MATHEMATICAL ASSOCIATION of AMERICA | American Mathematics Competitions AMC 10A 11 th Annual Contest AMC 12A 61 st Annual Contest AMC 10/12 TEACHERS’ MANUAL Instructions and Reporting Forms for School Contest Managers Tuesday, February 9, 2010 Please read this booklet completely upon receipt. Exams must be administered over a continuous 75-minute period to all students at the same time. DATES OF THE 2010 CONTESTS AMC 10/AMC 12 - Tuesday, February 9 &/orWednesday, February 24, 2010 AIME - Tuesday, March 16 orWednesday, March 31, 2010 USAMO - Tuesday &Wednesday, April 27-28, 2010 IMO - Astana, Kazakhstan, July 6-14, 2010
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TEACHERS’ MANUAL · 2013. 7. 12. · MATHEMATICAL ASSOCIATION of AMERICA | American Mathematics Competitions AMC 10A 11th Annual Contest AMC 12A 61st Annual Contest AMC 10/12 TEACHERS’
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MATHEMATICAL ASSOCIATION of AMERICA | American Mathematics Competitions
Table of ContentsImportant Procedures ........................................................................................................................... 4 I. Preliminary Instructions for Administering the AMC 10/AMC 12 ............................................... 4 II. Instructions For The Day of The AMC 10/AMC 12 ..................................................................... 4 III. Eligibility ..................................................................................................................................... 5
International Students & Non-Citizens in USA Schools ..............................................................................................5
IV. Team Score Identification .......................................................................................................... 5 V. School Results ............................................................................................................................ 5 VI. Policy Statements ....................................................................................................................... 6
Early Administration .......................................................................................................................................................................6Official Administration ...................................................................................................................................................................6Unofficial Administration ...............................................................................................................................................................6Contest Proctoring .........................................................................................................................................................................6One Contest per Date ....................................................................................................................................................................6Students with Visual or Learning Disabilities .................................................................................................................................6Sickness and Other Special Situations ..........................................................................................................................................6Questionable Scores .....................................................................................................................................................................6Follow-up Inquiries and Reexamination ........................................................................................................................................6Policy for Changes ..........................................................................................................................................................................6Refund/Credit Policy ......................................................................................................................................................................6Request for Student Names Policy ................................................................................................................................................6
VII. AIME Instructions ........................................................................................................................ 7AIME Rules for AMC 10/AMC 12 ....................................................................................................................................................7AIME School Manager ...................................................................................................................................................................7Second AIME Testing Date ............................................................................................................................................................7
VIII. USAMO Participant Selection .............................................................................................................. 7 IX. The MOSP Program ..................................................................................................................... 7 X. Regions of the AMC 10/AMC 12 .................................................................................................. 7 XI. Intramural and National Awards ................................................................................................ 8
Intramural Awards for Students .....................................................................................................................................................8Committee and Donor Awards for Students .................................................................................................................................8National School Awards ................................................................................................................................................................8Awards for Schools not Receiving a Cup .....................................................................................................................................8School Intramural Awards..............................................................................................................................................................8
XII. Contest A Certification ............................................................................................................... 9Certification by the Contest Manager: ...........................................................................................................................................9Certification by the Principal, official or person with comparable title: .......................................................................................9Service Questionnaire ..................................................................................................................................................................10XIII. Additional Forms used .........................................................................................................................................................11Additional Bundles Form .............................................................................................................................................................11Proof of Intent to Pay ....................................................................................................................................................................11Rescoring Request Form ...............................................................................................................................................................12Method of Payment: ....................................................................................................................................................................12B Contest Registration Form .........................................................................................................................................................13Publications Order Form ..............................................................................................................................................................14
XIV. Classroom Accessories ............................................................................................................. 15Publicity .......................................................................................................................................................................................15Letter to Parents (for Reproduction) ............................................................................................................................................16Hallway Promotional Flier ............................................................................................................................................................17AMC 10 Participation Certificates ................................................................................................................................................18AMC 12 Participation Certificates ................................................................................................................................................19Facsimile of AMC 10 Front Cover .................................................................................................................................................20Facsimile of AMC 12 Front Cover .................................................................................................................................................21AMC 10 Student Practice Questions ...........................................................................................................................................22AMC 12 Student Practice Questions ...........................................................................................................................................27
2. HandoutthestudentAnswerFormsandhavethestudentscompletethenon-answersectionsonthefrontandback.Have students use their full legal name, no nicknames or abbreviations. Have them pay special attention to marking their name and address accurately. RemindthemthatstudentnameslistedintheNationalSummarycomefromthisform.TheAMCOfficewillnotdoanyeditingoftheinformation.
4. Announce that the students may use scratch paper,graph paper, ruler, compass, and/ or a protractor.CALCULATORSARENOTPERMITTED.Wecannotassure that all participants in the contest have the sameaccesstotechnology.Inaddition,itisbecomingincreasinglydifficulttoensurethatstudentswillnotusecalculatorsascommunicationdevices.Forthesereasons,theAMCdoesnot allow theuseof calculatorson its examinations.Noproblemsonthecontestwillrequiretheuseofacalculator.
Important ProceduresFormat
TherearetwoofficialdatesfortheAMC10&AMC12.GivetheAMC10andAMC12atthesametimewithineachpar-ticipatingschoolonTuesday,February9,2010(AMC10-A&AMC12-A),orWednesday,February24,2010(AMC10-B&AMC12-B)inaconvenient75-minuteinterval,preferablyinthemorning.Allfourcontestsconsistof25questions.N O T E : Eachcorrectanswerscores6points, a blank scores 1.5 points andanincorrectanswerscores0points.TheAMC10andAMC12haveseveralquestionsincommon.Thestudentsingrades10andbelowshouldchoosebetweentheAMC10andAMC12.Studentsingrades11and12mayonlytaketheAMC12.Allrulesandawardsapplytobothcontestsforallschoolsandstudents.Anystudentwhomissedtheexammaytakeitunofficially,andwewillbehappytogradeit.Studentsmaytakethecontestbookletshomewiththemthedayofthecontest.
School FlierWeareagainprovidingaflieryoucanusetopromotethecontestswithinyourschool.Itisincludedonpage17ofthisManual.Ithasspaceleftforyoutoaddschoolinformation,suchasthelocationofthecontests,andwhotocontactlocallyformoreinformation.Ifyouwouldlikeacolorversion,visitourwebpagetodownloadapdfversionoftheflieryouneed:www.unl.edu/amc/
6. Encourageparticipationbystudentswhohavenottakenthecontestbefore,especiallyyoungerstudents,butmakesurestudentsknowwhattoexpect.Letthemknowabouttypicalscoresatyourschoollastyearattheirgradelevel. Showstudentsthenationalstatistics inourNationalSummaryof Results and Awards from last year. Tell them to setappropriategoalsforthemselves.
8. Make sureyouhavearranged to followall the rulesandprocedures in this manual. Early administration ofthe contests is never permitted, and will lead todisqualification. To assure the validity of the results wereport,wetakeourrulesveryseriously.
a. Inform the students that the contest may not bediscussedwithanyoneoutsideofyourschoolverbally,via email, phone, text, web, social networking site,copier or media of any type until after the contestperiod.
b. Studentsmaykeepthecontestbookletsandtakethemhome.
III. EligibilityAMC 12 Eligibility—Astudent inaprogram leading toahighschooldiploma,andunder19.5yearsofageonthedayofthecontest.AMC 10 Eligibility—Astudentinaprogramleadingtoahighschooldiploma,andunder17.5yearsofageonthedayofthecontest,andnotenrolledingrades11or12orequivalent.Please note:students in grades 11 & 12 cannot take the AMC 10.However,studentsingrades9&10maychoosewhichcontesttheytake.International Students & Non-Citizens in USA SchoolsUSandCanadianCitizensandInternationalStudentsresidingintheUnitedStates(withqualifyingscores)areeligibletotaketheUSAMO.Studentslearning“EnglishasaSecondLanguage”(ESL)mayuseabooknontechnicaldictionarybetweentheirnativelanguageandEnglish.Astudentmayusethedictionaryonlythefirsttimethathe/shetakestheAMC10/AMC12.Thedictionarymustbegiventotheschoolcontestmanagertoexamineandretainforthe24-hourperiodprecedingthecontest.Theproctormustan-nouncetootherstudentsthatthestudent(s)has/havebeengivenspecialpermissiontousethedictionaryduringthecontest.
IV. Team Score IdentificationTORECEIVEOFFICIALTEAMSTATUSANDAWARDS,ASCHOOLMUSTHAVEATLEASTTHREEPARTICI-PANTSONACONTESTDATE.Theteamscoreforaschoolisthesumofitsthreehigheststudentscoresandwillbedeter-minedbytheAMCOffice.ThescoreofUSAandCanadianteamsisusedtodetermineNationalSchoolawards.Inaddition,theteamscoreisusedtoselectthetop60schoolstoidentifyteacherswhoareeligible for theEdythMaySliffeAward forDistinguishedHighSchoolTeaching.
V. School ResultsTheAMCofficewillsendresultsbyemail(ifavailable)andfirstclassmailassoonastheanswerformsarescored.Ifyouhavenotreceivedyourresultsfromourofficewithin30daysaftertheAMC10/AMC12pleasecontactustoverifythatyouranswerformswereinfactreceived.Ifyouwouldliketoreceiveyourresultsbye-Mail,andhavenotpreviouslysentusyouremailaddress,sendamessage,includingyourname,schoolname,address,andCEEB#to:
5. Inform the students to, “Carefully read instructions 3and4onthecontestcover.”(seepages20and21ofthismanual). The AMC 10/AMC 12 has a scoring systemwhichhasimportantconsequencesforguessing.Unlessyouarefairlysureoftheanswer,itisbettertoleaveaquestionunansweredthantoguess.Sixpointsaregivenforacorrectanswer,1.5pointsforablankanswerand0pointsforanincorrectanswer.Ifastudentcanreducetheproblemtothreepossibleanswers,itisadvantageoustoguessoneofthethreepossibleanswers.Ifastudentcanonlyreduceto4possibleanswersbyeliminating1ofthepossibilities,thenitisnotadvantageoustoguess.
8. Students who finish the contest early may be dismissedprovided theywillbeunder the supervisionof a teacherduringtheremainderofthecontestperiod.
9. You(andotherteachers,iftherearemanyparticipants)shouldproctorcontinuallyasyouwouldforanyimportantcontest.Students whose eyes wander should be warned; studentscaughtcopyinganswersorcollaboratingmustbedisqualified.Trytoprovideasquietanenvironmentaspossible.
12. Pleasedonotgradetheanswerforms.TheyaretobesenttotheAMCofficeforgrading.Studentsmaycircletheiranswers on the contest booklet. However, the officialanswerswillbetheonesblackenedontheanswerform.
Official AdministrationTheAMC10-A/AMC12-AwillbegivenofficiallyonTuesdayFebruary9,2010.TheAMC10-B/AMC12-BwillbegivenofficiallyonWednesday,February24,2010.Onlyofficialpar-ticipants,theirschoolandtheirteacherareeligibleforNationalAwards. In addition, official participants are eligible for allintramuralawardsandforparticipationintheAIME.
Unofficial AdministrationIfyouareunable togiveContestAonTuesday,February9,2010because: a. yourschoolisclosed, b. yourschoolhasanacademicconflict, c. theclassperiodshavebeenshortenedduetoanassembly
orotherreason,thenyoumaygive thesecondversionof thecontests (AMC10-B/AMC 12-B) on the second official day, Wednesday,February24,2010(SeeContestBRegistrationFormonpage13).Youmaystilltakeeitherexamunofficiallyonlaterdates,but those contests will not be eligible for state and nationalawardsandwillnotbeeligibleforparticipationintheAIME.Unofficialparticipantsarestilleligibleforintramuralawards.It is important tonote that theonlydayseligible forofficialparticipationarethetwoofficialContestdays:February9,andFebruary24,2010.
One Contest per DateA student may take only one exam on a given day but canparticipateonbothcontestdatesiftheschoolregistersforbothcontests.Thehigherscorewillbeusedforindividualawards.
Students with Visual or Learning DisabilitiesTheAMC10/AMC12timelimitsetbytheCAMCforstudentswhoarevisuallyimpairedorlearningdisabledis120minutes.Ateacheroraschooladministratormayreadthequestionstothestudentandmarktheanswersasdirectedbythestudent.ThecostofaBrailleorLargePrintexamis$7.00shippingandhandlingplus$1.60perexamforboththeAMC10andAMC12.They are mailed separately and must be ordered no later than three weeks before the test.
Sickness and Other Special SituationsAstudentwhoissickoronafieldtriponthefirstcontestdaymayregisterandtakethealternateContestBonWednesday,February24.YOUMUSTREGISTERFORCONTESTBifyouhavenotalreadydoneso.(seepage13foraRegistrationForm).
Follow-up Inquiries and ReexaminationTheresultsofthiscontesthelpstoidentifystudentswithunusualmathematicalability.Toassurethatthispurposeisserved,theCAMC reserves the right to retest students before decidingwhether to grant official status to individual or team scores.Reexaminationwillberequestedwhen,afteraninquiry,thereisareasonablebasistodisbelieveascore.Officialstatuswillnotbegrantedifastudentorschooldoesnotagreetoarequestedretesting.
Policy for ChangesTheCAMCmay,fromtimetotime,changetheprogramrules,regulations,awardsandconditionsofparticipationinwholeorinpart.Wheneverpossibleyouwillbenotifiedofthesechangesaheadoftime.
Request for Student Names PolicyThefollowingstatementappearsonthestudentanswerformsfortheAMC10andAMC12:
The American Mathematics Competitions (AMC) receives re-quests from educational institutions and organizations for the names, addresses and grade levels of high scoring AMC 12 (or AMC 10) students. The optional personal data on ethnic origin and gender is used for recruiting and academic purposes..
Blacken this circle if you give the AMC permission to release this information to these organizations. (Your score will not be af-fected if you do not blacken the circle.)
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Receivinginformationisan“opt-in”decisionforeachindividualstudent.TheAMChandlesrequestsfrominstitutionsandorganizationsonacase-by-casebasisandevaluateseachindividuallyforap-propriateness.Weprovidelegitimateeducationalinstitutionsofalllevels,bothsecondaryandcollegiate/universitylevel,withone-timeuseofselectednamesandaddressesforpostalmail-ings.WealsoprovideprofessionalandscholarlyorganizationssuchasthoselistedascontributorstotheAMCwithone-timeuseofnamesandaddresses forpostalmailings,generally forprofessionalorcareerinformation.The only information we provide is the name, address, city,state,andzipcodenecessaryforapostalmailing.Wedonotlistindividualscoresorawards.
VII. AIME InstructionsThe28thannualAmericanInvitationalMathematicsExamina-tion(AIME)willbeheldonTuesday,March16,2010withasecondalternateexamgivenonthealternatedateofWednesday,March31,2010.Thesearetheonlydaystheexammaybetakenofficially.Youmaygivetheexamforpractice(unofficially),aftertheofficialdates.Wewillbepleasedtogradeitforyoubutyourstudentswillnotbeeligible to take theUSAMO/USAJMO.Thecontest isprovidedfreeofchargetoall thosetakingtheexamonthefirstdate,howeverthosetakingtheexamonthesecondalternatedatewillbechargedaprocessingfeetocoverexpediteddelivery.
AIME Rules for AMC 10/AMC 12Studentswhoscore100oraboveorfinishinthetop5%onthisAMC12orstudentswhoscore120oraboveorfinishinthetop1%onthisAMC10willqualifytotakethe28thannualAmeri-canInvitationalMathematicsExamination(AIME)onTuesday,March16,2010orWednesday,March31,2010.PLEASEreadthe followingparticipation rules to your students as soon asyoureceivetheAMC10/AMC12packagesopotentialAIMEstudentswillbeabletoplanaccordingly.
AIME School Manager1. TheAMCofficewill includeallmaterialsrelatingtothe
examination (including instructions for the exam) withyourAMC10/AMC12results.
anAIME solutionpamphlet, and a list of studentswhoqualifyfortheUSAMO.
7. All AMC 10/AMC 12 procedures for disqualification,follow-upinquiriesandreexaminationapplytotheAIMEasappropriate.
8. If you have students who you feel may qualify for theAIMEpleaseorderprioryearAIMEexamsandsolutionsfor practicenow. This way youwill have these practice
Second AIME Testing DateSituationsinwhichastudentmaytakeasecondversionoftheAIMEtobeheldonWednesday,March31,2010,keepingtheirUSAMOeligibilityopenare:1. SchoolisclosedonMarch16(i.e.springbreak,weather).2. Studentisoutofschooltheentiredayduetoattendanceat
anacademic/schoolrelatedevent.3. StudentisillandcannotattendschoolonMarch16.TherewillbeaprocessingfeeforthesecondAIMEasfollows:1-10students=$25,11+students=$50.Wewillneedyourpayment before the answer forms can be graded. A specialenvelopeandpaymentformwillbeincludedwithyourAIMEmaterial,ifyouhaveAIMEqualifiers.AllAIMEanswerformsmustarriveintheAMCofficebyApril2,2010.EmailrequestsforthesecondAIMEmaybesentto:
The USA Mathematical Olympiad (USAMO) is a two day,nine-hour,six-question,essay-proofexamination.SelectionfortheUSAMOwillbeexplainedintheAIMETeacherManual.Thegoalistoselectabout270ofthetopscorersfromthepriorAIMEandAMC12A,AMC12BtoparticipateintheUSAMO,and230topscorersfromtheAIMEandAMC10AandAMC10BconteststoparticipateintheUSAJMO.The USAMO/USAJMO is scheduled for Tuesday andWednesday,April27&28,2010atyourschool.Ifyoufeelyoumayhaveaqualifier,pleasearrangeforaspaceandproctorforthesedates.Thetop12scoringstudentsontheUSAMOwillbeinvitedtoattendanawardceremonyheldinWashington,D.C.,onJune7,2010.
IX. The MOSP ProgramTheMathematicalOlympiadSummerProgram(MOSP)isa3-week,academicchallengedesignedtobroadenparticipants’viewofmathematicswhilefosteringexcitementtowardfurthermathstudy.ItisheldeachyearattheUniversityofNebraska-LincolninJune-July.Invitedstudentsincludethetop12USA-MOwinners,12-18high-scoringUSAMOparticipants,whoarecurrentjuniorsandbelow,andanadditional25USAMO/USAJMOparticipants,grantfundingpermitting.Watchforfurtherdetailstobeannouncedinthe2010AIME/USAMOTeachers’ManualandontheAMCwebsiteatwww.unl.edu/amc.
X. Regions of the AMC 10/AMC 12TheUSAandCanadaarepartitionedintothefollowingregions.NationalAwardsaregiventoaminimumof10highscoringstudents and5 schools (basedon the team score) in eachoftheseregions.
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Region0 Connecticut, Maine, Massachusetts, New Hampshire,
4. NCTM Book* — A book, donated by the NationalCouncilofTeachersofMathematics, isawardedtothreehigh-scoringparticipantsineachoftheelevenregions.
5. Mu Alpha Theta Book* — Abook,donatedbyMuAlphaTheta,isawardedtooneparticipantwithahighscoreineachoftheelevenregions.
6. AMATYC Review* — A one-year subscription to theAmerican Mathematical Association of Two-Year Colleges Review is awarded to one student in each of the elevenregions.
7. CAMC Problem Book — OneProblem BookdonatedbyTheAmericanMathematicsCompetitions,isawardedtothetop-scoringstudentingrade9orlessineachoftheelevenregions.
National School AwardsTheAMCdivides theUnitedStatesandCanada intoelevenregions.Ineachoftheseregionsthefiveschoolswiththehigh-estteamscores(sumofthehighestthreescoresbyparticipants)arerecognizedbyDonororCommitteeAwards.1. Charles T. Salkind Memorial Silver Cup — This silver
2. William H. Fagerstrom Memorial Silver Cup — Thissilvercupisawardedtotheschoolwiththesecondhighestteamscore(inanyregion).
3. Committee Bronze Cups — IneachoftheregionsinwhichasilvercupisnotawardedtheCommitteeontheAmericanMathematicsCompetitionsprovidesaBronzeCuptotheschoolwiththehighestteamscore.Awards for Schools not Receiving a Cup
4. CAMC Mathematics Books* — In each of the elevenregions,fivebooksaredonatedbytheCommitteeontheAmericanMathematicsCompetitionstooneschoolhavingahighteamscore.
5. W. H. Freeman Books* — Ineachoftheelevenregions,asetofbooks,donatedbyW.H.FreemanandCompany,SanFranciscoisawardedtooneschoolhavingahighteamscore.
XII. Contest A CertificationTheAMC10andAMC12mustbeadministeredbyateacheroranadultnotrelatedtoanyoftheparticipants.Theadministrationofthecontestmusttakeplaceinapublicbuilding(e.g.school,library,church).PleasesendallAnswerFormsfromyourschoolorgroupatonetime.TheContestManagerandthePrincipal,VicePrincipal,orHeadmastermustsignthisformwhichistobereturnedwithyourstudentAnswerForms.
Certification by the Contest Manager:Icertifythatthefollowingstatementsaretrueorthat,ifthereareanyexceptions,Ihavecheckedtheboxatthebottomofthispageandhavelistedthemonaseparatepage.Iunderstandthattheabsenceofeithersignaturefromthisform,andaconsiderationoftheexceptionsmayresultinDISQUALIFICATIONofallscoresfromourschool.
USED-(seeSectionI.Item4).7. Participantshadexactly75minutesworkingtime.(Seepage6forStudentDisabilitiesPolicy)8. Nostudentswerepermittedtoproctororgradethecontest.9. The instructions relating to the opening of the “Complimentary Solutions Envelope” and/or Solution Packets were
Questionnaire to Help Us Serve You BetterThankyouinadvancefortakingthetimetoanswerthesequestionsforus.Answertothebestofyourability,andifyoudon’tknow,justgiveusa“bestestimate”.PleasefillinyouranswersintheovalsprovidedonthebackoftheschoolIDform.Thankyou!
ContestABundlesoften.................#_________@$16/bundle=............$__________ ASolutionsSetsoften(optional).......#_________@$6/set=....................$__________To order either the 2009-2010 AMC 10/12 Math Club Package or the 21st Century CD with pdf’s of all contests 2001-2009, download the Publications Order Form from the AMC website at www.unl.edu/amc/
Proof of Intent to PayThisdocumentisintendedtobeusedinlieuofpre-paymentwhencallingorfaxinginanorder.Pleaseindicateifyouwishtobebilledorwillbesendinga“checkinthemail”(tobereceivedwithin2weeksoforderoryouwillbebilled).Mailordersnotwishingtobebilledshouldincludeacheckwhenreturningthisform.Thepersonwhosignsthisformmustbeauthorizedtopaytheorderthatisplacedbytheteacher.
_____________________________________________ E-mail (for sending results) Please Print Clearly (Circle appropriate responses, below) Learning Center,Type of Public Private Home Outside SchoolGroup : School School School Class/GroupGrades: PreK K 1 2 3 4 5 6 7 8 9 10 11 12 13School Size: 0-200 201-400 401-1000 1001+
PAYMENT OPTIONS Do not send payment alone. The Registration Form must be included with your payment option. Checks sent without ap-propriate registration information cannot be processed and will be returned to sender.Check P.O. # ___________ Visa Master CardTerms - Payment in U.S. Funds only. Make checks payable to: MAA American Math Competitions
_____________________________________________ Name (Please Print) Give an address for mailing the charge receipt in “Billing Address” above
( )
Step 1 - Pick registration by current date, if you are Canadian,US nonadjacent, or Int’l please include additional shipping, below.Registration - Required One fee covers 10B/12B Registration and US contiguous 48 states shipping (choose 1) Registration/Expedited Shipping .....................$ 50.00 REQUIRED for Short Shipping window 2-day shipping OR Registration/Overnight Shipping ...................$ 60.00 REQUIRED after January 22 (Jan. 23--Feb 19). Overnight shipping U. S. Registration subtotal ................................. $______
Step 2 - Add extra shipping if not in US contiguous 48 statesAdditional Shipping Canada, Alaska, Guam, Hawaii, & Puerto Rico Additional shipping.. ..............$10.00B .. $______ OR
Step 3 - Indicate your selection of contests (and solutions)CONTEST BUNDLES of Ten -- AMC 10B -- Ten contests per bundle English 10B contests #____x $16/bundle .................. $_____ + Spanish 10B contests #____x $16/bundle .................. $_____ + French 10B contests #____x $16/bundle.................. $_____ + (optional) English 10B Solutions #____@ $ 6/set ....$_____ + AMC 12B -- Ten contests per bundle English 12B contests #____x $16/bundle .................. $_____ + Spanish 12B contests #____x $16/bundle .................. $_____ + French 12B contests #____x $16/bundle.................. $_____ + (optional) English 12B Solutions #____x $ 6/set ......$_____ = Contest Bundles subtotal .................$____ B ... $______Braille & Large Print Contests Priced individually, not available for International Schools: Braille AMC 10B #___ x 1.60/each ......................$_____ + LgPrint AMC 10B #___ x 1.60/each ......................$_____ + Braille AMC 12B #___ x 1.60/each ......................$_____ + LgPrint AMC 12B #___ x 1.60/each ......................$_____ + + Shipping (sent separately) ..........................$ 7.00 =Braille/Lg Print subtotal order by Feb 1 ..$_____ ..... $______
Step 4 - Decide if you want to include a Math Club PackageAMC 10/12 Math Club Package: Study Guide, CD, Web materials, see brochure. Sent separately; available Fall 2009With Shipping for Contiguous US 48 states ..... $ 25.00 = $______ Additional postage required for International/Overseas Addresses, please email AMC office: [email protected]
Step 5 - Add sub-totals for steps 1,2, 3, and 4
TOTAL ORDER . . . . . . . . . . . . . . . . .$______MUST BE PAID IN US FUNDS
MAA American Mathematics CompetitionsATTN: AMC 10/12 Registration
P.O. Box 81606Lincoln, NE 68501-1606
All orders Non-Refundable once shipped.Mail along with your payment or Purchase Order to
or fax to402-472-6087
2010 Registration Form - 10/12 B Wednesday,February 24, 2010
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Office use only
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The Mathematical Association of AmericaAmerican Mathematics Competitions
For additional publications please visit the MAA online store https://www.maa.org/EbusPPRO/
NEW!! MAA AMC T-Shirt — $15 each
Sports the 2008-2010 theme of “6 Decades of the American Mathematics Competitions.”
Black, 100% cotton, full-color design on back, blue and white logo on front left chest.
to:AMERICANMATHEMATICSCOMPETITIONS• PAYMENTINU.S. FUNDSONLY.• All 50 U.S states, only:Noadditionalfeeforshipping.• All other including Canadian & International Orders: Please include e-mail or fax number. AMC will send you a total order confirmation
with the shipping charge added to your order. When paying by Visa or MasterCard we need your charge card number, expiration date, and name of card holder. Unpaid Purchase Orders will be accepted. Payment in U.S. FUNDS ONLY.
NEW this year CD’s with all the Contests — $20 each1. AJHSME & AMC 8 (1985-2007) + worksheets (1999-2007)2. AHSME 1 (1950-1974)3. AHSME 2 (1975-1999)
OR . . . Buy the Math Club Package! — $25 eachThis includes a book with teaching ideas and club activities,and a CD which contains contests and solutions from the last10 years of the AMC 8 (98-07), AMC 10 A&B (00-08), and AMC12 A&B (99-08), plus AIMEs (01-08), and USAMOs (01-08). All the questions and solutions fromthese years are available in PDFform. We have also includedthe 275 AMC 8 worksheetsand 395 AMC 10 andAMC 12 worksheets wehave developed thus far.
sixty years of the American Mathematics Competitions
EachyeartheAMC10andAMC12areontheNational Association of Secondary School Principals Advisory List of Contests and Activities.TheAMCContestsaresponsoredbytheMathematicalAssocia-tionofAmerica,andare considered to be such a valuable stimulus to student interest in mathematics that 20 professional societies and organizations, including the NationalCouncilofTeachersofMathematics
andthose represented below, support the contests with financial contributions.
The Mathematical Association of AmericaAmerican Mathematics Competitions
CERTIFICATEAwarded to
for participating in the
American Mathematics Contest 10(AMC 10)
2010 ___________________________________ ___________________________________ Director Chair American Mathematics Competitions AMC 10 Subcommittee
The Mathematical Association of AmericaAmerican Mathematics Competitions
CERTIFICATEAwarded to
for participating in the
American Mathematics Contest 10(AMC 10)
2010 ___________________________________ ___________________________________ Director Chair American Mathematics Competitions AMC 10 Subcommittee
AMC 10 Participation Certificates
The Mathematical Association of AmericaAmerican Mathematics Competitions
CERTIFICATEAwarded to
for participating in the
American Mathematics Contest 12(AMC 12)
2010 ___________________________________ ___________________________________ Director Chair American Mathematics Competitions AMC 12 Subcommittee
The Mathematical Association of AmericaAmerican Mathematics Competitions
CERTIFICATEAwarded to
for participating in the
American Mathematics Contest 12(AMC 12)
2010 ___________________________________ ___________________________________ Director Chair American Mathematics Competitions AMC 12 Subcommittee
AMC 12 Participation Certificates
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Facsimile of AMC 10 Front Cover
Tuesday, FebruAry 9, 2010
INSTRUCTIONS1. DONOTOPENTHISBOOKLETUNTILYOURPROCTORTELLSYOU.2. This is a twenty-five questionmultiple choice test.Eachquestion is followedby answersmarkedA,B,C,DandE.Onlyoneoftheseiscorrect.3. Mark your answer to each problem on the AMC 10 Answer Form with a #2 pencil.Check the blackened circles for accuracy and erase errors and straymarks completely.Onlyanswersproperlymarkedontheanswerformwillbegraded.4. SCORING:Youwill receive 6points for each correct answer, 1.5points for each problemleftunanswered,and0pointsforeachincorrectanswer.5. No aids are permitted other than scratch paper, graph paper, rulers, protractors, and erasers. No calculators are allowed. No problems on the test will require the useofacalculator.6.Figuresarenotnecessarilydrawntoscale.7. Before beginning the test, your proctor will ask you to record certain information ontheanswerform.8. When your proctor gives the signal, begin working on the problems. You will have75 minutestocompletethetest.9. Whenyoufinishtheexam,sign your nameinthespaceprovidedontheAnswerForm.
MaTheMaTical associaTion of aMerica American Mathematics Competitions
The Committee on the American Mathematics Competitions (CAMC) reserves the right to re-examine students before deciding whether to grant official status to their scores. The CAMC also reserves the right to disqualify all scores from a school if it is determined that the required security procedures were not followed.
Students who score 120 or above or finish in the top 1% on this AMC 10 will qualify to take the 28th annual American Invitational Mathematics Examination (AIME) on Tuesday, March 16, 2010 or Wednesday, March 31, 2010. More details about the AIME and other information are on the back page of this test booklet.Thepublication, reproductionorcommunicationof theproblemsor solutionsof theAMC10during theperiodwhenstudentsareeligibletoparticipateseriouslyjeopardizestheintegrityoftheresults.Disseminationviacopier,telephone,e-mail,WorldWideWebormediaofanytypeduringthisperiodisaviolationofthecompetitionrules.Afterthecontestperiod,permissiontomakecopiesofproblemsinpaperorelectronicformincludingpostingonweb-pagesforeducationaluseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforprofitorcommercialadvantageandthatcopiesbearthecopyrightnotice.
21
Facsimile of AMC 12 Front Cover
Tuesday, FebruAry 9, 2010
INSTRUCTIONS1. DONOTOPENTHISBOOKLETUNTILYOURPROCTORTELLSYOU.2. This is a twenty-five questionmultiple choice test.Each question is followedby answersmarkedA,B,C,DandE.Onlyoneoftheseiscorrect.3. Mark your answer to each problem on the AMC 12 Answer Form with a #2 pencil.Check the blackened circles for accuracy and erase errors and straymarks completely.Onlyanswersproperlymarkedontheanswerformwillbegraded.4. SCORING:Youwill receive 6points for each correct answer, 1.5points for each problemleftunanswered,and0pointsforeachincorrectanswer.5. No aids are permitted other than scratch paper, graph paper, rulers, protractors, and erasers. No calculators are allowed. No problems on the test will require the useofacalculator.6.Figuresarenotnecessarilydrawntoscale.7. Before beginning the test, your proctor will ask you to record certain information ontheanswerform.8. When your proctor gives the signal, begin working on the problems. You will have75 minutestocompletethetest.9. Whenyoufinishtheexam,sign your nameinthespaceprovidedontheAnswerForm.
MaTheMaTical associaTion of aMerica American Mathematics Competitions
The Committee on the American Mathematics Competitions (CAMC) reserves the right to re-examine students before deciding whether to grant official status to their scores. The CAMC also reserves the right to disqualify all scores from a school if it is determined that the required security procedures were not followed.
Students who score 100 or above or finish in the top 5% on this AMC 12 will qualify to take the 28th annual American Invitational Mathematics Examination (AIME) on Tuesday, March 16, 2010 or Wednesday, March 31, 2010. More details about the AIME and other information are on the back page of this test booklet.Thepublication, reproductionorcommunicationof theproblemsor solutionsof theAMC10during theperiodwhenstudentsareeligibletoparticipateseriouslyjeopardizestheintegrityoftheresults.Disseminationviacopier,telephone,e-mail,WorldWideWebormediaofanytypeduringthisperiodisaviolationofthecompetitionrules.Afterthecontestperiod,permissiontomakecopiesofproblemsinpaperorelectronicformincludingpostingonweb-pagesforeducationaluseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforprofitorcommercialadvantageandthatcopiesbearthecopyrightnotice.
Triangle ABC has a right angle at B. Point D is the
foot of the altitude from B, AD = 3, and DC = 4.
What is the area of �ABC ?
A
B C
D
3
4
(A) 4√
3 (B) 7√
3 (C) 21 (D) 14√
3 (E) 42
2009 AMC 10 A, Problem #10—“�ADB and �BDC are similar.”
Solution
Answer (B): By the Pythagorean Theorem, AB2 = BD2 + 9, BC2 =BD2 + 16, and AB2 + BC2 = 49. Adding the first two equations andsubstituting gives 2 · BD2 + 25 = 49. Then BD = 2
√3, and the area of
�ABC is 12· 7 · 2
√3 = 7
√3.
OR
Because �ADB and �BDC are similar, BD3
= 4BD
, from which BD =
2√
3. Therefore the area of �ABC is 12· 7 · 2
√3 = 7
√3 .
Difficulty: Medium-hard
NCTM Standard: Geometry Standard: explore relationships (including congruence and similarity)among classes of two- and three-dimensional geometric objects, make and test conjectures aboutthem, and solve problems involving them.
Mathworld.com Classification: Geometry > Plane Geometry > Triangles > Special Triangles >Other Triangles > Right Triangle
Triangle ABC has a right angle at B. Point D is the
foot of the altitude from B, AD = 3, and DC = 4.
What is the area of �ABC ?
A
B C
D
3
4
(A) 4√
3 (B) 7√
3 (C) 21 (D) 14√
3 (E) 42
2009 AMC 10 A, Problem #10—“�ADB and �BDC are similar.”
Solution
Answer (B): By the Pythagorean Theorem, AB2 = BD2 + 9, BC2 =BD2 + 16, and AB2 + BC2 = 49. Adding the first two equations andsubstituting gives 2 · BD2 + 25 = 49. Then BD = 2
√3, and the area of
�ABC is 12· 7 · 2
√3 = 7
√3.
OR
Because �ADB and �BDC are similar, BD3
= 4BD
, from which BD =
2√
3. Therefore the area of �ABC is 12· 7 · 2
√3 = 7
√3 .
Difficulty: Medium-hard
NCTM Standard: Geometry Standard: explore relationships (including congruence and similarity)among classes of two- and three-dimensional geometric objects, make and test conjectures aboutthem, and solve problems involving them.
Mathworld.com Classification: Geometry > Plane Geometry > Triangles > Special Triangles >Other Triangles > Right Triangle
23
AMC 10 Student Practice Questions continued
At Jefferson Summer Camp, 60% of the children play
soccer, 30% of the children swim, and 40% of the
soccer players swim. To the nearest whole percent,
what percent of the non-swimmers play soccer?
(A) 30% (B) 40% (C) 49% (D) 51% (E) 70%
2009 AMC 10 A, Problem #18—“Of the 60 soccer players, 40% or 60× 40
100 = 24 are alsoswimmers.”
Solution
Answer (D): For every 100 children, 60 are soccer players and 40 arenon-soccer players. Of the 60 soccer players, 40% or 60× 40
100= 24 are also
swimmers, so 36 are non-swimmers. Of the 100 children, 30 are swimmersand 70 are non-swimmers. The fraction of non-swimmers who play socceris 36
70≈ .51, or 51%.
Difficulty: Medium-hard
NCTM Standard: Algebra Standard: use mathematical models to represent and understandquantitative relationships.
Mathworld.com Classification: Number Theory > Arithmetic > Fractions > Percent
24
AMC 10 Student Practice Questions continued
Segment BD and AE intersect at C, as shown,
AB = BC = CD = CE, and ∠A = 52∠B. What is
the degree measure of ∠D ?
A
B
C
D
E
(A) 52.5 (B) 55 (C) 57.5 (D) 60 (E) 62.5
2009 AMC 10 B, Problem #9—“Notice that �ABC and �CDE are isosceles.”
Solution
Answer (A): Because �ABC is isosceles, ∠A = ∠C. Because ∠A =52∠B, we have 5
2∠B + 5
2∠B + ∠B = 180◦, so ∠B = 30◦. Therefore
∠ACB = ∠DCE = 75◦. Because �CDE is isosceles, 2∠D + 75◦ =180◦, so ∠D = 52.5◦.
Difficulty: Medium
NCTM Standard: Geometry Standard: analyze properties and determine attributes of two- andthree-dimensional objects.
Points A and C lie on a circle centered at O, each of
BA and BC are tangent to the circle, and �ABC
is equilateral. The circle intersects BO at D. What
is BDBO ?
(A)
√2
3(B)
1
2(C)
√3
3(D)
√2
2(E)
√3
2
2009 AMC 10 B, Problem #16—“�BCO is a right triangle with a 30◦ angle at B .”
Solution
Answer (B): Let the radius of the circle be r. Because �BCO is a righttriangle with a 30◦ angle at B, the hypotenuse BO is twice as long as OC,so BO = 2r. It follows that BD = 2r − r = r, and
BD
BO=
r
2r=
1
2.
A
B
C
D
O
Difficulty: Medium-hard
NCTM Standard: Geometry Standard: analyze properties and determine attributes of two- andthree-dimensional objects.
2009 AMC 10 B, Problem #20—“Angle Bisector Theorem.”
Solution
Answer (B): By the Pythagorean Theorem, AC =√
5. By the AngleBisector Theorem, BD
AB= CD
AC. Therefore CD =
√5·BD and BD+CD =
2, from which
BD =2
1 +√
5=
√5 − 1
2.
OR
Let DE be an altitude of �ADC. Then note that �ABD is congruentto �AED, and so AE = 1. As in the first solution AC =
√5. Let
x = BD. Then DE = x, EC =√
5 − 1, and DC = 2 − x. Applying the
Pythagorean Theorem to �DEC yields x2 +(√
5 − 1)2
= (2 − x)2, from
which x =√
5−12
.
Difficulty: Medium-hard
NCTM Standard: Geometry Standard: explore relationships (including congruence and similarity)among classes of two- and three-dimensional geometric objects, make and test conjectures aboutthem, and solve problems involving them.
2009 AMC 12 A, Problem #3—“Find one third of the difference between 1
4 and 34, then
calculate the required number.”
Solution
Answer (B): The number is
1
4+
1
3
(3
4− 1
4
)=
1
4+
1
3· 1
2=
1
4+
1
6=
5
12.
Difficulty: Medium-easy
NCTM Standard: Number and Operations Standard: compare and contrast the properties ofnumbers and number systems, including the rational and real numbers, and understand complexnumbers as solutions to quadratic equations that do not have real solutions.
Mathworld.com Classification: Number Theory > Arithmetic > Fractions > Fraction
28
AMC 12 Student Practice Questions continued
Functions f and g are quadratic, g(x) = −f(100−x),
and the graph of g contains the vertex of the graph of
f . The four x-intercepts on the two graphs have x-
coordinates x1, x2, x3, and x4, in increasing order, and
x3−x2 = 150. The value of x4−x1 is m+n√
p, where
m, n, and p are positive integers, and p is not divisible
by the square of any prime. What is m + n + p ?
(A) 602 (B) 652 (C) 702 (D) 752 (E) 802
2009 AMC 12 A, Problem #23—“Write f(x) in “completing-the-square” form, thenexpress g(x) in terms of f(x).”
Solution
Answer (D): Let (h, k) be the vertex of the graph of f . Because the graphof f intersects the x-axis twice, we can assume that f(x) = a(x−h)2 + k
with −ka
> 0. Let s =√
−ka
; then the x-intercepts of the graph of f are
h ± s. Because g(x) = −f(100 − x) = −a(100 − x − h)2 − k, it followsthat the x-intercepts of the graph of g are 100 − h ± s.The graph of g contains the point (h, k); thus
k = f(h) = g(h) = −a(100 − 2h)2 − k,
from which h = 50 ±√
22
s. Regardless of the sign in the expression for h,the four x-intercepts in order are
50−s
(
1 +
√2
2
)
< 50−s
(
1 −√
2
2
)
< 50+s
(
1 −√
2
2
)
< 50+s
(
1 +
√2
2
)
.
Because x3 − x2 = 150, it follows that 150 = s(2 −√
2), that is s =
150(1 +
√2
2
). Therefore x4 − x1 = s(2 +
√2) = 450 + 300
√2, and then
m + n + p = 450 + 300 + 2 = 752.
Difficulty: Hard
NCTM Standard: Algebra Standard: understand and perform transformations such asarithmetically combining, composing, and inverting commonly used functions, using technology toperform such operations on more-complicated symbolic expressions.
Ten women sit in 10 seats in a line. All of the 10 get
up and then reseat themselves using all 10 seats, each
sitting in the seat she was in before or a seat next to
the one she occupied before. In how many ways can
the women be reseated?
(A) 89 (B) 90 (C) 120 (D) 210 (E) 2238
2009 AMC 12 B, Problem #21—“Try cases for 1 woman, 2 women, 3 women first.”
Solution
Answer (A): Let Sn denote the number of ways that n women in n seatscan be reseated so that each woman reseats herself in the seat she occupiedbefore or a seat next to it. It is easy to see that S1 = 1 and S2 = 2. Nowconsider the case with n ≥ 3 women, and focus on the woman at theright end of the line. If this woman sits again in this end seat, then theremaining n − 1 women can reseat themselves in Sn−1 ways. If this endwoman sits in the seat next to hers, then the former occupant of this newseat must sit on the end. Then the remaining n − 2 women can seatthemselves in Sn−2 ways. Thus for n ≥ 3, Sn = Sn−1 + Sn−2. Therefore(S1, S2, . . . , S10) = (1, 2, 3, 5, 8, 13, 21, 34, 55, 89), which are some of thefirst few terms of the Fibonacci Sequence. Thus S10 = 89.
Difficulty: Hard
NCTM Standard: Number and Operations Standard: develop an understanding of permutationsand combinations as counting techniques.
Mathworld.com Classification: Discrete Mathematics > Recurrence Equations >Fibonacci Number
30
AMC 12 Student Practice Questions continued
For how many values of x in [0, π] is sin−1(sin 6x) =
cos−1(cos x) ?
Note: The functions sin−1 = arcsin and cos−1 =
arccos denote inverse trigonometric functions.
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7
2009 AMC 12 B, Problem #24—“If 0 ≤ x ≤ π/12 , then sin−1(sin 6x) = 6x.”
Solution
Answer (B): Let f(x) = sin−1(sin 6x) and g(x) = cos−1(cos x). If0 ≤ x ≤ π, then g(x) = x. If 0 ≤ x ≤ π/12 , then f(x) = 6x.Note also that sin
(6(
π6− x
))= sin 6x, sin
(6(
π3− x
))= − sin 6x, and
sin(6(
π3
+ x))
= sin 6x, from which it follows that f(π6− x) = f(x),
f(π3− x) = −f(x), and f(π
3+ x) = f(x). Thus the graph of y = f(x)
has period π3
and consists of line segments with slopes of 6 or −6 andendpoints at ((4k + 1) π
12, π
2) and ((4k + 3) π
12,−π
2) for integer values of k.
The graphs of f and g intersect twice in the interval [0, π6] and twice more
in the interval [π3, π
2]. If π
2< x ≤ π, then g(x) = x > π
2, so the graphs of
f and g do not intersect.
Difficulty: Hard
NCTM Standard: Algebra Standard: understand and compare the properties of classes offunctions, including exponential, polynomial, rational, logarithmic, and periodic functions.
Mathworld.com Classification: Calculus and Analysis > Special Functions > TrigonometricFunctions > Trigonometric Functions
31
For how many values of x in [0, π] is sin−1(sin 6x) =
cos−1(cos x) ?
Note: The functions sin−1 = arcsin and cos−1 =
arccos denote inverse trigonometric functions.
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7
2009 AMC 12 B, Problem #24—“If 0 ≤ x ≤ π/12 , then sin−1(sin 6x) = 6x.”
Solution
Answer (B): Let f(x) = sin−1(sin 6x) and g(x) = cos−1(cos x). If0 ≤ x ≤ π, then g(x) = x. If 0 ≤ x ≤ π/12 , then f(x) = 6x.Note also that sin
(6(
π6− x
))= sin 6x, sin
(6(
π3− x
))= − sin 6x, and
sin(6(
π3
+ x))
= sin 6x, from which it follows that f(π6− x) = f(x),
f(π3− x) = −f(x), and f(π
3+ x) = f(x). Thus the graph of y = f(x)
has period π3
and consists of line segments with slopes of 6 or −6 andendpoints at ((4k + 1) π
12, π
2) and ((4k + 3) π
12,−π
2) for integer values of k.
The graphs of f and g intersect twice in the interval [0, π6] and twice more
in the interval [π3, π
2]. If π
2< x ≤ π, then g(x) = x > π
2, so the graphs of
f and g do not intersect.
Difficulty: Hard
NCTM Standard: Algebra Standard: understand and compare the properties of classes offunctions, including exponential, polynomial, rational, logarithmic, and periodic functions.
Mathworld.com Classification: Calculus and Analysis > Special Functions > TrigonometricFunctions > Trigonometric Functions
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