Mathletes.• Recognize the achievements of club members at a school awards program.• Have a students-versus-teachers Countdown Round, and invite the student body to watch.• Solicit donations from local businesses to be used as prizes in practice competitions.• Plan retreats or field trips for the Mathletes to area college campuses or hold an annual reunion.• Take photos at club meetings, coaching sessions or competitions and keep a scrapbook.• Distribute MATHCOUNTS shirts to participating students.• Start a MATHCOUNTS summer-school program.• Encourage teachers of lower grades to participate in mathematics enrichment programs.MATHCOUNTS 2010–2011 9
  • MATHCOUNTS COMpETITION prOgrAM . . . A MOrE DETAILED LOOk The MATHCOUNTS Foundation administers its math enrichment, coaching and competition program with a grassroots network of more than 17,000 volunteers who organize MATHCOUNTS competitions nationwide. Each year more than 500 local competitions and 56 “state” competitions are conducted, primarily by chapter and state societies of the National Society of Professional Engineers. All 50 states, the District of Columbia, Puerto Rico, Guam, Virgin Islands, U.S. Department of Defense schools and U.S. State Department schools worldwide participate in MATHCOUNTS. Here’s everything you need to know to get involved. prEpArATION MATErIALSThe annual MATHCOUNTS School Handbook provides the basis for coaches and volunteers to coachstudent Mathletes on problem-solving and mathematical skills. Coaches are encouraged to make maximumuse of MATHCOUNTS materials by incorporating them into their classrooms or by using them withextracurricular math clubs. Coaches also are encouraged to share this material with other teachers at theirschools, as well as with parents.The 2010–2011 MATHCOUNTS School Handbook is in two volumes. Volume I contains over 100math problems, and Volume II contains 200 math problems. As always, these FREE, challenging andcreative problems have been written to meet the National Council of Teachers of Mathematics’ Standardsfor grades 6-8. Volume I is being sent directly to every U.S. school with seventh- and/or eighth-gradestudents and to any other school that registered for the MATHCOUNTS Competition Program last year.This volume also is available to schools with sixth-grade students. Volume II of the handbook will beprovided to schools free of charge, too. However, Volume II will be sent only to those coaches whoregister for the MATHCOUNTS Club Program or the MATHCOUNTS Competition Program.In addition to the great math problems, be sure to take advantage of the following resources that areincluded in the 2010-2011 MATHCOUNTS School Handbook: Vocabulary and Formulas are listed on pages 20-21. A Problem Index is provided on page 61 to assist you in incorporating the MATHCOUNTS School Handbook problems into your curriculum. This index organizes the problems by topic and includes a difficulty rating for each problem. Difficulty Ratings on a scale of 1-7, with 7 being the most difficult, are explained on page 47.A variety of additional information and resources can be found on the MATHCOUNTS website, atwww.mathcounts.org, including problems and answers from the previous year’s Chapter and StateCompetitions, the MATHCOUNTS Coaching Kit, MATHCOUNTS Club Program resources, forums andlinks to state programs. When you sign up for the Club Program or Competition Program (and you havecreated a User Profile on the site), you will receive access to even more free resources that are not visibleor available to the general public. Be sure to create a User Profile as soon as possible, and then visit theCoaches section of the site.The MATHCOUNTS OPLET, which contains MATHCOUNTS School Handbook and competitionproblems from the last 10 years, is a wonderful resource. Once a 12-month subscription is purchased,the user can create customized worksheets, flash cards and Problems of the Day by using this databaseof problems. For more information, see page 8 or go to www.mathcounts.org/oplet and check out somescreen shots of the MATHCOUNTS OPLET. A 12-month subscription can be purchased online. New forthis year are step-by-step solutions to selected problems in OPLET.10 MATHCOUNTS 2010-2011
  • Additional coaching materials and novelty items may be orderedthrough Sports Awards. An order form, with information on thefull range of products, is available in the MATHCOUNTS Storesection at www.mathcounts.org/store or by calling Sports Awardstoll-free at 800-621-5803. A limited selection of MATHCOUNTSmaterials also is available at www.artofproblemsolving.com. COACHINg STUDENTSThe coaching season begins at the start of the school year. The sooner you begin your coaching sessions,the more likely students still will have room in their schedules for your meetings and the more preparationthey can receive before the competitions.Though Volume I has plenty of problems to get your students off to a great start, be sure to request yourcopy of Volume II of the 2010-2011 MATHCOUNTS School Handbook as soon as possible (see page63). Do not hold up the mailing of your Volume II because you are waiting for a purchase order tobe processed or a check to be cut by your school for the registration fee. Fill out your RegistrationForm and send in a photocopy of it without payment. We immediately will mail your Club in a BoxResource Kit (which contains Volume II of the MATHCOUNTS School Handbook) and credit youraccount once your payment is received with the original Registration Form.The original problems found in Volumes I and II of the MATHCOUNTS School Handbook are dividedinto three sections: Warm-Ups, Workouts and Stretches. Each Warm-Up and Workout contains problemsthat generally survey the grades 6-8 mathematics curricula. Workouts assume the use of a calculator;Warm-Ups do not. The Stretches are collections of problems centered around a specific topic.The problems are designed to provide Mathletes with a large variety of challenges and prepare themfor the MATHCOUNTS competitions. (These materials also may be used as the basis for an excitingextracurricular mathematics club or may simply supplement the normal middle school mathematicscurriculum.)Answers to all problems in the handbook include codes indicating level of difficulty. The difficulty ratingsare explained on page 47.wArM-UpS AND wOrkOUTSThe Warm-Ups and Workouts are on pages 27-35 and are designed to increase in difficulty as students gothrough the handbook.For use in the classroom, Warm-Ups and Workouts serve as excellent additional practice for themathematics that students already are learning. In preparation for competition, the Warm-Ups can beused to prepare students for problems they will encounter in the Sprint Round. It is assumed that studentswill not be using calculators for Warm-Up problems. The Workouts can be used to prepare students forthe Target and Team Rounds of competition. It is assumed that students will be using calculators forWorkout problems. Along with discussion and review of the solutions, it is recommended that Mathletesbe provided with opportunities to present solutions to problems as preparation for the Masters Round.All of the problems provide students with practice in a variety of problem-solving situations and may beused to diagnose skill levels, to practice and apply skills or to evaluate growth in skills.STrETCHPage 36 presents the Systems of Equations Stretch. The problems cover a variety of difficulty levels. ThisStretch, and the two included in Volume II, may be incorporated in your students’ practice at any time.ANSwErSAnswers to all problems can be found on pages 47-50.MATHCOUNTS 2010–2011 11
  • SOLUTIONSComplete solutions for the problems start on page 51. These are only possible solutions. You or yourstudents may come up with more elegant solutions.SCHEDULEThe Stretches and What About Math? problems can be used at any time. The following chart is therecommended schedule for using the Warm-Ups and Workouts. (Volumes I and II of the handbook arerequired to complete this schedule.) September Warm-Ups 1-2 Workout 1 October Warm-Ups 3-6 Workouts 2-3 November Warm-Ups 7-10 Workouts 4-5 December Warm-Ups 11-14 Workouts 6-7 January Warm-Ups 15-16 Workout 8 MATHCOUNTS School Competition Warm-Ups 17-18 Workout 9 February Selection of competitors for Chapter Competition MATHCOUNTS Chapter CompetitionTo encourage participation by the greatest number of students, postpone selection of your school’s officialcompetitors until just before the local competition.On average, MATHCOUNTS coaches meet with Mathletes for an hour one or two times a week at thebeginning of the year and with increasing frequency as the competitions approach. Sessions may be heldbefore school, during lunch, after school, on weekends or at other times, coordinating with your school’sschedule and avoiding conflicts with other activities.Here are some suggestions for getting the most out of the Warm-Ups and Workouts at coaching sessions:• Encourage discussion of the problems so that students learn from one another.• Encourage a variety of methods for solving problems.• Have students write problems for each other.• Use the MATHCOUNTS Problem of the Week. Currently, this set of problems is posted every Monday on the MATHCOUNTS website at www.mathcounts.org/potw.• Practice working in groups to develop teamwork (and to prepare for the Team Round).• Practice oral presentations to reinforce understanding (and to prepare for the Masters Round).• Take advantage of additional MATHCOUNTS coaching materials, such as previous years’ competitions, to provide an extra challenge or to prepare for competition.• Provide refreshments and vary the location of your meetings to create a relaxing, enjoyable atmosphere.• Invite the school principal to a session to offer words of support.• Recruit volunteers. Volunteer assistance can be used to enrich the program and expand it to more students. Fellow teachers can serve as assistant coaches. Individuals such as MATHCOUNTS alumni and high school students, parents, community professionals and retirees also can help.12 MATHCOUNTS 2010-2011
  • OFFICIAL rULES AND prOCEDUrESThe following rules and procedures govern all MATHCOUNTS competitions. The MATHCOUNTSFoundation reserves the right to alter these rules and procedures at any time. Coaches are responsiblefor being familiar with the rules and procedures outlined in this handbook. Coaches should bring anydifficulty in procedures or in student conduct to the immediate attention of the appropriate chapter, stateor national official. Students violating any rules may be subject to immediate disqualification. rEgISTrATIONTo participate in the MATHCOUNTS Competition Program, a school representative is required tocomplete and return the Registration Form (available at the back of this handbook and on our website,at www.mathcounts.org) along with a check, money order, purchase order or credit card authorization.Your registration must be postmarked no later than December 10, 2010, and mailed to MATHCOUNTSRegistration, P.O. Box 441, Annapolis Junction, MD 20701. The team registration fee is $90. Theindividual registration fee is $25 per student. Reduced fees of $40 per team and $10 per individualare available to schools entitled to receive Title I funds. Registration fees are nonrefundable. SeeRegistration Form on page 63 for additional details. Do not hold up the mailing of your Volume II because you are waiting for a purchase order to be processed or a check to be cut by your school for the registration fee. Fill out your Registration Form and send in a photocopy of it without payment. We immediately will mail your Club in a Box Resource Kit (which contains Volume II of the MATHCOUNTS School Handbook) and credit your account once your payment is received with the original Registration Form.By completing the Registration Form, the coach attests to the school administration’s permission toregister students for MATHCOUNTS.Academic centers or enrichment programs that do not function as students’ official school of record arenot eligible to register.Registration in the Competition Program entitles a school to send students to the local competition andearns the school Bronze Level Status in the MATHCOUNTS Club Program. Additionally, registeredschools will receive two mailings: ♦ The first, immediate mailing will be the Club in a Box Resource Kit, which contains Volume II of the 2010–2011 MATHCOUNTS School Handbook, the Club Resource Guide with 10 club meeting ideas and other materials for the MATHCOUNTS Club Program. ♦ The second mailing will include the School Competition Kit (with instructions, School Competition & Answer Key, recognition ribbons and student participation certificates) and a catalog of additional coaching materials. The first batch of School Competition Kits will be mailed in early November, and additional mailings will occur on a rolling basis to schools sending in the Registration Forms later in the fall.Your Registration Form must be postmarked by December 10, 2010. In some circumstances, lateregistrations might be accepted, at the discretion of MATHCOUNTS and the local coordinator. Thesooner you register, the sooner you will receive your School Competition Kit and can start preparing yourteam. Once processed, confirmation of your registration will be available through the registration databasein the Registered Schools section of the MATHCOUNTS website (www.mathcounts.org). Other questionsabout the status of your registration should be directed to MATHCOUNTS Registration, P.O. Box 441,Annapolis Junction, MD 20701. Telephone: 301-498-6141. Your state or local coordinator will be notifiedof your registration, and you then will be informed of the date and location of your local competition.If you have not been contacted by mid-January with competition details, it is your responsibility tocontact your local coordinator to confirm that your registration has been properly routed and that yourschool’s participation is expected. Coordinator contact information is available in the Find a Coordinatorsection of www.mathcounts.org. Questions about a local or state program should be addressed to yourcoordinator.MATHCOUNTS 2010–2011 13
  • ELIgIBLE pArTICIpANTSStudents enrolled in the sixth, seventh or eighth grade are eligible to participate in MATHCOUNTScompetitions. Students taking middle school mathematics classes who are not full-time sixth-, seventh- oreighth-graders are not eligible. Participation in MATHCOUNTS competitions is limited to three years foreach student, though there is no limit to the number of years a student may participate in the school-basedcoaching phase.School Registration: A school may register one team of four and up to six individuals for a total of10 participants. You must designate team members versus individuals prior to the start of the chapter(local) competition (i.e., a student registered as an “individual” may not help his or her school teamadvance to the next level of competition).Team Registration: Only one team (of up to four students) per school is eligible to compete.Members of a school team will participate in the Sprint, Target and Team Rounds. Members of a schoolteam also will be eligible to qualify for the Countdown Round (where conducted). Team members willbe eligible for team awards, individual awards and progression to the state and national levels based ontheir individual and/or team performance. It is recommended that your strongest four Mathletes formyour school team. Teams of fewer than four will be allowed to compete; however, the team score will becomputed by dividing the sum of the team members’ scores by 4 (see “Scoring” on page 18 for details).Consequently, teams of fewer than four students will be at a disadvantage.Individual Registration: Up to six students may be registered in addition to or in lieu of a schoolteam. Students registered as individuals will participate in the Sprint and Target Rounds but not the TeamRound. Individuals will be eligible to qualify for the Countdown Round (where conducted). Individualsalso will be eligible for individual awards and progression to the state and national levels.School Definitions: Academic centers or enrichment programs that do not function as students’ officialschool of record are not eligible to register. If it is unclear whether an educational institution is considereda school, please contact your local Department of Education for specific criteria governing your state.School Enrollment Status: A student may compete only for his or her official school of record. Astudent’s school of record is the student’s base or main school. A student taking limited course workat a second school or educational center may not register or compete for that second school or center,even if the student is not competing for his or her school of record. MATHCOUNTS registration is notdetermined by where a student takes his or her math course. If there is any doubt about a student’s schoolof record, the local or state coordinator must be contacted for a decision before registering.Small Schools: MATHCOUNTS will no longer distinguish between the sizes of schools for CompetitionProgram registration and competition purposes. Every “brick-and-mortar” school will have the sameregistration allowance of up to one team of four students and/or up to six individuals. A school’sparticipants may not combine with any other school’s participants to form a team when registering orcompeting.Homeschools: Homeschools in compliance with the homeschool laws of the state in which they arelocated are eligible to participate in MATHCOUNTS competitions as individuals only and in accordancewith all other rules. Homeschool coaches must complete an affidavit verifying that students from thehomeschool are in the sixth, seventh or eighth grade and that the homeschool complies with applicablestate laws. Completed affidavits must be submitted to the local coordinator prior to competition.Virtual Schools: Any virtual school interested in registering students must contact the MATHCOUNTSnational office at 703-299-9006 before December 10, 2010 for registration details. Any student registeringas a virtual school student must register as an individual and compete in the MATHCOUNTS ChapterCompetition assigned according to the student’s home address. Additionally, virtual school coachesmust complete an affidavit verifying that the students from the virtual school are in the sixth, seventh oreighth grade and that the virtual school complies with applicable state laws. Completed affidavits must besubmitted to the local coordinator prior to competition.14 MATHCOUNTS 2010-2011
  • Substitutions by Coaches: Coaches may not substitute team members for the State Competitionunless a student voluntarily releases his or her position on the school team. Additional requirements/documentation on substitutions (such as requiring parental release or requiring the substitution request tobe submitted in writing) are at the discretion of the state coordinator. Coaches may not make substitutionsfor students progressing to the State Competition as individuals. At all levels of competition, studentsubstitutions are not permitted after on-site competition registration has been completed. A student beingadded to a team need not be a student who was registered for the Chapter Competition as an individual.Religious Observances: A student who is unable to attend a competition due to religious observancesmay take the written portion of the competition up to one week in advance of the scheduled competition.In addition, all competitors from that student’s school must take the exam at the same time. Advancetesting will be done at the discretion of the local and state coordinators and under proctored conditions.If the student who is unable to attend the competition due to a religious observance is not part of theschool team, then the team has the option of taking the Team Round during this advance testing or onthe regularly scheduled day of the competition with the other teams. The coordinator must be madeaware of the team’s decision before the advance testing takes place. Students who qualify for an officialCountdown Round but are unable to attend will automatically forfeit one place standing.Special Needs: Reasonable accommodations may be made to allow students with special needs toparticipate. A request for accommodation of special needs must be directed to local or state coordinatorsin writing at least three weeks in advance of the local or state competition. This written request shouldthoroughly explain a student’s special need as well as what the desired accommodation would entail. Manyaccommodations that are employed in a classroom or teaching environment cannot be implemented inthe competition setting. Accommodations that are not permissible include, but are not limited to, grantinga student extra time during any of the competition rounds or allowing a student to use a calculator forthe Sprint or Countdown Rounds. In conjunction with the MATHCOUNTS Foundation, coordinatorswill review the needs of the student and determine if any accommodations will be made. In making finaldeterminations, the feasibility of accommodating these needs at the National Competition will be taken intoconsideration. LEvELS OF COMpETITIONMATHCOUNTS competitions are organized at four levels: school, chapter (local), state and national.Competitions are written for the sixth- through eighth-grade audience. The competitions can be quitechallenging, particularly for students who have not been coached using MATHCOUNTS materials. Allcompetition materials are prepared by the national office.The real success of MATHCOUNTS is influenced by the coaching sessions at the school level. Thiscomponent of the program involves the most students (more than 500,000 annually), comprises thelongest period of time and demands the greatest involvement.SCHOOL COMPETITION: In January, after several months of coaching, schools registered for theCompetition Program should administer the School Competition to all interested students. The SchoolCompetition is intended to be an aid to the coach in determining competitors for the Chapter (local)Competition. Selection of team and individual competitors is entirely at the discretion of coaches andneed not be based solely on School Competition scores. The School Competition is sent to the coachof a school, and it may be used by the teachers and students only in association with that school’sprograms and activities. The current year’s School Competition questions must remain confidentialand may not be used in outside activities, such as tutoring sessions or enrichment programs withstudents from other schools. For additional announcements or edits, please check the Coaches’ Forumin the Coaches’ section on the MATHCOUNTS website before administering the School Competition.It is important that the coach look upon coaching sessions during the academic year as opportunities todevelop better math skills in all students, not just in those students who will be competing. Therefore, it issuggested that the coach postpone selection of competitors until just prior to the local competitions.MATHCOUNTS 2010–2011 15
  • CHAPTER COMPETITIONS: Held from February 1 through February 28, 2011, the ChapterCompetition consists of the Sprint, Target and Team Rounds. The Countdown Round (official or just forfun) may or may not be included. The chapter and state coordinators determine the date and administrationof the local competition in accordance with established national procedures and rules. Winning teams andstudents will receive recognition. The winning team will advance to the State Competition. Additionally,the two highest-ranking competitors not on the winning team (who may be registered as individuals or asmembers of a team) will advance to the State Competition. This is a minimum of six advancing Mathletes(assuming the winning team has four members). Additional teams and/or Mathletes also may progress atthe discretion of the state coordinator. The policy for progression must be consistent for all chapters withina state.STATE COMPETITIONS: Held from March 1 through March 27, 2011, the State Competitionconsists of the Sprint, Target and Team Rounds. The Countdown Round (official or just for fun) andthe optional Masters Round may or may not be included. The state coordinator determines the date andadministration of the State Competition in accordance with established national procedures and rules.Winning teams and students will receive recognition. The four highest-ranked Mathletes and the coach ofthe winning team from each State Competition will receive an all-expense-paid trip to the National Competition.RAYTHEON MATHCOUNTS NATIONAL COMPETITION: Held Friday, May 6, 2011, inWashington, D.C., the National Competition consists of the Sprint, Target, Team, Countdown andMasters Rounds. Expenses of the state team and coach to travel to the National Competition will be paidby MATHCOUNTS. The national program does not make provisions for the attendance of additionalstudents or coaches. All national competitors will receive a plaque and other items in recognition of theirachievements. Winning teams and individuals also will receive medals, trophies and college scholarships. COMpETITION COMpONENTSMATHCOUNTS competitions are designed to be completed in approximately three hours:The SPRINT ROUND (40 minutes) consists of 30 problems. This round tests accuracy, with time periodallowing only the most capable students to complete all of the problems. Calculators are not permitted.The TARGET ROUND (approximately 30 minutes) consists of 8 problems presented to competitorsin four pairs (6 minutes per pair). This round features multistep problems that engage Mathletes inmathematical reasoning and problem-solving processes. Problems assume the use of calculators.The TEAM ROUND (20 minutes) consists of 10 problems that team members work together to solve.Team member interaction is permitted and encouraged. Problems assume the use of calculators.Note: Coordinators may opt to allow those competing as individuals to create a “squad” to take the TeamRound for the experience, but the round should not be scored and is not considered official.The COUNTDOWN ROUND is a fast-paced oral competition for top-scoring individuals (based onscores in the Sprint and Target Rounds). In this round, pairs of Mathletes compete against each other andthe clock to solve problems. Calculators are not permitted.At Chapter and State Competitions, a Countdown Round may be conducted officially, unofficially (forfun) or omitted. However, the use of an official Countdown Round will be consistent for all chapterswithin a state. In other words, all chapters within a state must use the round officially in order for anychapter within a state to use it officially. All students, whether registered as part of a school team or as anindividual competitor, are eligible to qualify for the Countdown Round.An official Countdown Round is defined as one that determines an individual’s final overall rank inthe competition. If the Countdown Round is used officially, the official procedures as established by theMATHCOUNTS Foundation must be followed.If a Countdown Round is conducted unofficially, the official procedures do not have to be followed.Chapters and states choosing not to conduct the round officially must determine individual winners on thesole basis of students’ scores in the Sprint and Target Rounds of the competition.16 MATHCOUNTS 2010-2011
  • In an official Countdown Round, the top 25% of students, up to a maximum of 10, are selected tocompete. These students are chosen based on their individual scores. The two lowest-ranked studentsare paired, a question is projected and students are given 45 seconds to solve the problem. A studentmay buzz in at any time, and if he or she answers correctly, a point is scored; if a student answersincorrectly, the other student has the remainder of the 45 seconds to answer. Three questions are readto the pair of students, one question at a time, and the student who scores the higher number of points(not necessarily 2 out of 3) captures the place, progresses to the next round and challenges the next-higher-ranked student. (If students are tied after three questions (at 1-1 or 0-0), questions continue tobe read until one is successfully answered.) This procedure continues until the fourth-ranked Mathleteand his or her opponent compete. For the final four rounds, the first student to correctly answerthree questions advances. The Countdown Round proceeds until a first-place individual is identified.(More detailed rules regarding the Countdown Round procedure are identified in the Instructions sectionof the School Competition Booklet.) Note: Rules for the Countdown Round change for the NationalCompetition.The Masters Round is a special round for top individual scorers at the state and national levels. In thisround, top individual scorers prepare an oral presentation on a specific topic to be presented to a panelof judges. The Masters Round is optional at the state level; if this round is held, the state coordinatordetermines the number of Mathletes that participate. At the national level, four Mathletes participate.(Participation in the Masters Round is optional. A student declining to compete will not be penalized.)Each student is given 30 minutes to prepare his or her presentation. Calculators may be used. Thepresentation will be 15 minutes—up to 11 minutes may be used for the student’s oral response to theproblem, and the remaining time may be used for questions by the judges. This competition valuescreativity and oral expression as well as mathematical accuracy. Judging of presentations is based onknowledge, presentation and the responses to judges’ questions. ADDITIONAL rULESAll answers must be legible.Pencils and paper will be provided for Mathletes by competition organizers. However, students maybring their own pencils, pens and erasers if they wish. They may not use their own scratch paper.Use of notes or other reference materials (including dictionaries and translation dictionaries) is notpermitted.Specific instructions stated in a given problem take precedence over any general rule or procedure.Communication with coaches is prohibited during rounds but is permitted during breaks. Allcommunication between guests and Mathletes is prohibited during competition rounds. Communicationbetween teammates is permitted only during the Team Round.Calculators are not permitted in the Sprint and Countdown Rounds, but they are permitted in theTarget, Team and Masters Rounds. When calculators are permitted, students may use any calculator(including programmable and graphing calculators) that does not contain a QWERTY (typewriter-like)keypad. Calculators that have the ability to enter letters of the alphabet but do not have a keypad in astandard typewriter arrangement are acceptable. Personal digital assistants (PDAs) are not consideredto be calculators and may not be used during competitions. Students may not use calculators toexchange information with another person or device during the competition.Coaches are responsible for ensuring that their students use acceptable calculators, and studentsare responsible for providing their own calculators. Coordinators are not responsible for providingMathletes with calculators or batteries before or during MATHCOUNTS competitions. Coaches arestrongly advised to bring backup calculators and spare batteries to the competition for their teammembers in case of a malfunctioning calculator or weak or dead batteries. Neither the MATHCOUNTSFoundation nor coordinators shall be responsible for the consequences of a calculator’s malfunctioning.MATHCOUNTS 2010–2011 17
  • Pagers, cell phones, iPods® and other MP3 players should not be brought into the competitionroom. Failure to comply could result in dismissal from the competition.Should there be a rule violation or suspicion of irregularities, the MATHCOUNTS coordinator orcompetition official has the obligation and authority to exercise his or her judgment regarding thesituation and take appropriate action, which might include disqualification of the suspected student(s)from the competition. SCOrINgCompetition scores do not conform to traditional grading scales. Coaches and students should view anindividual written competition score of 23 (out of a possible 46) as highly commendable.The individual score is the sum of the number of Sprint Round questions answered correctly and twicethe number of Target Round questions answered correctly. There are 30 questions in the Sprint Round and8 questions in the Target Round, so the maximum possible individual score is 30 + 2(8) = 46.The team score is calculated by dividing the sum of the team members’ individual scores by 4 (even ifthe team has fewer than four members) and adding twice the number of Team Round questions answeredcorrectly. The highest possible individual score is 46. Four students may compete on a team, and thereare 10 questions in the Team Round. Therefore, the maximum possible team score is((46 + 46 + 46 + 46) ÷ 4) + 2(10) = 66.If used officially, the Countdown Round yields final individual standings. The Masters Round is acompetition for the top-scoring individuals that yields a separate winner and has no impact on progressionto the National Competition.Ties will be broken as necessary to determine team and individual prizes and to determine whichindividuals qualify for the Countdown Round. For ties among individuals, the student with the higherSprint Round score will receive the higher rank. If a tie remains after this comparison, specific groupsof questions from the Sprint and Target Rounds are compared. For ties among teams, the team with thehigher Team Round score, and then the higher sum of the team members’ Sprint Round scores, receivesthe higher rank. If a tie remains after these comparisons, specific questions from the Team Round willbe compared. Note: These are very general guidelines. Competition officials receive more detailedprocedures.In general, questions in the Sprint, Target and Team Rounds increase in difficulty so that the most difficultquestions occur near the end of each round. In a comparison of questions to break ties, generally thosewho correctly answer the more difficult questions receive the higher rank.Protests concerning the correctness of an answer on the written portion of the competition mustbe registered with the room supervisor in writing by a coach within 30 minutes of the end of eachround. Rulings on protests are final and may not be appealed. Protests will not be accepted during theCountdown or Masters Round. rESULTS DISTrIBUTIONCoaches should expect to receive the scores of their students and a list of the top 25% of students andtop 40% of teams from their coordinators. In addition, single copies of the blank competition materialsand answer keys may be distributed to coaches after all competitions at that level nationwide have beencompleted. Coordinators must wait for verification from the national office that all such competitionshave been completed, before distributing blank competition materials and answer keys. Both theproblems and answers from Chapter and State Competitions will be posted on the MATHCOUNTSwebsite following the completion of all competitions at that level nationwide (Chapter-early March;State-early April). The previous year’s problems and answers will be taken off the website at that time.Student competition papers and answers will not be viewed by nor distributed to coaches, parents,students or other individuals. Students’ competition papers become the confidential property of theMATHCOUNTS Foundation.18 MATHCOUNTS 2010-2011
  • FORMS OF ANSWERSThe following list explains acceptable forms for answers. Coaches should ensure that Mathletes arefamiliar with these rules prior to participating at any level of competition. Judges will score competitionanswers in compliance with these rules for forms of answers.All answers must be expressed in simplest form. A “common fraction” is to be considered a fraction ain the form ± b , where a and b are natural numbers and GCF(a, b) = 1. In some cases the term “common Afraction” is to be considered a fraction in the form B , where A and B are algebraic expressions and A andB do not share a common factor. A simplified “mixed number” (“mixed numeral,” “mixed fraction”) is to abe considered a fraction in the form ± N b , where N, a and b are natural numbers, a < b and GCF(a, b) = 1.Examples: 2 4Problem: Express 8 divided by 12 as a common fraction. Answer: 3 Unacceptable: 6Problem: Express 12 divided by 8 as a common fraction. Answer: 3 2 Unacceptable: 12 8 , 11 2Problem: Express the sum of the lengths of the radius and the circumference of a circle with a diameter 1 1+ 2 π of 4 as a common fraction in terms of π. Answer: 8Problem: Express 20 divided by 12 as a mixed number. Answer: 1 2 3 Unacceptable: 1 12 , 8 5 3Ratios should be expressed as simplified common fractions unless otherwise specified. Examples:Simplified, Acceptable Forms: 2 , π , 6 1 7 3 4 −π Unacceptable: 3 1 , 3 , 3.5, 2:1 2 4Radicals must be simplified. A simplified radical must satisfy: 1) no radicands have a factor whichpossesses the root indicated by the index; 2) no radicands contain fractions; and 3) no radicals appear inthe denominator of a fraction. Numbers with fractional exponents are not in radical form. Examples:Problem: Evaluate 15 × 5 . Answer: 5 3 Unacceptable: 75Answers to problems asking for a response in the form of a dollar amount or an unspecifiedmonetary unit (e.g., “How many dollars...,” “How much will it cost...,” “What is the amount ofinterest...”) should be expressed in the form ($) a.bc, where a is an integer and b and c are digits.The only exceptions to this rule are when a is zero, in which case it may be omitted, or when b and c areboth zero, in which case they may both be omitted. Examples:Acceptable: 2.35, 0.38, .38, 5.00, 5 Unacceptable: 4.9, 8.0Units of measurement are not required in answers, but they must be correct if given. When aproblem asks for an answer expressed in a specific unit of measure or when a unit of measure is providedin the answer blank, equivalent answers expressed in other units are not acceptable. For example, if aproblem asks for the number of ounces and 36 oz is the correct answer, 2 lbs 4 oz will not be accepted. Ifa problem asks for the number of cents and 25 cents is the correct answer, $0.25 will not be accepted. 2Do not make approximations for numbers (e.g., π, 3 , 5 3 ) in the data given or in solutions unlessthe problem says to do so.Do not do any intermediate rounding (other than the “rounding” a calculator performs) whencalculating solutions. All rounding should be done at the end of the calculation process.Scientific notation should be expressed in the form a × 10n where a is a decimal, 1 < |a| < 10, and n is aninteger. Examples:Problem: Write 6895 in scientific notation. Answer: 6.895 × 103Problem: Write 40,000 in scientific notation. Answer: 4 × 104 or 4.0 × 104An answer expressed to a greater or lesser degree of accuracy than called for in the problem will notbe accepted. Whole number answers should be expressed in their whole number form.Thus, 25.0 will not be accepted for 25 nor vice versa.The plural form of the units will always be provided in the answer blank, even if the answerappears to require the singular form of the units.MATHCOUNTS 2010–2011 19
  • VOCABULARY AND FORMULASThe following list is representative of terminology used in the problems but should not be viewed asall‑inclusive. It is recommended that coaches review this list with their Mathletes.abscissa degree measure interior angle of a polygonabsolute value denominator intersectionacute angle diagonal of a polygon inverse variationadditive inverse (opposite) diagonal of a polyhedron irrational numberadjacent angles diameter isoscelesalgorithm difference lateral surface areaalternate interior angles digit lateral edgealternate exterior angles digit‑sum lattice point(s)altitude (height) direct variation LCMarea dividend linear equationarithmetic mean divisible meanarithmetic sequence divisor median of a set of database 10 edge median of a trianglebinary endpoint midpointbisect equation mixed numberbox‑and‑whisker plot equiangular mode(s) of a set of datacenter equidistant multiplechord equilateral multiplicative inversecircle evaluate (reciprocal)circumference expected value natural numbercircumscribe exponent numeratorcoefficient expression obtuse anglecollinear exterior angle of a polygon octagoncombination factor octahedroncommon denominator factorial odds (probability)common divisor Fibonacci sequence opposite of a number (additivecommon factor finite inverse)common fraction formula ordered paircommon multiple frequency distribution ordinatecomplementary angles frustum origincomposite number function palindromecompound interest GCF parallelconcentric geometric mean parallelogramcone geometric sequence Pascal’s trianglecongruent height (altitude) pentagonconvex hemisphere percent increase/decreasecoordinate plane/system hexagon perimetercoordinates of a point hypotenuse permutationcorresponding angles image of a point (under a perpendicularcounting numbers transformation) planarcounting principle improper fraction polygoncube inequality polyhedroncylinder infinite series prime factorizationdata inscribe prime numberdecimal integer principal square root20 MATHCOUNTS 2010-2011
  • prism remainder system of equations/probability repeating decimal inequalitiesproduct revolution tangent figuresproper divisor rhombus tangent lineproper factor right angle termproper fraction right circular cone terminating decimalproportion right circular cylinder tetrahedronpyramid right polyhedron total surface areaPythagorean Triple right triangle transformationquadrant rotation translationquadrilateral scalene triangle trapezoidquotient scientific notation triangleradius segment of a line triangular numbersrandom semicircle trisectrange of a data set sequence unionrate set unit fractionratio similar figures variablerational number simple interest vertexray slope vertical anglesreal number slope‑intercept form volumereciprocal (multiplicative solution set whole number inverse) sphere x‑axisrectangle square x‑coordinatereflection square root x‑interceptregular polygon stem‑and‑leaf plot y‑axisrelatively prime sum y‑coordinate supplementary angles y‑interceptThe list of formulas below is representative of those needed to solve MATHCOUNTS problems butshould not be viewed as the only formulas that may be used. Many other formulas that are useful inproblem solving should be discovered and derived by Mathletes. CIRCUMFERENCE SURFACE AREA & VOLUME Circle C = 2 × � × r = � × d Sphere SA = 4 × π × r 2 AREA Sphere V = 4 × π × r 3 3 Square A = s 2 Rectangular prism V = l × w × h Rectangle A = l × w = b × h Circular cylinder V = π × r 2 × h Parallelogram A = b × h Circular cone V = 1 × π × r 2 × h 3 Trapezoid A = 1 (b1 + b2) × h 2 Pyramid V= 1×B×h 3 Circle A = π × r 2 1 Triangle A= 2×b×h Pythagorean Theorem c2 = a 2 + b2 Triangle A = s ( s − a )( s − b)( s − c) Counting/ n! n Cr = Combinations (r !)((n − r )!) s2 3 Equilateral triangle A= 4 1 Rhombus A= 2 × d1 × d2MATHCOUNTS 2010–2011 21
  • 22 MATHCOUNTS 2010-2011
  • MATHCOUNTS CLUB prOgrAM . . . A MOrE DETAILED LOOk MATHCOUNTS recognizes that math clubs can play an important role in shaping students’ attitudes and abilities. In an effort to support existing math clubs and their coaches, as well as encourage the formation of new math clubs, MATHCOUNTS offers the MATHCOUNTS Club Program (MCP). Whether you are starting a new program or continuing a tradition of strong math clubs in your school, MATHCOUNTS understands thechallenges involved in such a commitment. MATHCOUNTS also understands the meaningful rewards ofcoaching a math club and the strong impact you can have on students through organized math activities.The MATHCOUNTS Club Program is designed to provide schools with the structure and activities tohold regular meetings of a math club. Depending on the level of student and teacher involvement, aschool may receive a recognition plaque or banner and be entered in a drawing for prizes.The Club Program may be used by schools as a stand-alone program, a curriculum supplement forclassroom work or part of the student preparation for the MATHCOUNTS Competition Program. CLUB MATErIALSWhen a school registers a club in the MATHCOUNTS Club Program, the school will be sent a Club ina Box Resource Kit, containing (1) further details on the Club Program, (2) the Club Resource Guide,which outlines structured club activities, (3) the Silver Level Status Packet, complete with five SilverLevel Challenges for your students, (4) game board to supplement an activity in the Club ResourceGuide, (5) 12 MATHCOUNTS pencils and (6) a MATHCOUNTS tote bag for the coach. Additionally,solutions to all Silver Level Challenges, a sixth (optional) Silver Level Challenge and an Ultimate MathChallenge will be made available online for use by math club students.The Club Resource Guide and the Silver Level Challenges are the backbone of the MCP. The ClubResource Guide contains 10 meeting activity ideas you can use as a basis for planning enjoyable,instructive get-togethers for your math club throughout the year. There is no particular order in whichthe meeting ideas must be used, and all activities are optional. In addition to the meeting ideas, six SilverLevel Challenges are provided. Club sponsors should work these Challenges into their meeting plans, ascompletion of at least five of these Challenges is required for Silver Level Status.In addition to the materials sent in the Club in a Box Resource Kit, a special MCP Members Only page(www.mathcounts.org/club) will become visible and available to any coach signing up a club in theMCP or registering students in the Competition Program. (Coaches also must create a User Profile onmathcounts.org to gain access to this members-only page.) This web page provides solutions to all SilverLevel Challenges (released one per month), one additional (optional) Silver Level Challenge, over 30meeting plans or ideas from previous years and all of the resources necessary to conduct any of thesuggested meeting ideas. rULES, prOCEDUrES AND DEADLINESgETTINg STArTED 1. The MATHCOUNTS Club Program is open only to sixth-, seventh- and eighth-grade students in U.S.-based schools. 2. The club coach must complete and submit the MATHCOUNTS Registration Form to register the school’s math club. By selecting Option 1 or Option 2 on this form, your school has reached Bronze Level Status and will be recognized on the MATHCOUNTS website. (The MATHCOUNTS Registration Form is available at the back of this book and online at www.mathcounts.org/club.)MATHCOUNTS 2010–2011 23
  • 3. Shortly after MATHCOUNTS receives your Registration Form, Volume II of the 2010–2011 MATHCOUNTS School Handbook and the Club in a Box Resource Kit will be sent to you. 4. Begin recruiting club members and spreading the word about your first club meeting. The Club Resource Guide included in the Club in a Box Resource Kit contains many helpful ideas for starting a club program. 5. Start using the handbook problems and materials in the Club in a Box Resource Kit with the students in your math club. Among other items, the Resource Kit includes structured club activities, the Silver Level Status Packet, pencils for your students and game boards that accompany an activity in the Club Resource Guide.ATTAININg SILvEr LEvEL STATUS 1. Though it is hoped that more than 12* students will participate in your math club and tackle the Silver Level Challenges, your school must have at least 12* students who each participate in at least 5 of the 6 Challenges. Silver Level Challenge solutions will be available each month from September through February. 2. Each of the Silver Level Challenges (and their answer keys) will be available on the MCP Members Only page in the Club Program section of www.mathcounts.org. 3. Once your school club has 12* students who have each fulfilled the Silver Level requirement of completing at least five of the six Silver Level Challenges, complete the Participant Verification Form/ Application for Silver Level Status with the names of those students and your contact information. This form is available in the Club in a Box Resource Kit, in the Club Resource Guide and on the MCP Members Only page of the MATHCOUNTS website. MATHCOUNTS Foundation 4. Submit your Participant Verification Form/Application for Silver Silver Level – Club Program Level Status via fax or mail to the address shown here. (Please 1420 King Street submit your form only once.) Alexandria, VA 22314 Fax: 703-299-5009 5. Deadline: Your Participant Verification Form/Application for Silver Level Status must be received by March 4, 2011 for your school to be eligible for the prize drawing ($250 gift cards). 6. The winners of the drawing will be notified by April 29, 2011. A list of the winners will be posted online. MATHCOUNTS will send the prizes to the winners by April 29, 2011. 7. In May, MATHCOUNTS will send a plaque to all Silver Level Schools in recognition of their achievement. Your Participant Verification Form/Application must be received by April 15, 2011.ATTAININg gOLD LEvEL STATUS 1. Your math club first must attain Silver Level Status in the Club Program. 2. Once a club has reached Silver Level Status, MATHCOUNTS will e-mail the coach the Ultimate Math Challenge. (The first e-mails will go out on February 11, 2011. Schools attaining Silver Level Status after this date will receive their Ultimate Math Challenge within one week of attaining Silver Level Status.) MATHCOUNTS Foundation 3. Once your students have completed the Ultimate Math Challenge, Gold Level – Club Program mail students’ completed Challenges (maximum of 20 per school) 1420 King Street and the Verification Form/Application for Gold Level Status to Alexandria, VA 22314 the address shown here. More specific details and the Verification Form/Application for Gold Level Status will be provided when the Ultimate Math Challenge is e-mailed to the coach.* Minimum club participation based on number of students at your school. Please see Frequently Asked Questions on pages 25-26for more information on Club Program eligibility.24 MATHCOUNTS 2010-2011
  • 4. MATHCOUNTS will score the students’ Ultimate Math Challenges and determine if your math club has attained Gold Level Status. Every math club member may take the Ultimate Math Challenge. However, no more than 20 completed Challenges per school may be submitted. At least 12* of the submitted Ultimate Math Challenges each must have 80% or more of the problems answered correctly for a school to attain Gold Level Status. 5. Deadline: Your Verification Form/Application for Gold Level Status and your students’ completed Challenges must be received by March 25, 2011 for your school to be eligible for the prize drawing ($500 gift cards, trip to the National Competition). 6. The winners of the drawing will be notified by April 29, 2011. A list of the winners will be posted online. MATHCOUNTS will send the prizes to the winners by April 29, 2011. 7. In May, MATHCOUNTS will send a banner and a plaque to all Gold Level Schools in recognition of their achievement. (Gold Level Schools will not receive the Silver Level plaque, too.) Your Participant Verification Form/Application must be received by April 15, 2011. FrEqUENTLY ASkED qUESTIONSWho is eligible to participate?Anyone eligible for the MATHCOUNTS Competition Program is eligible to participate in the ClubProgram. The Club Program is open to all U.S. schools with sixth-, seventh- or eighth-grade students.Schools (brick-and-mortar, virtual and homeschools) with 12* or fewer students in each of the sixth,seventh and eighth grades are permitted to combine for the purpose of reaching Silver or Gold LevelStatus. Please note that no schools may combine to create a team for the MATHCOUNTS CompetitionProgram. See page 14 of this handbook for updated details on eligibility requirements for theMATHCOUNTS Competition Program.How many students can participate?There is no limit to the number of students who may participate in the MATHCOUNTS Club Program.Encourage every interested sixth-, seventh- and eighth-grade student to get involved.What if our school has more than one math club?MATHCOUNTS encourages all math clubs in a school to make use of the MATHCOUNTS Club Programmaterials. However, each school may have only one officially registered club in the MATHCOUNTSClub Program. Therefore, it is recommended that schools combine their clubs when working towardmeeting the requirements of Silver or Gold Level Status.What does it cost to participate?Nothing. There is no fee to participate in the Club Program. Similar to the MATHCOUNTS SchoolHandbook, the Club in a Box Resource Kit is free to all eligible schools that request them.Can a school participate in both the Club Program and the Competition Program?Yes. A school may choose to participate in the Club Program, the Competition Program or both.Since these programs can complement each other, any school that registers for the MATHCOUNTSCompetition Program (Option 2 on the MATHCOUNTS Registration Form) will automatically be signedup for the Club Program and will be sent the Club in a Box Resource Kit.How is the Club Program different from the Competition Program?The Club Program does not include a school-versus-school competition with the opportunity for topperformers to advance. There are no fees to participate in the Club Program, and recognition is focusedentirely on the school and the math club.What if our school has a small student population?The number of required participants in your club is based on the number of students at your school. ♦ For schools with fewer than a total of 50 students in the sixth, seventh and eighth grades, the* Minimum club participation based on number of students at your school. Please see Frequently Asked Questions on pages 25-26for more information on Club Program eligibility.MATHCOUNTS 2010–2011 25
  • ♦ For schools with a total of 50-99 students in the sixth, seventh and eighth grades, the minimum number of participants needed to satisfy the Silver and Gold Level requirements is 8. ♦ For schools with a total of 100 students or more in the sixth, seventh and eighth grades, the minimum number of participants needed to satisfy the Silver and Gold Level requirements is 12.What are the different levels of the program? Level Requirements School Receives 1. Choose Option 1 (Club Program) or □ Recognition on www.mathcounts.org Option 2 (Club Program and Competition □ Club in a Box Resource Kit Program) and fill in the required information □ Volume II of the 2010–2011 MATHCOUNTS BRONZE on the MATHCOUNTS Registration Form School Handbook, containing over 200 math (p. 63), in the 2010–2011 MATHCOUNTS problems School Handbook or online at www.mathcounts.org. 2. Submit the form to MATHCOUNTS. 1. Depending on the size of your school, at □ Recognition on www.mathcounts.org least 12* members of the math club each □ Certificates for students must take at least 5 of the 6 Silver Level □ Plaque identifying your school as a Silver SILVER Challenges (available online and in the Silver Level MATHCOUNTS School Level Status Packet). □ Entry to a drawing** for one of ten $250 gift 2. The Participant Verification Form/Application cards for student recognition for Silver Level Status must be received by MATHCOUNTS by March 4, 2011 (available in the Club in a Box Resource Kit, in the Club Resource Guide and on the MCP Members Only page in the Club Program section of www.mathcounts.org) for entry to drawings. **Schools going on to reach Gold Level Status will be 3. Applications will be accepted through included in the Silver Level drawing but will receive only the April 15, 2011 for plaque recognition. Gold Level plaque. 1. Achieve Silver Level Status. □ Recognition on www.mathcounts.org 2. At least 12* members of the math club each □ Certificate for students must score 80% or higher on the Ultimate □ Banner and plaque identifying your school as GOLD Math Challenge (available in February/ a Gold Level MATHCOUNTS School e-mailed to coaches of Silver Level □ Entry to a drawing for: Schools). 1) One of five $500 gift cards for student 3. The completed Ultimate Math Challenges recognition and Verification Form/Application for 2) Grand Prize: $500 gift card for student Gold Level Status must be received by recognition and a trip for four students and MATHCOUNTS by March 25, 2011. (Any the coach to witness the 2011 Raytheon number of students may take the Challenge, MATHCOUNTS National Competition in but a maximum of 20 completed Ultimate Washington, D.C. (May 5-8, 2011) Challenges may be submitted per school.) 3. Applications will be accepted through April 15, 2011 for plaque and banner recognition.* Minimum club participation based on number of students at your school. Please see Frequently Asked Questions on pages 25-26for more information on Club Program eligibility. MATHCOUNTS reserves the right to modify the MATHCOUNTS Club Program guidelines as necessary.26 MATHCOUNTS 2010-2011
  • Warm-Up 1 $1. ��������� If a total of twenty million dollars is to be divided evenly among four million people, how much money, in dollars, will each person receive? Y2. ��������� Triangle XYZ has side XZ = 6 cm. Segment YA is perpendicular sq cm to XZ. If AY is 4 cm, what is the area of triangle XYZ? X Z A $3. ��������� Wilhelmina went to the store to buy a few groceries. When she paid for the groceries with a $20 bill, she correctly received $4.63 back in change. How much did the groceries cost? 4. ��������� How many multiples of 8 are between 100 and 175? multiples5. ��������� A three-digit integer is to be randomly created using each of the digits 2, 3 and 6 once. What is the probability that the number created is even? Express your answer as a common fraction. 6. ��������� In the following arithmetic sequence, what is the value of m ? -2, 4, m, 16, . . . 27. ��������� A particular fraction is equivalent to 3 . The sum of its numerator and denominator is 105. What is the numerator of the fraction?8. ��������� What is the least natural number that has four distinct prime factors? 9. ��������� The perimeter of a particular rectangle is 24 inches. If its length and width are areas positive integers, how many distinct areas could the rectangle have? 10. �������� Trapezoid ABCD and trapezoid EFGH are congruent. What is the difference sq units between the area of triangle ABC and the area of triangle EFJ? D C H J G A B E FMATHCOUNTS 2010-2011 27
  • Warm-Up 2 Cartons of Milk Sold per Day 401. ��������� The bar graph shows the number of cartons of milk sold per cartons Number Sold day last week at Jones Junior High. What is the positive 30 difference between the mean and median number of cartons 20 of milk sold per day over the five-day period? 10 0 Mon Tues Wed Thurs Fri Day B2. ��������� inches An equilateral triangle PBJ that measures 2 inches on each side is P J cut from a larger equilateral triangle ABC that measures 5 inches on each side. What is the perimeter of trapezoid PJCA? A C 13. ��������� boys Currently, 4 of the members of a local club are boys, and there are 80 members. If no one withdraws from the club, what is the minimum number of boys that would need to join to make the club 3 boys? 14. ��������� If William has 3 pairs of pants and 4 shirts, and an outfit consists of 1 pair of pants outfits and 1 shirt, how many distinct outfits can William create? 5. ��������� Jessica reads 30 pages of her book on the first day. The next day, she reads another days 36 pages. On the third day, she reads another 42 pages. If she continues to increase the number of pages she reads each day by 6, how many days will it take her to read a book that has 270 pages? 6. ��������� A basket of fruit contains 4 oranges, 5 apples and 6 bananas. If you choose a piece of fruit at random from the basket, what is the probability that it will be a banana? Express your answer as a common fraction. 5 117. ��������� What number is halfway between 8 and 16 ? Express your answer as a common fraction. pounds8. ��������� Farmer Fred read that his crop needs a fertilizer that is 8% nitrate for optimal yield. He needs to apply 4 pounds of nitrate per acre. If his field has 188 acres, how many pounds of fertilizer will Fred use? 9. ��������� Four squares are cut from the corners of a rectangular sheet of cardboard. It is then sq in folded as shown to make a box that is 15 inches long, 8 inches wide and 2 inches tall. What was the area of the original piece of cardboard? degrees10. �������� Two consecutive angles of a regular octagon are bisected. What is the degree measure of each of the acute angles formed by the intersection of the two angle bisectors? 28 MATHCOUNTS 2010-2011
  • Workout 11. ��������� Sweaters cost $49 at a local department store, and the catalog department of $ the same store sells them for $44 online. The local department store charges 7% tax, and the catalog department charges $3.50 per sweater for shipping (but does not charge tax). How much would a person save by buying 6 sweaters through the catalog department rather than at the department store? 2. ��������� If Jordan passed mile marker 138 at 5:08 pm and then passed mph mile marker 216 at 6:20 pm, what was her average speed in miles per hour? 3. ��������� A group of 5 women and 8 men weigh a total of 1921 pounds. If the women of the pounds group have an average weight of 109 pounds, what is the average weight of the men in the group? 4. ��������� On planet Wobble, 1 womp is equal in value to 3 wamps, and 2 wamps are equal in womps value to 5 wemps. How many womps are equal in value to 15 wemps? 5. ��������� How many positive three-digit perfect cubes are even? cubes questions6. ��������� Billy was playing a trivia game. According to the rules, Billy would receive 25 points for each question he answered correctly but would lose 50 points for each question he answered incorrectly. At the end of the game, Billy had a total of 450 points. He had 5 times as many questions correct as incorrect. If Billy answered every question, how many questions were asked in the game? 7. ��������� What is the value of the following expression, expressed as a repeating decimal? ( 1 9 1 ) + 7 − 2 98. ��������� Gary chooses a two-digit positive integer, adds it to 200, and squares the result. What is the largest number Gary can get?9. ��������� The sale price of Kara’s dress was $43.20 after a 40% discount. What was the $ original price of the dress? degrees10. �������� The supplement of an angle is 5.5 times the complement of the angle. What is the measure of the angle? MATHCOUNTS 2010-2011 29
  • Warm-Up 31. ��������� Thirty percent of twice a number is 30. What was the original number? 2. ��������� A bouncy ball bounces 16 feet high on its first inches bounce and exactly half as high on the next bounce. For each later bounce, the ball bounces half as high as on the previous bounce. How many inches high does the ball bounce on its seventh bounce? 3. ��������� Tamika’s dad bought a Mustang convertible for $12,000 to celebrate his first job in $ 1978, which paid him $15,000 a year. If Tamika is to use the same fraction of her salary as her dad to buy a Mustang convertible, which now sells for $32,000, what will her first job have to pay? 4. ��������� A rectangular prism has dimensions 4 by 6 by x. If the total surface area of the prism is 248 square units, what is the value of x? 5. ��������� How many ways can the letters in FACTOR be rearranged so that the first and last ways letters are vowels? 6. ��������� If a # b = (a + b)2, what is the value of 3 # 1? 7. ��������� The triangular array of positive integers shown continues indefinitely, with each row containing one entry more than the row above it. What is the sum of the two integers directly above 100? 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 points8. ��������� In a class of 20 students, the average (arithmetic mean) score on a test is 84 points. If 6 students each scored 100 points and 4 students each scored 50 points, what is the average of the scores of the remaining students? a −2b9. ��������� If 3a − 4b = 5 , what is the ratio of a to b? Express your answer as a common fraction. y 6 I (3, 5)10. �������� If triangle WIN is reflected over the x-axis ( , ) to create triangle W’I’N’, what will be the 4 W (-1, 2) N (11, 2) coordinates of point I’? Express your answer as an ordered pair. x -2 2 4 6 8 10 -230 MATHCOUNTS 2010-2011
  • Warm-Up 4 Potato Chip Prices 15 9 Key 16 9 15 | 9 means $1.591. ��������� A survey of the cost of one-pound bags of potato chips bags 17 5 9 at five local supermarkets is shown in the stem-and-leaf plot. How many bags of chips cost less than the median 18 8 9 price of a bag of chips? 19 9 9 9 9 9 20 9 21 5 5 8 9 22 5 9 9 23 9 24 25 92. ��������� Vivian has 45 songs on her MP3-player, 15 of which are alternative rock and the rest more rap songs of which are rap. Malik has 60 songs on his MP3-player in the same ratio as Vivian’s songs. How many more rap songs does Malik have on his MP3-player than Vivian has on hers? pckgs3. ��������� Lashae will plant pansies and violets around the edge of her garden. She must use the same number of each plant, and she must use all of the flowers she buys. If pansies are sold in packages of 6 and violets are sold in packages of 10, what is the least number of packages of violets she could buy, given that she buys at least 1 package? 14. ��������� 1 1 The first four rows of Pascal’s triangle are shown to the left. If the first 1 2 1 two numbers in a particular row of Pascal’s triangle are 1 and 10, what is 1 3 3 1 the fourth number of the next row? 5. ��������� In the figure, the gray small squares and the white small squares are all the same size, and the outer corners of the outer gray squares lay on the edge of the largest square. What fraction of the largest square is shaded? Express your answer as a common fraction. %6. ��������� Last July there were 56 dogs for sale in the newspaper ads, and this July there were 84 dogs for sale. What is the percent increase in the number of dogs for sale? 7. ��������� What is the value of the expression 3x + 4y 3 if x = 3 and y = −2? 8. ��������� Points A, B and C are collinear, with B between A and C. If the distance from point A units to point B is 8 more than twice the distance from point B to point C, and the length of segment AC is 5 times the length of segment BC, how long is segment AC? 9. ��������� The sum of two numbers is 3 times their positive difference. What is the ratio of the smaller number to the larger number? Express your answer as a common fraction. 10. �������� The net shown is composed of 6 congruent squares and has a units3 perimeter of 56 units. When the net is folded into a cube, what is the volume of the cube? MATHCOUNTS 2010-2011 31
  • Workout 21. ��������� Sara has 30 pounds of dried bananas that she wants to divide into small bags and $ sell. If each small bag holds 5 of a pound and will sell for $1.58, how much 8 will Sara bring in from sales if she sells all the bags? integers2. ��������� How many five-digit integers are perfect squares? $3. �������� Four friends (Amanda, Barbara, Charles and Dylan) invested $3000, $5000, $7000 and $9000, respectively, in a certain start-up company. Profits will be divided in the same ratio as the investment. If the business made a profit of $72,000 this year, how much of the profit will the largest investor receive? 4. ��������� The mean score on Mr. Pascal’s first semester exam for his 28 students is points 73 points. If he removes the lowest score of 19 points, what is the adjusted class average (arithmetic mean)? 5. ��������� The sum of four consecutive even integers is 596. What is the product of the least and greatest of these integers? N6. ��������� In this Number Wall, you add the numbers next to each other and 65 34 write the sum in the block directly above the two numbers. What 15 number will be in the block labeled N? 9 7 A B7. ��������� Chord AB of circle O is 6 units long and 4 units from center O. sq units What is the area of circle O? Express your answer in terms of �. 4 {O8. ��������� Three fair six-sided number cubes, each with faces labeled 0, 2, 4, 6, 8, 10, are rolled. What is the probability that the sum of the numbers rolled is greater than 20? Express your answer as a common fraction. 9. ��������� In the patterns of dots with P1, P2, P3 and P4, each successive pattern of dots has one more row and one more column. What is the ratio of the number of dots in P50 to the number of dots in P25? Express your answer as a common fraction. P1 P2 P3 P410. �������� The population of Circletown has been growing at an people annual rate of 5% for the last 5 years. The population is now 3105 people. What was the population of Circletown 5 years ago? Express your answer to the nearest whole welcome to number. Circletown32 MATHCOUNTS 2010-2011
  • Warm-Up 51. ��������� If Minh picks a letter of the alphabet at random, what is the probability it occurs in the name “Hattiesburg, Mississippi”? Express your answer as a common fraction. Total Sales By Grade %2. ��������� The graph shows the amount of sales made by each Grade 5: Grade 8: 750 grade at Memorial Middle School in the annual 1250 fundraiser. What percent of the total sales was made by the Grade 7 class? Express your answer to the nearest tenth. Grade 6: Grade 7: 1000 5003. ��������� What is the integer value of 0 − 5 + 10 − 15 + 20 − 25 + 30 − . . . + 240? 4. ��������� In this additive magic square the sum of the three numbers in each z row, in each column and along each diagonal is the same. What is the 11 6 value of z? 105. ��������� What is the area of the quadrilateral with vertices at (1, 1), (4, 1), (7, 5) and (4, 5)? sq units 6. ��������� If a # b = a b - b a, what is the value of 3 # 5? %7. ��������� Coach Kennedy had the starters of his basketball team practice free throws. Each player had to shoot 20 free throws. On Wednesday, 4 of the 5 starters did the free throw drill and averaged making 15 out of 20 attempted baskets. When the fifth starter did his drill, the team average dropped to 14 made out of 20 attempted. What percent of his free throws did the fifth starter make? ft per sec8. ��������� William is traveling at a rate of 75 miles per hour. How fast is William traveling in feet per second? 9. ��������� To create a more affordable line of her gourmet chocolates, Kindra pounds mixes 10 pounds of chocolate worth $1.50 per pound with another type of chocolate, worth $9.00 per pound. She wishes to make a mixture of chocolate worth $4.00 per pound. How many pounds of the $9.00 chocolate must she add to create her new product? inches10. �������� A solid is formed by placing a square pyramid with a base of 3 inches by 3 inches on top of a cube with edges of 3 inches. The volume of this combined solid is 54 cubic inches. What is the height of the pyramid? MATHCOUNTS 2010-2011 33
  • Warm-Up 61. ��������� What is the smallest positive integer that is divisible by 2, 5, 6 and 9? 2. ��������� What is the value of (0.07)3? Express your answer as a decimal to the nearest millionth. a b3. ��������� If x = b and y = a , then what is the square of the product of x and y ? 4. ��������� What is the only number that when added to its reciprocal is equal to 2? 5. ��������� Two squares have centers at the origin of a coordinate plane points and sides parallel to the axes. The smaller square has an area of 18 square units, and the larger square has an area of 50 square units. How many points with only integer coordinates are outside the smaller square region and inside the larger square region? $6. ��������� Pedro earns $7.50 an hour at his job at Matrix Cinemas. If he works 20 hours this week and his employer withholds a total of 22% of his weekly pay for taxes and Social Security, what is his take-home pay for the week? pm7. ��������� Mary leaves New York City at 9:00 am, traveling to Charlotte, NC, at an average rate of 55 miles per hour. Simba leaves one hour later than Mary and follows Mary’s route at an average rate of 65 miles per hour. At what time will Simba catch up to Mary? ( , )8. ��������� What ordered pair of positive integers (r, s) satisfies the equation 5r + 6s = 47, such that r > s ? 9. ��������� What is the minimum number of 3-inch by 5-inch index cards needed to completely index cards cover a 3-foot by 4-foot rectangular desktop without cutting the index cards? E F10. �������� sq ft Points A, B, C and D are the centers of four circles, and they A B are also intersection points of these circles, as shown. Each circle has a radius of 6 feet and is tangent to two sides of square EFGH. What is the area of square EFGH? D C H G34 MATHCOUNTS 2010-2011
  • Workout 31. ��������� When the repeating decimal 0.38 is expressed as a common fraction, what is the sum of the numerator and denominator? 2. ��������� The sum of three different two-digit prime numbers is 79. The largest of the numbers is 43. The difference of the other two is 10. What is the product of the three numbers? 3. ��������� Five wooden disks are numbered -3, -2, 1, 4 and 7. If two disks are chosen at random, without replacement, what is the probability that their product is negative? Express your answer as a common fraction. 4. ��������� The digits 1, 3, 5, 6, 7 and 9 are each used once and only once to form three two- digit prime numbers. What is the largest possible sum that can be formed? 5. ��������� A cylinder and a rectangular prism have the same volume. If the length, width and height of the rectangular prism are doubled and the radius of the cylinder is doubled, what is the ratio of the volume of the new cylinder to the volume of the new rectangular prism? Express your answer as a common fraction. y B ( , )6. ��������� A quadrilateral is formed by connecting the vertices A(5, 0), B(2, 6), C(–9, 0) and D(2, –6). What would be the coordinates of point C if you moved it to the right just enough to C A transform quadrilateral ABCD into rhombus ABCD? Express x your answer as an ordered pair. D7. ��������� Three stoplights on different streets each operate on their own independent am schedules, as follows: the first stoplight is red 1 minute out of every 2 minutes (1 minute red, then 1 minute green), the second is red 2 minutes out of every 3 minutes (2 minutes red, then 1 minute green) and the third is red 3 minutes out of every 5 minutes (3 minutes red, then 2 minutes green.) At 9:00 am each stoplight turns red. The lights are either red or green (don’t worry about yellow). What time is it when the next 1-minute segment of time in which all three stoplights are red begins? 8. ��������� If Sara makes 80% of the free throws she attempts, what is the probability that she misses exactly two of her next three free throws? Express your answer as a common fraction. sq units9. ��������� A 4-unit by 4-unit grid has this pattern shaded in the design of a company’s logo. What is the area of the shaded portion of the logo? 10. �������� What is the 2011th term of the arithmetic sequence −4, −1, 2, 5, . . . , where each term after the first is 3 more than the preceding term? MATHCOUNTS 2010-2011 35
  • Systems of Equations StretchFor problems 1-3, solve each of the following systems of equations. Describe the method you used. What other methods could you have used to solve each system? Express each answer as an ordered pair. Express any non-integer value as a common fraction.1. 2x + 3y = 11 2. 2 x + 3 y = 11 3. 2x 2 + 3y 2 = 11 3x - y = 11 3 −y 1 = 11 3x 2 - y 2 = 11 x In problems 4 and 5, for what value of k does the linear system have no solution? Express any non-integer value as a common fraction.4. kx + 3y = 11 5. 2x + ky = 11 3x - y = 11 3x - y = 11 6. Given that 2a + b = 19, 2c + d = 37 and b + d = 24, what is the value of a + b + c + d ?7. Given that a + b = 29 and ab = 204, what is the value of a 2 + b 2? 8. Solve each of the following linear systems. Express your answer as an ordered pair.a. 5x + 6y = 7 b. x + 2y = 3 8x + 9y = 10 4x + 5y = 6 Many problems can be solved using a linear system of equations. For problems 9 and 10, use linear systems of equations to help you solve. 9a. For each equation below (i, ii, iii and iv), does there exist a solution (x, y) with positive integers x and y? In each case, either find all pairs of integers satisfying the equation or explain why none exist. Hint for the first equation: x 2 – y 2 = (x + y)(x – y), so x + y = 12 and x – y = 4 is one case to check. i. x 2 – y 2 = 48 ii. x 2 – y 2 = 23 iii. x 2 – y 2 = 45 iv. x 2 – y 2 = 90 b. In general, for what type of integers, n, does x 2 – y 2 = n have at least one solution? 10. A shipping clerk wishes to determine the weights of each of five boxes. Each box weighs a different integer amount less than 100 kg. Unfortunately the only scales available measure weights in excess of 100 kg. The clerk therefore decides to weigh the boxes in pairs so that each box is weighed with every other box. The weights for the 10 pairs of boxes are (in kilograms) 110, 112, 113, 114, 115, 116, 117, 118, 120 and 121. From this information the clerk can determine the weight of each box. What are the weights of each of the five boxes?36 MATHCOUNTS 2010-2011
  • What About Math?In an effort to bridge the gap between the knowledge students gain from MATHCOUNTS and the importantapplications of this knowledge, MATHCOUNTS is providing a What About Math? section in Volume I of theMATHCOUNTS School Handbook. This year the problems are from the fields of engineering and statistics.Written by actual professionals in these fields, these problems illustrate how students can put the skills they arelearning today into practice in the near future.This year’s problems and solutions were written for MATHCOUNTS by members of the Association forUnmanned Vehicle Systems International (AUVSI), The Actuarial Foundation (TAF) and the ConsumerElectronics Association (CEA). AUVSI Foundation Writers/Editors: Wendy Amai, Sandia National Laboratories; Dr. Kenneth Berry, Assistant Director of the Science and Engineering Education Center, University of Texas at Dallas; Angela Carr, Web Services Manager, AUVSI; Dr. Deborah A. Furey, DARPA Program Manager; Lisa Marron, Sandia National Laboratories and Dr. Dave Novick, Sandia National Laboratories TAF Writers/Editors: Charles Cicci, ACAS, MAAA; Kevin Cormier, FCAS, MAAA; Jeremy Fogg; Mubeen Hussain; Jeremy Scharnick, FCAS, MAAA; John Stokesbury, FSA, FCA, EA, MAAA; Pete Rossi, FSA, FCA, CERA, MAAA and Patricia Teufel, FCAS, MAAA. CEA Writers/Editors: Bill Belt; Deepak Joseph; Brian Markwalter, PE and Dave WilsonASSOCIATION FOR UNMANNED VEHICLE SYSTEMS INTERNATIONAL FOUNDATION(WWW.AUVSIFOUNDATION.ORG)The AUVSI Foundation is a nonprofit, charitable organization that was established to support the educationalinitiatives of the Association for Unmanned Vehicle Systems International (AUVSI). The AUVSI Foundation’sprimary objective is to develop educational programs that attract and equip students for careers in robotics. To dothis, the AUVSI Foundation provides K-12 and university level students with hands-on educational activities thathighlight the world of robotics while emphasizing STEM curriculum–science, technology, engineering and math.Currently, the AUVSI Foundation directly sponsors five annual advanced robotics competitions that attractnearly 200 student teams, representing more than 2000 students from schools around the world. Last year, nearly$100,000 in prize money was awarded through these student competitions. For more information on the AUVSIFoundation and its educational programs, please visit www.auvsifoundation.org.THE ACTUARIAL FOUNDATION (WWW.ACTUARIALFOUNDATION.ORG)The Actuarial Foundation (TAF) was founded in 1994 as the charitable arm of the actuarial profession. TAF’smission is to develop, fund and execute education and research programs that serve the public by harnessingthe talents of actuaries. Through the Foundation’s youth education math activities, actuaries assist with thedevelopment of math materials that help students develop a love for math and embrace mathematical conceptsin a fun and engaging way in order for them to understand how math is used in the “real world.” If you likemath – consider becoming an actuary. “Actuary” was rated the number two profession in the most recent JobsRated Almanac. The actuarial profession’s ability to continue providing competent service in the future requiressupporting mathematics achievement among today’s students. The Actuarial Foundation is proud to support andsponsor MATHCOUNTS!CONSUMER ELECTRONICS ASSOCIATION (WWW.CE.ORG)The Consumer Electronics Association (CEA) is the preeminent trade association promoting growth in the$165 billion U.S. consumer electronics industry. More than 2000 companies enjoy the benefits of CEAmembership, including legislative advocacy, market research, technical training and education, industrypromotion, standards development and the fostering of business and strategic relationships. CEA also sponsorsand manages the International CES–The Global Stage for Innovation. All profits from CES are reinvested intoCEA’s industry services. MATHCOUNTS 2010–2011 37
  • What About Math? AUVSI Foundation1a. �������� Robots often have two treads (like tank treads) instead of four wheels. You can kg calculate the amount of weight a tread or wheel is carrying per square inch of tread or wheel by dividing the weight of the robot by the total area of the tread or wheel on the ground. If a robot weighs 120 kg and each of the robot’s 2 treads touches an area of ground that is 30 cm long and 5 cm wide, how much weight is carried over each square centimeter of tread? Express your answer as a decimal to the nearest tenth. kg1b. �������� Compare this to the same robot with 4 wheels (instead of tread) that each touch a section of ground that is 3 cm × 5 cm. How much weight is on each square centimeter of the wheels?1c. �������� A particular robot wheel is 6 cm in diameter. The motor attached to the wheel runs cm at a speed of 120 rpm (rotations per minute). How many centimeters will the robot travel in 30 seconds? Express your answer as a decimal to the nearest tenth. kg2. ��������� The buoyancy of an object is defined as the mass of the displaced fluid. A particular underwater robot that has a mass of 10 kg has a volume equivalent to the right circular cylinder shown. The density of water is 1000 kg/m3, and the volume of a right circular cylinder is defined as V = πr2h. For d = 9 cm and h = 100 cm (use 3.14 for π), determine if h this robot will sink or float. How much flotation will need to be added (or mass will need to be added/removed) to get this robot to be neutrally buoyant in water? Express your answer as a decimal to the nearest ten thousandth. Note: d For an object to be neutrally buoyant, the mass of the object must be equal to the mass of an identical volume of water.3a. �������� An unmanned aerial vehicle (UAV) carries 45 gallons of fuel. It burns fuel at a rate miles of 3 gph (gallons/hour) and flies at a speed of 160 mph (miles/hour). We want to fly it to a location X, fly it in the area for 1 hour, and then fly it back to its launching point. What is the greatest number of miles away from the UAV’s launching point can location X be?3b. �������� How far away can location X be if we want the UAV to arrive back home with a fuel miles reserve of 0.5 hour?38 MATHCOUNTS 2010-2011
  • 4a. �������� A communication network can carry 128,000,000 bits per second. A digital video frames per sec camera on the network makes pictures that are 640 pixels wide by 480 pixels high. Each pixel is composed of 3 bytes. A byte is 8 bits. A picture is called a “frame.” How many frames per second from the camera can this network carry? Express your answer to the nearest whole number.4b. �������� We can increase the frame rate if we decrease the size of the picture. If we pixels wanted to send back square frames at a rate of 30 frames/second, how many pixels wide would a square frame (same size high as wide) be? Express your answer as a decimal to the nearest tenth.5a. �������� Obstacles such as fallen trees, fences and ditches can be difficult for wheeled meters robotic vehicles to overcome. Imagine that a specially equipped robot could jump over such things. At the peak of the jump, the robot’s vertical velocity is 0 m/s. Standard gravity on Earth, g, is 9.8 m/s2. If the time it takes to fall from the peak is 1.7 s, how high did the robot jump? Height is equal to ½(g)(t)2, where g = gravity and t = time to fall to the ground, in seconds. Express your answer as a decimal to the nearest hundredth.5b. �������� How fast, in meters per second, is the robot traveling when it hits the ground? meters per second Velocity = (g)(t). Express your answer as a decimal to the nearest hundredth.6. ��������� In 2003, NASA launched the rover Spirit onto the surface of Mars to explore our min sec planetary neighbor. Near the end of January 2010, Mars was 99.33 million km away from Earth. Assume the speed of communication is at the speed of light, which is 300,000,000 m/s. If a researcher on Earth requested information from Spirit about its temperature, how long did the researcher have to wait for a reply after sending the request? Express your answer in the form x minutes y seconds, where x and y are whole numbers. 7. ��������� A robot has a battery life of 3 hours while carrying 0 pounds. The robot’s maximum feet velocity is 20 feet per minute. For every pound that the robot needs to carry, the total amount of battery time the robot will operate while traveling at maximum velocity decreases by 8 minutes. Starting with a fully charged battery, how many feet can the robot travel at maximum velocity while carrying 10 pounds? 5 ft8. ��������� A small ground robot has a telescoping pounds lifting arm with a motor that can pick up a 3 ft 20-pound weight when it is 3 feet away. The motor can support a maximum moment, or torque, of 60 foot-pounds (moment = weight × distance). How much weight, in pounds, could the arm lift at 5 feet away? 20 lb ? lbMATHCOUNTS 2010-2011 39
  • 9. ��������� Torque is a measure of “rotational force,” and it is equal to force times distance. foot- pounds A motor at the end of an 8-inch robot arm is lifting a 3-pound weight. How much torque is the motor generating? Note: Torque is measured in foot-pounds. 10a. ������� Gears perform two important duties. First, they can increase or decrease the speed of a motor. Second, they can decrease or increase the power transferred. Gears with the same size teeth are usually described not by their physical size but by the number of teeth around the circumference. The gear ratio is the relationship between the numbers of teeth on two gears that are meshed. Gear ratio = (number of teeth on output)/(number of teeth on input). When multiple gears are involved, the gear ratios from each pair of meshed gears are multiplied to calculate the gear train’s gear ratio. 60 teeth 25 teeth 40 teeth 30 teeth 20 teeth 120 rpm What is the gear ratio of the gear train shown if the rightmost gear is the input? Express your answer as a common fraction.10b. ������� The speed of an output gear is equal to (input speed) × (1 ÷ gear ratio). If a motor rpm with 120 rpm (revolutions per minute) is attached to the leftmost gear (input), what is the speed of the final rightmost gear that is the output?40 MATHCOUNTS 2010-2011
  • What About Math? The Actuarial Foundation1. ��������� The probability that a particular state will have neither a hailstorm nor a tornado in a given month is 55%. In the same period, the probability of a hailstorm is 35% and the probability of a tornado is 25%. If the probability of a hailstorm and the probablity of a tornado are independent but not mutually exclusive, what is the probability of both a hailstorm and a tornado occurring in a given month? Express your answer as a decimal to the nearest hundredth.Use the following information for problems 2 and 3.The amount of damage (in millions of dollars) that could be caused by an earthquake in the fictitious country of Shakalot is shown in the table below. Dollar Value of Damage Probability (millions) 5 10% 10 15% 25 20% 50 30% 75 20% 100 4% 150 1% $2. ��������� What is the expected (mean) damage amount, in millions of dollars, for an million earthquake in Shakalot? Express your answer as a decimal to the nearest tenth. 3. ��������� What is the interquartile range, in millions of dollars, for this data? $ million4. ��������� An insurance company has sold 10,000 policies. The policyholders are classified policy- holders using gender (Male or Female) and age (Young or Old). Of these policyholders, 3000 are Old, 4000 are Young Males and 4000 are Female. How many of the company’s policyholders are Old Females?5. ��������� Sneaky Joe has a loaded die with the numbers 1, 2 and 3 each having a probability $ of 1/4 of being rolled and the numbers 4, 5 and 6 each having a probability of 1/12 of being rolled. No one but Joe knows about the loaded die. Sneaky Joe offers to take the following bet with you: You roll the die. If the result is an even number, you win $x ; if it’s an odd number, you lose and pay Joe $5. How much should Sneaky Joe pay you for a win to make this a fair bet?MATHCOUNTS 2010-2011 41
  • 6. ��������� A certain disease is expected to infect 1 out of every 10,000 individuals in a % country. A test for the disease is 99.5% accurate. It never gives a false indication when it is negative, so 0.5% of the people who take the test will get inaccurate readings, all of which will be false positives (meaning that the people test positive but do not have the disease). Let us suppose you test positive; what is the probability that you actually have the disease? Express your answer as a percent to the nearest whole number. Use the following information for problems 7–9. Frequency = the ratio of the number of claims to the number of policies Severity = the mean size of a group of claims Pure Premium = Frequency × Severity Charged Premium = Pure Premium + Expenses + ProfitSuppose there are 50 policies that generate the following five claims in 2009: $7500 $5000 $3000 $11,500 $80007. ��������� What is the frequency? Express your answer as a decimal to the nearest claims/ policies hundredth.8. ��������� What is the severity of the five claims? $ $9. ��������� How much premium should be charged if 15% of the charged premium is to be used for expenses and 5% of the charged premium is for profit? days10. �������� AutoMakers, Inc. is a car manufacturer and currently has three operational plants cleverly named Plant A, Plant B and Plant C. Plant A can produce 100 cars a day. Plant B can produce 80 cars a day. Plant C can produce only 70 cars a day. If all three plants are running at full production capacity, how many full days are needed to produce 1600 cars? 42 MATHCOUNTS 2010-2011
  • What About Math? Consumer Electronics Association Signal must reach at least this far.A provider of wireless Internet service will offer service to a neighborhood that is shaped like a square and is 1 km (1000 m) long on each side. The service provider will be using 1 kma tower that is located precisely in the middle of the square. The signal from the tower will travel an equal distance in all Neighborhooddirections, producing a circular coverage area. 1 kmTo function correctly, a consumer’s wireless Internet device must receive a signal from the wireless Internet service provider that is at least 1 microwatt (1 x 10-6 watt) in strength. Radio frequency signals lose their strength as they travel over a distance according to the following formula: Received signal strength in watts = Transmitted signal strength in watts ÷ [4πd/λ]2, where λ is the wavelength of the wireless Internet signal in meters and d is the distance from the transmitter to the receiver in meters. The wavelength (λ) of a radio frequency signal in meters is equivalent to the speed of light in meters per second divided by the frequency of the signal in hertz (Hz). The speed of light is approximately 3 × 108 meters per second. The wireless Internet service provider will be using the frequency 900 megahertz (MHz), which is equivalent to 900,000,000 Hz.1. ��������� How strong does the signal sent by the transmitter have to be in order to watts provide at least one microwatt to every receiver within the 1-square-kilometer neighborhood? Express your answer to the nearest whole number. Note: Disregard the height of the transmitters. An electric circuit in a house goes from the circuit breaker box in the basement to the kitchen. In the kitchen there are four outlets connected to this circuit. These are the only things in this circuit. The wire in the wall from the basement to the kitchen is able to handle 15 amps of current without becoming too hot. The circuit breaker for this circuit in the basement protects the wiring in the wall from damage. The ciruit breaker will trip if more than 15 amps of current flows through the circuit. The house has standard 120-volt electric service. In the kitchen there is one thing connected to the outlets in the circuit—an 1100-watt toaster. The power required for operating the toaster will be pulled from the total power available to the circuit. The homeowner wants to purchase a coffeemaker and connect it to another outlet in the same circuit. 2. ��������� What is the maximum power, in watts, that the coffeemaker can consume without watts causing the circuit breaker in the basement to trip when both the coffeemaker and the toaster are operating simultaneously? Note: Power = voltage x current (that is, watts = volts × amperes). MATHCOUNTS 2010-2011 43
  • Plug-in hybrid electric vehicles (PHEVs) combine one or more electric motors and a gas or diesel engine for propulsion. Energy is stored in a battery pack for the electric motor and in the gas tank for the combustion engine. The combination allows short trips to be made entirely on electric power and raises the effective fuel efficiency of the car. The battery in a typical PHEV stores 9000 watt-hours (W-h) of energy. Some additional information you will need: A house has standard 120-volt electric service. power = voltage × current (that is, watts = volts × amperes)energy = power × time (that is, watt-hours = watts × hours) A standard wall outlet with a 15-ampere circuit breaker can supply 12 amperes for continuous charging, which avoids tripping the circuit breaker (circuit breakers are set 25% above the maximum continuous current a circuit is designed to carry). Electric vehicle engineers call charging from a standard outlet Level 1 charging. 3. ��������� How long will it take to charge a PHEV with a 9000-Wh-battery from a standard hours wall outlet in your home? Express your answer as a decimal to the nearest hundredth.Level 2 charging uses a 240-volt outlet that typically supplies 32 amperes of current. 4. ��������� How long will it take to charge the same 9000-Wh-PHEV (discussed in question 3) hours with a Level 2 charging outlet? Express your answer as a decimal to the nearest hundredth. Televisions use electric power while turned off and while turned on. The small amount of power used when a TV is off allows the TV to respond to a command from a remote control. Most of the power used when the TV is turned on goes toward producing the light for the picture we see on the screen. The average TV spends 5 hours a day turned on and 19 hours a day turned off. A 55-inch TV uses 197 watts/hour when it is on and 0.4 watt/hour when it is off. Thus, a 55-inch TV uses 992.6 watts per day, which can be calculated by adding together the power used while the TV is on and the power used while the TV is off each day. The power used while the TV is on is found by multiplying the number of hours the TV is on times the power used per hour while it is on. A corresponding formula applies for the power used while the TV is off. 5. ��������� How many watts per day does a small TV use if it uses 33 watts/hour while on and watts 0.4 watt/hour while off? Assume that the TV is on for 5 hours and off for 19 hours. Express your answer as a decimal to the nearest tenth. 44 MATHCOUNTS 2010-2011
  • Digital displays (such as TVs and computer monitors) are built according to standard dimensions(width and height) and resolutions (pixels). The width and height dimensions of a display can alsobe expressed as a ratio called the aspect ratio. The aspect ratio of a display is the ratio of thewidth of the image to its height, expressed as two numbers separated by a colon. The resolution ofa display is the number of distinct pixels (smallest elements of a screen) that can be displayed ineach dimension.For example, a display that has resolution of 800 × 600 pixels, or 800 pixels in the width dimensionand 600 pixels in the height dimension has an aspect ratio of 4:3. This is because the aspect ratio800:600 can be reduced to 4:3.6. ��������� If a high-definition digital TV has a resolution of 1920 × 1080 pixels, what is its aspect ratio? Express your answer in the form a:b, where a and b are positive integers and have no common factors other than 1.Internet speed to a home is 25 megabits per second (Mbps) peak throughput. Viewing or using certain common websites takes up varying bandwidth as shown in the list below:(a) Viewing MySpace takes up 1 Mbps(b) Using Pandora takes up 5 Mbps(c) Viewing Facebook takes up 2 Mbps(d) Viewing Amazon On Demand takes up 24 Mbps (e) Viewing YouTube-HD takes up 12 MbpsFor example: Viewing Myspace (1 Mbps) and listening to Pandora (5 Mbps) at the same time uses 1 Mbps + 5 Mbps = 6 Mbps of bandwidth of the 25 Mbps available. This is because bandwidth usage is additive. If a group of websites exeeds the available bandwidth, they cannot all be usedsimultaneously.7. ��������� What is the maximum number of distinct websites from the above list that can be websites simultaneously used at the home?MATHCOUNTS 2010-2011 45
  • 46 MATHCOUNTS 2010-2011
  • ANSWERS TO HANDBOOK AND WHAT ABOUT MATH? PROBLEMSIn addition to the answer, we have provided a difficulty rating for each problem. Our scale is 1-7, with 7 being the most difficult. These are only approximations, and how difficult a problem is for a particular student will vary. Below is a general guide to the ratings: Difficulty 1/2/3 - One concept; one- to two-step solution; appropriate for students just starting the middle-school curriculum. 4/5 - One or two concepts; multistep solution; knowledge of some middle-school topics is necessary. 6/7 - Multiple and/or advanced concepts; multistep solution; knowledge of advanced middle-school topics and/or problem-solving strategies is necessary. Warm-Up 1Answer Difficulty1. 5 or 5.00 (1) 4. 9 (1) 7. 42 (2) 22. 12 (1) 5. 3 (1) 8. 210 (1) 3. 15.37 (1) 6. 10 (1) 9. 6 (3) 10. 0 (2) Warm-Up 2Answer Difficulty1. 1* (1) 4. 12 (1) 7. 21 (2) 322. 13 (1) 5. 6 (1) 8. 9400 (2) 23. 10 (2) 6. (1) 9. 228 (2) 5 10. 45 (3) Workout 1Answer Difficulty1. 29.58 (2) 4. 2 (2) 7. 0.031746 (2)2. 65 (3) 5. 2 (2) 8. 89,401 (1)3. 172 (2) 6. 36 (3) 9. 72 or 72.00 (3) 10. 70 (4)* The plural form of the units is always provided in the answer blank, even if the answer appears to require the singular form of the units.MATHCOUNTS 2010-2011 47
  • Warm-Up 3Answer Difficulty1. 50 (2) 4. 10 (3) 7. 140 (2)2. 3 (2) 5. 48 (3) 8. 88 (2) 93. 40,000 or (2) 6. 16 (2) 9. (4) 7 40,000.00 10. (3, −5) (2) Warm-Up 4Answer Difficulty1. 6 (1) 4. 165 (4) 7. −23 (4) 12. 10 (2) 5. (3) 8. 20 (3) 2 13. 3 (1) 6. 50 (2) 9. 2 (3) 10. 64 (2) Workout 2Answer Difficulty1. 75.84 (2) 4. 75 (2) 7. 25π (4) 352. 217 (3) 5. 22,192 (2) 8. 216 (4) 513. 27,000 or (2) 6. 132 (2) 9. (3) 13 27,000.00 10. 2433 (3)48 MATHCOUNTS 2010-2011
  • Warm-Up 5Answer Difficulty 61. (1) 4. 7 (3) 7. 50 (3) 132. 14.3 (2) 5. 12 (3) 8. 110 (3)3. 120 (3) 6. 118 (2) 9. 5 (4) 10. 9 (4) Warm-Up 6Answer Difficulty1. 90 (2) 4. 1 (3) 7. 3:30 (3)2. 0.000343 (2) 5. 24 (3) 8. (7, 2) (4)3. 1 (2) 6. 117 or 117.00 (2) 9. 116 (3) 10. 324 (3) Workout 3Answer Difficulty1. 137 (3) 4. 211 (3) 7. 9:06 (3) 1 122. 12,857 (2) 5. (4) 8. 125 (5) 2 33. (3) 6. (-1, 0) (3) 9. 4 (3) 5 10. 6026 (3) MATHCOUNTS 2010-2011 49
  • System of Equations Stretch Answer1. (4, 1) 5. - 2 9a. i. (13, 11), (8, 4), (7, 1) 3 1 ii. (12, 11)2. ( 4 , 1) 6. 40 iii. (23, 22), (9, 6), (7, 2)3. (±2, ±1) 7. 433 iv. no solution 9b. n is odd or a multiple of 4.4. -9 8a. (-1, 2) 8b. (-1, 2) 10. 54, 56, 58, 59, 62 What About Math? - AUVSI Foundation Answer 1a. 0.4; b. 2; c. 1131.0 4a. 17; b. 421.6 7. 2000 2. 3.6415 5a. 14.16; b. 16.66 8. 12 3a. 1120; b. 1080 6. 11 min 2 sec 9. 2 1 10a. ; b. 60 2 What About Math? - TAF Answer 1. 0.15 4. 1000 7. 0.10 2. 42.5 5. 7 or 7.00 8. 7000 or 7000.00 3. 40 6. 2 9. 875 or 875.00 10. 7 What About Math? - CEA Answer 1. 711 3. 6.25 5. 172.6 2. 700 4. 1.17 6. 16:9 7. 450 MATHCOUNTS 2010-2011
  • 60 MATHCOUNTS 2010-2011
  • PROBLEM INDEXIt is difficult to categorize many of the problems in the MATHCOUNTS School Handbook. It isvery common for a MATHCOUNTS problem to straddle multiple categories and cover severalconcepts. This index is intended to be a helpful resource, but since each problem has been placedin exactly one category, the index is not perfect.Code: WU 1-7; 2 refers to Warm-Up 1, problem 7; difficulty rating 2. See the explanation ofdifficulty ratings on page 47.Algebraic Number Coordinate Percents & ProblemExpressions Theory Geometry Fractions Solving& Equations WU 1-4; 1 WU 3-10; 2 WU 1-7; 2 (Misc.)WO 1-6; 3 WU 1-8; 1 WU 5-5; 3 WO 1-9; 3 WO 1-1; 2WU 3-6; 2 WU 2-7; 2 WU 6-5; 3 WU 4-6; 2 WO 1-4; 2WU 3-9; 4 WO 1-5; 2 WO 3-6; 3 WO 2-10; 3 WO 2-1; 2WU 4-7; 4 WU 4-4; 4 WU 6-6; 2 WU 5-9; 4WU 5-6; 2 WO 2-2; 3WU 6-3; 2 WO 2-6; 2WU 6-7; 3 WU 6-1; 2 Logic Pattern GeneralWU 6-8; 4 WU 6-4; 3 WU 2-5; 1 Math*Stretch* WO 3-2; 2 WO 1-2; 3 Recognition WO 3-4; 3 WO 1-8; 1 WU 3-2; 2 WU 1-1; 1 WU 3-7; 2 WO 2-5; 2 WU 1-3; 1 WU 4-3; 1 WO 3-7; 3 WO 1-7; 2Sequences WU 5-4; 3 WU 3-1; 2 Solid WU 4-9; 3& Series Geometry WU 6-9; 3 WU 5-3; 3WU 1-6; 1 WU 2-9; 2 Proportional WU 6-2; 2WO 2-9; 3 WU 3-4; 3 Reasoning WO 3-1; 3WO 3-10; 3 WU 4-10; 2 Probability, WU 2-3; 2 WU 5-10; 4 Counting & WU 2-8; 2 WO 3-5; 4 Combinatorics WU 3-3; 2 StatisticsMeasurement WU 1-5; 1 WU 4-2; 2 WU 2-1; 1 WU 2-4; 1 WO 2-3; 2 WO 1-3; 2WU 1-2; 1 WU 2-6; 1 WU 5-8; 3 Plane WU 3-8; 2WU 1-9; 3 WU 3-5; 3 Geometry WU 4-1; 1WO 1-10; 4 WO 2-8; 4WU 4-8; 3 WU 1-10; 2 WO 2-4; 2 WU 2-2; 1 WU 5-1; 1 WU 5-2; 2 WU 2-10; 3 WO 3-3; 3 WU 5-7; 3 WU 4-5; 3 WO 3-8; 5 WO 2-7; 4 WU 6-10; 3 WO 3-9; 3 For an additional 200 problems, simply request Volume II of the MATHCOUNTS School Handbook by using the Registration Form provided on the last page of this book. MATHCOUNTS 2010-2011 61
  • 2010-2011 Club and Competition Program RegistrationWhy do you need this form?• Sign Up a Math Club and Receive the Club in a Box Resource Kit [FREE]• Register Your School to Participate in the MATHCOUNTS Competition Program [Competition Registration Deadline is Dec. 10, 2010]MATHCOUNTS CLUB PROGRAMThe MATHCOUNTS Club Program (MCP) may be used by schools as a stand-alone program, orit may be incorporated into preparation for the MATHCOUNTS Competition Program. The MCPprovides schools with the structure and activities for regular meetings of a math club. Depending onthe level of student and teacher involvement, a school may receive a recognition plaque or bannerand may be entered in a drawing for prizes. Open to schools with sixth-, seventh- and eighth-gradestudents, the Club Program is free to all participants.For the fourth year running, the grand prize in the Gold Level drawing is an all-expense-paid tripfor the club coach and four students to watch the National Competition as our VIP guests. The 2011Raytheon MATHCOUNTS National Competition will be held in Washington, D.C. All schools thathave successfully completed the Ultimate Math Challenge by the March 25, 2011 deadline will attainGold Level Status and will be entered in the grand-prize drawing.MATHCOUNTS COMPETITION PROGRAMMATHCOUNTS proudly presents the 28th consecutive year of the MATHCOUNTS CompetitionProgram, consisting of a series of School, Local (Chapter), State and National Competitions. Morethan 6,000 schools from 56 U.S. states and territories will participate in this unique mathematicalbee. The final 224 Mathletes® will travel to Washington, D.C., to compete for the prestigious title ofNational Champion at the 2011 Raytheon MATHCOUNTS National Competition, to be held May 5-8.Schools may register via www.mathcounts.org or on the following registration form.Do not hold up the mailing of your Volume II because you are waiting for a purchase order to beprocessed or a check to be cut by your school for the registration fee. Process your RegistrationForm through your system, but send in a photocopy of it directly to MATHCOUNTS withoutpayment. We immediately will mail your Club in a Box Resource Kit (which contains Volume IIof the MATHCOUNTS School Handbook) and credit your account once your payment is receivedwith the original Registration Form. Note: Registration fees are non-refundable.*SPECIAL COACHES COMMUNITY ON MATHCOUNTS.ORGRegistered Competition Program and/or Club Program coaches will receive special access to theCoaches Community on www.mathcounts.org. If you have not already done so, you will need tocreate a new User Profile and allow MATHCOUNTS to verify your User Profile request.Exceptions to the refund policy will be considered in the event of an extenuating circumstance. All refunds are provided at the discretion of*MATHCOUNTS.MATHCOUNTS 2010–2011 62
  • REGISTRATION FORM: 2010-2011 School Year (HB1) Mail or fax this completed form (with payment if choosing Option 2) to: MATHCOUNTS Registration Registration questions should be directed to the MATHCOUNTS Registration Office at 301-498-6141. P.O. Box 441, Annapolis Junction, MD 20701 Competition Registration (Option 2) must be postmarked by Dec. 10, 2010. Fax: 301-206-9789 Teacher/Coach’s Name _________________________________ Principal’s Name _______________________________________ School Name __________________________________________________________________  Previous MATHCOUNTS School School Mailing Address _______________________________________________________________________________________ City, State ZIP ________________________________________ County/District ________________________________________ If your school is overseas: My school is a  DoDDS OR  State Department sponsored school. Country _________________ Teacher/Coachs Phone __________________________________ MATHCOUNTS Chapter (if known)________________________________ Teacher/Coach’s E-mail _______________________________________________________________________________________  I would like to receive education-focused information from select MATHCOUNTS strategic partners. Alternative E-mail Address _____________________________________________________________________________________ Please ship my materials to the  School address above or the  Alternative mailing address below. Alternative Mailing Address ____________________________________________________________________________________ School Type?  Public  Charter  Religious  Private  Homeschool*  Virtual* *May not register as a team Academic centers or enrichment programs that do not function as students official school of record are not eligible to register. By checking a box above, you agree to adhere to this eligibility requirement. How did you hear about MATHCOUNTS?  MATHCOUNTS Mailing  Colleague  Conference  mathcounts.org  Other ___________________________ Total # of students in your math club: ______ Avg. # of problems used per month: ______ If you use MATHCOUNTS problems with your classes... How many students work MATHCOUNTS problems during class? ______ Avg. # of problems per month: ______  REGISTER MY MATH CLUB for the MATHCOUNTS Club Program, and send meOption 1 the Club in a Box Resource Kit with Volume II of the 2010-2011 MATHCOUNTS School Handbook, which contains over 200 math problems. (There is NO COST for the Club Program.) Please see pages 23-26 in Vol. 1 of the School Handbook or visit www.mathcounts.org/club for rules and procedures. By checking the box above, you agree that you have read said rules and procedures.  REGISTER MY SCHOOL for the MATHCOUNTS Competition and ClubOption 2 Programs, and send me the Club in a Box Resource Kit and Volume II of the 2010-2011 MATHCOUNTS School Handbook. Please see pages 10-21 in Vol. 1 of the School Handbook or visit www.mathcounts.org/competition for rules and procedures. By checking the box above, you agree that you have read said rules and procedures. Homeschools must register students as individuals. Competition Registration Fees: Rate Title I Rate*  Team Registration (up to 4 students) 1 @ $90 = $ ______ 1 @ $40 = $ ______  Individual Registration(s): # of students ____ (max. of 6) @ $25 each = $ ______ @ $10 each = $ ______ By completing this registration form, you attest to the school administration’s Total Due = $ ______ Total Due = $ ______ permission to register students for MATHCOUNTS under this school’s name. * Principals signature is required for verifying that school qualifies for Title I fees: For your school to qualify for the Title I reduced fee rate, at least 40% of the student body must be enrolled in the Free and Reduced Lunch Program. Refer a colleague and save... Turn the page to learn how you can save on your registration fees! Payment:  Check  Money order  Purchase order # _____________ (p.o. must be included)  Credit card Name on card: _______________________________________________________________  Visa  MasterCard Signature: _________________________________________ Card #: ___________________________________ Exp. __________ Make checks payable to the MATHCOUNTS Foundation. Payment must accompany this registration form. All registrations will be confirmed with an invoice indicating payment received or payment due. Invoices will be sent to the school address provided. If a purchase order is used, the invoice will be sent to the address on the purchase order. Payment for purchase orders must include a copy of the invoice. Registration confirmation may be obtained at www.mathcounts.org. MC1011-HB1
  • MATHCOUNTS Referral ProgramHelp us spread the word about MATHCOUNTS Programs! Refer a middle school not currently participating in theMATHCOUNTS Competition Program.Over 90% of MATHCOUNTS coaches said they would refer MATHCOUNTS to a colleague. Why not take advantage of areferral and get a portion of your registration fees reimbursed? MATHCOUNTS is once again utilizing the referral program toencourage schools to register for the Competition Program.If you know of a school that is not participating in the MATHCOUNTS Competition Program and should be, please enter the schoolinformation in the last section below. You may also send the coach to our registration link (www.mathcounts.org/competition). If theschool you refer registers a team of four for the Competition Program, your school and the referred school each will receive a $10refund on your teams registration fees!* With enough successful referrals, your school could be reimbursed for its entire registrationfee! To qualify as a referred school, a school may not have participated in the MATHCOUNTS Competition Program for theprevious two school years.Incentive for MATHCOUNTS Club Program SchoolsSchools that have participated only in the MATHCOUNTS Club Program (and not in the Competition Program) may also takeadvantage of the incentive program. If you choose to register a team of four for the MATHCOUNTS Competition Programthis school year, you will receive a refund of $10. Please check the appropriate box below indicating that (1) you were aMATHCOUNTS Club Program-only school in the past and (2) you are registering a team of four for the Competition Programthis year.Please check all that apply: Previous MATHCOUNTS Club Program-only school and would like to receive a $10 refund on my registration. NEW school that was REFERRED to MATHCOUNTS and would like to receive a $10 refund on my registration.If you were referred, please provide the referring school name and teachers name below so we may give proper credit.School Name: ___________________________________________________________________________________Teachers Name: _________________________________________________________________________________ Returning school REFERRING another school.Please list the school(s) you wish to refer. When a school registers, you will receive your $10 refund. 1 Teachers Name ___________________________________________________________________ School Name _____________________________________________________________________ School Mailing Address (if known) ____________________________________________________ City/State ________________________________________________________________________ E-mail __________________________________________________________________________ 2 Teachers Name ___________________________________________________________________ School Name _____________________________________________________________________ School Mailing Address (if known) ____________________________________________________ City/State ________________________________________________________________________ E-mail __________________________________________________________________________ 3 Teachers Name ___________________________________________________________________ School Name _____________________________________________________________________ School Mailing Address (if known) ____________________________________________________ City/State ________________________________________________________________________ E-mail __________________________________________________________________________*Refund not to exceed total amount of registration fees. Refunds will be distributed before the end of the MATHCOUNTS program year.