teachers’ manual · 2013. 7. 12. · mathematical association of america | american mathematics...

MATHEMATICAL ASSOCIATION of AMERICA | American Mathematics Competitions

AMC 10A 11th Annual ContestAMC 12A 61st Annual Contest

AMC 10/12

TEACHERS’MANUAL

Instructions and Reporting Forms forSchool 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 AMC10/AMC12 -Tuesday,February9&/orWednesday,February24,2010 AIME -Tuesday,March16orWednesday,March31,2010 USAMO -Tuesday&Wednesday,April27-28,2010 IMO -Astana,Kazakhstan,July6-14,2010

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Dear Contest Managers:

I am very pleased you will be providing the opportunity for your students to participate in the 61st

annual American Mathematics Contest 12 (AMC 12) or its sister contest, the 11th annual American

Mathematics Contest 10 (AMC 10). I believe that you and your students will find these contests to

be both interesting and challenging. Again this year the Committee on the American Mathematics

Competitions and panelists under the leadership of Committee Chairs LeRoy Wenstrom for the

AMC 10 and Bernardo Abrego for the AMC 12 developed a total of 4 contests (two AMC 10 and

two AMC 12). I extend special thanks to both of them for the effort involved in developing these

excellent contests.

This year, we continue to include in this manual several handouts, included in Section XIV:

* Worksheets for contest preparation (page 22-30)

* A handout for parents on why mathematics is important (page 16)

* A sample Press Release (page 15)

* A Certificate of Participation (pages 18-19)

You may reproduce these pages for your students.

Very sincerely,

Dr.StevenR.DunbarDirector,MAAAmericanMathematicsCompetitions

MATHEMATICAL ASSOCIATION of AMERICAAmerican Mathematics Competitions

University of Nebraska – Lincoln P.O. Box 880658 1740 Vine Street Lincoln, NE 68588-0658

Phone: 402.472.2257 Fax: 402.472.6087 [email protected] www.unl.edu/amc

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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

Our Sponsors ....................................................................................................................................... 32

Any forms not available here can be downloaded from our web site: www.unl.edu/amc/

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I. Preliminary Instructions for Administering the AMC 10/AMC 12

1. InformstudentsfarinadvanceaboutthedatefortheAMC10/AMC12andobtainasupplyofNo.2leadpencils.

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.

3. Thenameoftheschool,city,stateandpostalcodemustbestampedorwrittenoneachstudentanswerform.Thisisveryimportantbecausethereisnootherschoolidentificationontheanswerforms.Pleasedonotusestick-onlabelstoprovidethisinformation.

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.

Answer FormsTheAMC10andAMC12eachhavetheirowncolorcodedanswerform(AMC10-orangeandAMC12-red).ContestBan-swerformshaveinstructionsinblackink.Pleasebecarefultomatchthecorrectanswerformwiththeappropriatecontestwhenpassingoutthepapers.

AIME QualificationStudentswhoscore100oraboveorfinishinthetop5%onthisAMC12orstudentswhoscore120oraboveorfinishinthetop1%onthisAMC10willqualifytotakethe28thannualAmericanInvitationalMathematicsExamination(AIME)onTuesday,March16orWednesday,March31.

ResultsWewillsendyourschool’sresultsbyemail(ifavailable)andfirstclassmailassoonastheanswerformsarescored.Ifyouhavenotreceivedyourresultsfromourofficewithin30daysafterthecontest,pleasecontactustoverifythatyouranswerformswereinfactreceived.AllstudentAnswerFormsareheldfor90daysaftertheexamdate,thentheyarerecycled.

School FlierWeareagainprovidingaflieryoucanusetopromotethecontestswithinyourschool.Itisincludedonpage17ofthisManual.Ithasspaceleftforyoutoaddschoolinformation,suchasthelocationofthecontests,andwhotocontactlocallyformoreinformation.Ifyouwouldlikeacolorversion,visitourwebpagetodownloadapdfversionoftheflieryouneed:www.unl.edu/amc/

5. Reviewpasttestsandsolutionsorinformyourstudentshowtoordercopiesforthemselves.ThePublicationsOrderFormcanbeprintedfromtheAMCWebpageatwww.unl.edu/amc,(orcall,faxorwritetotheAMCOfficeforacopy).

6. Encourageparticipationbystudentswhohavenottakenthecontestbefore,especiallyyoungerstudents,butmakesurestudentsknowwhattoexpect.Letthemknowabouttypicalscoresatyourschoollastyearattheirgradelevel. Showstudentsthenationalstatistics inourNationalSummaryof Results and Awards from last year. Tell them to setappropriategoalsforthemselves.

7. RemindstudentsthedaybeforethecontestaboutthetimeandplaceoftheAMC10/AMC12.Alsotellthemyourplaniftheschoolshouldsuddenlyclose.Allstudentsmusttakethecontestatthesametime,eitherinonegrouporinseparateclassroomsandbeunderthesupervisionofaproctor.

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.

II. Instructions For The Day of The AMC 10/AMC 12

IftodayisnotyetTuesday,February9,2010STOPEVERY-THING.Undernocircumstancesisthecontesttobegivento

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takeintoconsiderationthefactthattherewillbeschoolstakingthecontestinotherlocationsatdifferenttimes.

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:

hstran@amc.unl.eduResultsarenotofficialuntilyoureceivethepostalmailedpapercopyofyourreport,andthatshouldbenolongerthan30daysaftertheAMC10/AMC12.

anyonebeforetheofficialday,norshouldthecontestpackage(s)beopenedbeforethatdate.1. SigntheCertificationFormtocertifythatthepackagewas

openedwithinanhourbeforethecontest.2. Whenthestudentsarrive,seatthemsotheyareseparated

byanemptyspaceifpossible.3. Hand out the Answer Forms which have been partially

completed by the students (AMC 10-orange/AMC 12-red).

4. Asyouhandoutthepapers,tellthestudentsnottoopenthecontestbooklet.Theyshouldthenreadtheentirefrontcover.Givethem5minutestodothis.

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.

6. Informtheparticipantstheymaynottalkoraskanyquestionsduringthecontest,andtheymustdotheirownwork.

7. Remindstudentstheyhave75minutes,thentellthemtoBEGIN.(Seepage6forstudentdisabilitiespolicy)

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.

10. Announcewhenthereare30minutesremainingandwhenthereare5minutesremaining.

11. Whentimeisup,tellthestudentstoSTOPandhavethemsigntheirnameinthespaceprovidedontheStudentAnswerForm.CollecttheAnswerFormsasquicklyaspossible.

12. Pleasedonotgradetheanswerforms.TheyaretobesenttotheAMCofficeforgrading.Studentsmaycircletheiranswers on the contest booklet. However, the officialanswerswillbetheonesblackenedontheanswerform.

13. Fantheforms,makingsurenonearestucktogether,placetheSchoolIDFormontop.ThereisonlyoneIDFORMtobeusedwithalltheAMC10/AMC12answerforms.

14. YOURSCHOOL’SCEEBNUMBERISTHENUMBERWRITTENONTHEFRONTOFTHEAMC10/AMC12REPORTENVELOPE.

15. CompletetheCertificationForm(onlyoneformisneeded)andplaceitontopoftheSchoolIDFormandanswerformsandplaceallintheReportEnvelope.SealandsenditbyFirstClass(trackablerecommended)within24hoursorassoonaspossible.Pleaseaffixtheproperpostagebeforemailing.

16. Pleasenote:AftertheAnswerFormshavebeendeliveredtotheschoolofficetobemailed,youmaydiscussthecontestwithyourstudentsunderthefollowingconditionswhich

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Pleaseretainthisreportforfuturereference,eitherinyourfilesorwiththestudentcounselor’soffice.Manystudentscitethesescoresoncollegeapplications.

VI. Policy StatementsEarly Administration

AdministrationonanearlierdateisNEVERpermittedandwillleadtodisqualification.Suchanadministrationwouldjeopar-dizethevalidityofallscoresfromotherparticipatingschools.

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.

Contest ProctoringTheproctoringoftheAMC10/AMC12contestshouldbeby:1stpreference,amathteacherattheschool2ndpreference,ateacheroradministratorattheschool3rdpreference,acollegeoruniversityteacherofmathematicsor

aresponsibleadultwhoisamathcluborteamcoach,andnotrelatedtoanyoftheparticipants

4th preference, a responsible adult not related to any of theparticipantssuchaslibrarian,clergy,etc.

• Theproctoringofthecontestmusttakeplaceinapublicbuild-ing,(E.g.schoollibrary,collegeoruniversity,church,etc.).

• Allstudentsmusttakethecontestatthesametime,eitherinonegrouporinseparateclassroomsandbeunderthesupervisionofaproctor.

• Theproctorshouldnotberelatedtoanyoftheparticipants• Ifaparentwishestohelp,theycanarrangeroomreservations

andset-up,helpwithstudentsign-up,providetreats,etc.

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).

Questionable ScoresIfitiscleartotheContestManagerfrompersonalobservationthatastudenthascheated,thentheManagermustdisqualifythestudent.IftheContestManagerreceivesanaccusationofcheating,orobtainsotherindirectevidenceofcheating,thentheManagermustholdbackthestudent’spaperandimmediatelyreportallthefactsofthesituationtotheAMCDirector,whoinconjunctionwiththeChairoftheCAMC,willdeterminewhattodo.UNDERNOCIRCUMSTANCESmaytheschooldecideonitsowntoacceptaquestionablescore,norshouldaschoolretestthestudentinquestionbeforereceivinginstructionsfromtheAMCOffice.

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.

Refund/Credit PolicyIfyourschoolisunabletotakethecontests,pleaseusethemate-rialsaspracticesetsforthenextyear.Donotreturnthem.WECANNOTGIVEREFUNDSORCREDITSAFTERTHECONTESTMATERIALSARESHIPPED.

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.

2. AllquestionsorproblemsconcerningtheAIMEshouldbedirectedtotheAMCoffice(800-527-3690).

3. The AIME is a three-hour examination. Each of its 15questions requires a three digit integer answer and eachcorrectanswerwillreceiveone(1)point.

4. Calculatorsarenotallowed.5. TheAIMEAnswerFormsaresentdirectlytotheAMC

officeforgradingandprocessing.6. Eachparticipatingschoolwillreceiveareportoftheirresults,

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

materialsonhandwhenyoureceiveyourAMC10/AMC12results.

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:

[email protected],youcancalltheAMCofficeat1-800-527-3690.Pleasehaveyourschoolidentificationnumber(CEEB)andchargecardinfor-mationavailablebeforecalling.E-Mailrequestsshouldincludetheschool’sCEEBnumber,andcompletemailingaddress.UndernocircumstancescanastudenttakebothAIME’s.

VIII. USAMO/USAJMOParticipant Selection

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,

Pennsylvania,RhodeIsland,Vermont1 NewJersey,NewYork2 Delaware,DistrictofColumbia,Maryland,NorthCarolina,

SouthCarolina,Virginia,WestVirginia3 Alabama, American Embassy and APO/FPO Schools,

Florida,Georgia,PuertoRico,VirginIslands4 Indiana,Michigan,Ohio5 Arkansas,Iowa,Kansas,Minnesota,Nebraska,NorthDa-

kota,Oklahoma,SouthDakota,Wisconsin6 Illinois,Kentucky,Missouri,Tennessee7 Louisiana,Mississippi,Texas8 Alaska,Arizona,Colorado,Guam,Hawaii,Idaho,Mon-

tana,Nevada,NewMexico,Oregon,Utah,Washington,Wyoming

9 California10 Canada

XI. Intramural and National AwardsIntramural Awards for Students

TheAMCOfficewillsendyouIntramuralawards,alongwithyourschoolresults.Yourregistrationfeeentitlesyoutopins,medals, certificates, andacopyof theNationalSummaryofResultsandAwards.Anorderformforadditionalintramuralawardswillbeincludedwithyourresults.CommitteeandDonorAwardswillbesentlater.1. AMC 10 & AMC 12 School Winner Pin — Foreach

school making a report on three or more students theMathematicsAwardPinisawardedtothestudentwiththehighestscore.

2. AMC 10 & AMC 12 Honor Roll of Distinction Pin — TheHonorRollofDistinctionPinisawardedtothetop1%ofallparticipatingstudents.

3. Gold Medal — TheGoldMedalisawardedtoparticipantswhoplacefirstintheirrespectiveschoolforfourconsecutiveyears.

4. Silver and Bronze Medals — Thesemedalsareawardedtoparticipantswhoplacefirstintheirschoolforthreeandtwoconsecutiveyears,respectively.

5. AMC 10/12 Certificate of Distinction — ACertificateofDistinctionisawardedtothosestudentswhoqualifytotaketheAmericanInvitationalMathematicsExamination(AIME).

6. AMC 12 Certificate of Achievement — Awarded tostudentsingrade10andbelowwhoscorea90oraboveontheAMC12contest.

7. AMC 10 Certificate of Achievement — Awarded tostudentsingrade8andbelowwhoscorea90oraboveontheAMC10contest.

8. School Certificate of Honor — Awardedtoschoolswithateamscoreof400orgreaterontheAMC12only.

9. School Certificate of Merit — Awardedtoschoolswithateamscorebetween300and399ontheAMC12only.Committee and Donor Awards for Students

AllcommitteeanddonorawardsarebasedontheAMC12scores.1. Large Plaque — Alargeplaque,donatedbytheCommittee

ontheAmericanMathematicsCompetitions,isawardedto

eachstudentwithaperfectscore.2. Small Plaque — Asmallplaque,donatedbytheCommittee

ontheAmericanMathematicsCompetitions,isawardedtothestudentwiththehighestscoreineachregioninwhichtherewasnoperfectscore.

3. Mathematics Magazine* — Aone-yearsubscriptiontotheMathematics Magazine,donatedbytheCommitteeoftheAmericanMathematicsCompetitions,isawardedtoamaximumoffourparticipantswithhighscoreineachoftheelevenregions.

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

cup,isawardedtotheschoolwiththehighestteamscore(inanyregion).

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.

6. Mathematics Magazine*—Ineachoftheelevenregions,aone-yearsubscriptionisdonatedbytheCommitteeontheAmericanMathematicsCompetitionstooneormoreschoolshavingahighteamscore.

9

AMC 10 2010 AMC 12

XII. Contest A CertificationTheAMC10andAMC12mustbeadministeredbyateacheroranadultnotrelatedtoanyoftheparticipants.Theadministrationofthecontestmusttakeplaceinapublicbuilding(e.g.school,library,church).PleasesendallAnswerFormsfromyourschoolorgroupatonetime.TheContestManagerandthePrincipal,VicePrincipal,orHeadmastermustsignthisformwhichistobereturnedwithyourstudentAnswerForms.

Certification by the Contest Manager:Icertifythatthefollowingstatementsaretrueorthat,ifthereareanyexceptions,Ihavecheckedtheboxatthebottomofthispageandhavelistedthemonaseparatepage.Iunderstandthattheabsenceofeithersignaturefromthisform,andaconsiderationoftheexceptionsmayresultinDISQUALIFICATIONofallscoresfromourschool.

1. Icertifythattheexampackage(s)wereretainedintheirsealedconditionwithinanhourofthestartofthecontest.2. Iacceptforourschooltherulesandproceduresdescribedonthispageandpages4-7,andacceptthatfailuretofollowthese

rulesandproceduresmayresultinDISQUALIFICATIONfromofficialstandingofallscoresfromourschool.3. ThecontestwasheldonTuesday,February9,2010.4. TheAMC10A/AMC12Aweregivenatthesametime.5. Theparticipantswerecontinuallymonitoredduringthecontest,andtheywereseparatedbyanemptyspace,ifpossible.6. Noaidswerepermittedotherthanscratchpaper,graphpaper,ruler,compass,andprotractor.NOCALCULATORSWERE

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

followed.10. Afterthecontest,theanswerformswerekeptsecureandnochangesweremadeintheanswers.11. Noparentassistedintheadministrationofthecontest.12. IhavefollowedalltherulesasstatedinthisTeachers’Manual.

Signature_____________________________________ E-mail(pleaseprint)____________________________________Day&DateTestwasgiven ___________________________________________________________________________NameofSchool____________________________________________________________________________________City ___________________________ State/Province _______________________________________________Telephone(______)___________________________ SchoolCEEBNumber__________________

Certification by the Principal, official or person of comparable title:IagreethattheContestManagerfollowedalltherulesandprocedureslistedabove.

Signature_____________________________________ Date_________________________________________

Title_________________________________________

PLEASEINDICATETHEEXACTNUMBEROFAMC10A/AMC12AANSWERFORMSRETURNEDFORGRADING.____________

EXCEPTIONS

10

Questionnaire to Help Us Serve You BetterThankyouinadvancefortakingthetimetoanswerthesequestionsforus.Answertothebestofyourability,andifyoudon’tknow,justgiveusa“bestestimate”.PleasefillinyouranswersintheovalsprovidedonthebackoftheschoolIDform.Thankyou!

1. Whatistheaveragenumberofstudentsinyourschool’smathclasses? 0)1-10 1)11-20 2)21-25 3)26-30 4)31-40 5)40+

2. Whatgradelevelsarecoveredinyourschool?(markallthatapply) 0)K-3 1)4-5 2)6 3)7-8 4)9-10 5)11-12

3. Howmanystudentsattendyourschool? 0)1-99 1)100-399 2)400-699 3)700-999 4)1,000-2,999 5)3,000+

4. Howmany“MiddleSchool”programs(schools)arethereinyourschooldistrict? 0)1 1)2-5 2)6-10 3)11-15 4)16-20 5)21+

5. HowdidyoufindoutabouttheAmericanMathematicsCompetitions? 0)Student/Parent 1)Fellowteacher 2)Administrator 3)Directemail 4)Directpostalmail 5)Ourschoolhashistoricallyparticipated

6. Whoorwhatdoyourelyonforinformationaboutactivitiesorcompetitionsyourschoolcanparticipatein?(markallthatapply) 0)Student/Parent 1)Administrators 2)Myownresearchonline 3)Directemail 4)Directpostalmail 5)Wordofmouthfromotherteachers

7. Ingeneral,dootherpeopleyoutalktoknow/haveheardabouttheAmericanMathematicsCompetitionsanditspurpose? 0)No,neverheardoftheAMC 1)Yes,butonlyknowit’samathcompetition 2)Yes,knowabouttheAMC

8. HowdoyouprefertoreceiveinformationabouttheAmericanMathematicsCompetitionsanditscontests?(markallthatapply) 0)Email 1)Postalmail 2)Iprefertogotothewebsitemyself

9. Doyouuseanyofthefollowingsocialnetworkingsitestokeepupdatedontopicsimportanttoyou?(markallthatapply) 0)Facebook 1)Myspace 2)Twitter 3)Blogger/Wordpress/Livejournal 4)MobileTexts 5)None

10. DoyouthinktheAmericanMathematicsCompetitionsshouldhaveapresenceonanyoftheabovementionedsites? 0)Yes,allofthem 1) Yes,someofthem 2)Idon’tknow/don’tcare 3)No,definintelynot

11

XIII. Additional Forms used - (if the form you need is not here, please see our web site)

Additional Bundles FormPleasefillintheinformationbelowandFAXyourorder.TheadministratororauthorizedpersonoftheschoolagreestopaytheAmericanMathematicsCompetitionsforthefollowingmaterials:

SchoolName______________________________________________________ CEEB#________________Address__________________________________________________________________________________City_________________________________________________State______Zip ______________________Teacherplacingtheorder_____________________________________Email____________________________ ContestABundlesoften...................#_________@$16/bundle=.............$__________ ASolutionsSetsoften(optional).......#_________@$6/set=.....................$ ____________

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/

Postage/handlingFee(seechartbelow)...............................................................................................$__________Total................................................................................................................................................. $_________

P.O.Number___________________ VISA/MC#:___________________________________________________

Name(PleasePrint): _______________________________________________________________________Address:___________________________________________________________________________________________Exp.Date:_________________

Proof of Intent to PayThisdocumentisintendedtobeusedinlieuofpre-paymentwhencallingorfaxinginanorder.Pleaseindicateifyouwishtobebilledorwillbesendinga“checkinthemail”(tobereceivedwithin2weeksoforderoryouwillbebilled).Mailordersnotwishingtobebilledshouldincludeacheckwhenreturningthisform.Thepersonwhosignsthisformmustbeauthorizedtopaytheorderthatisplacedbytheteacher.

BILLEDNameofPersonAuthorizedtoPay(pleaseprint):__________________________________________________

Signature:_______________________________________________________________________________

Title: _______________________________ Date: ____________________________

AMC 10AMC 12

*OrdersafterFebruary1stadd$10.00additionalfor1-dayService.

1.VISA&MasterCardaccepted.2.Makecheckspayableto: AMERICANMATHEMATICSCOMPETITIONS3.PAYMENTINU.S.FUNDSONLY.4.U.S.A.: OrderTOTAL ShippingCharge* $10.00--$40.00 $7.00 $40.01--$50.00 $9.00 $50.01--$75.00 $12.00 $75.01--UP $15.005.OUTSIDEUSA:Addadditional$10totheaboveUSAshippingcosts.

AMC ORDERING -- TERMSFAX:402-472-6087orPHONE:800-527-3690

Please Send Your Order To:

AmericanMathematicsCompetitionsATTN:AMC10/12AdditionalBundles

P.O.Box81606Lincoln,NE68501-1606

XIII. Additional Forms used

12

Rescoring Request Form

Iwouldliketohavethefollowingstudent’sanswerformrescored.Iunderstandthatthereisa$10.00chargeforeachstudentanswerformrescored

$ 10.00/each

Teacher’sName_____________________________________________CEEB#___________________________

SchoolName_____________________________________________________________________________

Address _____________________________________________________________________________

City__________________________________________________________ State ________ Zip_________

Email_________________________________________________________ Fax_______________________

Method of Payment: Check(USfundsonly)madepayableandmailedwiththisformtothe: AMERICANMATHEMATICSCOMPETITIONS UniversityofNebraska-Lincoln P.O.Box81606 Lincoln,NE68501-1606 ChargetoVisa/Mastercard#:_________________________________________________________________ Nameoncard(print):___________________________________________________________________ Signed ______________________________________________________________________________ ExpirationDate:__________________________Telephone:____________________________________

FAXto:402/472-6087

StudentName __________________________________________________________________ $ _____________Circlecontesttaken:AMC10-AAMC10-BAMC12-AAMC12-BAIMEIAIMEII

StudentName __________________________________________________________________ $ _____________Circlecontesttaken:AMC10-AAMC10-BAMC12-AAMC12-BAIMEIAIMEII

GrandTotal____________________________________________________________________ $ _____________

13

PLEASE PRINTHigh School CEEB# (6 digits):__ __ __ __ __ __

_____________________________________________ Contest Manager

_____________________________________________ School Name

_____________________________________________ School Street Address

_____________________________________________ City State Zip

_____________________________________________ School Phone #

BILLING ADDRESS (if different from school):PROCTOR ADDRESS (homeschool):

_____________________________________________ _____________________________________________ City State Zip

_____________________________________________ Phone #

_____________________________________________ 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

_____________________________________________ VISA/MC # Exp. Date: _____________

_____________________________________________ 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

International Schools additional Shipping ($70 maximum) # Exam bundles ____x $7.00/bundle ....................$______

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

( )

Office use only

14

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.

Current sizes: Small, Medium, Large and X-Large

• VISAandMasterCardaccepted.• Paymentbycheckwillalsobeaccepted.Makecheckspayable

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.

• PricesgooduntilSeptember1,2010.• Pleaseallow2weeksfordelivery.

FAX 402-472-6087 or call 1-800-527-3690PleaseSendYourOrderTo:

MAA American Mathematics Competitions ATTN: AMC Publications P.O. Box 81606 Lincoln, NE 68501-1606

MAA AMC Ordering -- TERMS

www.unl.edu/amc/ [email protected]

Shipping AddressPLEASE FILL IN ALL THE INFORMATION

If other than a school, please strike out school and any other irrelevant headers and fill in the

name, and ship-to address.

(Please Print)

Name: __________________________________________

School Name: ______________________________________

Street Address: _____________________________________

____________________________________________

City: _______________________ State: ______ Zip: ________

Phone #: ____(____________) ________________________

E-mail address: _____________________________________

CHARGE INFORMATION:

Card Name (Print): ___________________________________

VISA/MC #: ________________________________________

Exp. Date: ___________

E-mail address: _____________________________________

Address: _________________________________________

_______________________________________________

CD --AJHSME & AMC 8 (1985-2007) ......# ____ @ $20/ea = $ ______CD --AHSME 1 (1950-1974)........................# ____ @ $20/ea = $ ______CD --AHSME 2 (1975-1999)........................# ____ @ $20/ea = $ ______AMC8 Math Club Package......................# ____ @ $25/ea = $ ______AMC10/12 Math Club Package.............# ____ @ $25/ea = $ ______MAA AMC T-Shirt --60thAnniversary ..# ____ @ $15/ea = $ ______ Current Sizes : Small___ Medium___ Large___ X-Large___Postage & Handling (See TERMS)......................................... $ ______GRAND TOTAL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ ______

PAYABLEinU.S.FundsONLY

ALLOW 2-3 WEEKS FOR DELIVERY

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

www.unl.edu/amc/

15

PublicityThesampleContestAnnouncementnewsreleasebelow,shouldbepreparedanddistributedtothenewspapers,radioandtelevisionstationsinyourregion.Tomakepreparationofthenewsreleaseeasier,visitourwebsite,anddownloadthetextfromtheon-linecopyoftheAMC10/12Teachers’Manual.SeetheAMCwebsite,orthe2009SummaryofResultsforstatisticsandfiguresfromthe2009contest.Statisticsandfiguresforthe2010ContestwillbeavailableonourwebsiteinMarch2010.

(SchoolorSchoolDistrict)FORIMMEDIATERELEASE

(School)STUDENTSPARTICIPATEINNATIONALMATHEMATICSCOMPETITION

(#)studentsat(School)participatedinthe61stannualAmericanMathematicsContest12,and/orthe11thannualAmericanMathematicsContest10.ThecontestswereheldonTuesday,February9,2010(orWednesday,February24,2010).Thestudentscompetedforlocal,regionalandnationalstudentandschoolawards.Thecontest,whichcovershighschoolmathematics,isgiveninparticipatingschools.Itspurposeistospurinterestinmathematicsanddeveloptalentthroughtheexcitementoffriendlycom-petitionatproblemsolvinginatimedformat.In2009over220,000studentsfrom4,200schoolsparticipatedintheAMC10&AMC12contestincluding(_#_)studentsfrom(_#_)schoolsin(State).Topscorersat(school)were(________,_______,etc.).AccordingtoProf.StevenDunbar,whoservesasDirectoroftheAmericanMathematicsCompetitions,theAMC12(firstof-feredin1950),andtheAMC10(firstofferedin2000),arepartofaseriesofcontestssponsoredeachyearbyTheMathematicalAssociationofAmerica,throughtheirprogram,theAmericanMathematicsCompetitions.TheAMCofferstheonlymathcom-petitionseriesinthecountryleadingtotheUnitedStatesofAmericaMathematicalOlympiad(USAMO)andtheMathematicalOlympiadSummerProgram(MOSP).Fromthisgroupofstudents,theAMCsendsthehighlycompetitiveUSATeamtotheprestigiousannualInternationalMathematicalOlympiad.TheAMCprogramincludes: AmericanMathematicsContest8(AMC8) Grades6-8 November AmericanMathematicsContest10(AMC10) Grades10&below 2datesinFebruary AmericanMathematicsContest12(AMC12) Grades12&below 2datesinFebruary AmericanInvitationalMathematicsExamination(AIME) Allwhoqualify 2datesinMarch-April USAMathematicalOlympiad(USAMO) Allwhoqualify late-April MathematicalOlympiadSummerProgram(MOSP) QualifythruUSAMO June InternationalMathematicalOlympiad(IMO) TopsixfromUSAMO,MOSP JulyTheAMCislocatedattheUniversityofNebraska-Lincoln.andreceivessupportfromtheAkamaiFoundation,AcademyofAp-pliedSciences,AmericanMathematicalAssociationofTwoYearColleges,AmericanMathematicalSociety,AmericanStatisticalAssociation,ArtofProblemSolving,AwesomeMath,Canada/USAMathcamp,CasualtyActuarialSociety,IDEAMath,InstituteforOperations,ResearchandtheManagementSciences,MathPath,MathZoomAcademy,MuAlphaTheta,NationalCouncilofTeachersofMathematics,PiMuEpsilon,SocietyofActuaries,U.S.A.MathTalentSearch,W.H.FreemanandCompany,WolframResearchInc.TheContestsaregivenacrosstheU.S.A,Canada,andinmanyschoolsabroad.

Detailsconcerningthe2010AMC10/12contestsforHighSchool,aswellastherestofAMC’sprogramsareavailableontheAMCwebsite:www.unl.edu/amc/.

ForfurtherinformationcontacttheAMC--telephone:800-527-3690,email:[email protected].

XIV. Classroom Accessories

February, 2010

Dear Parent or Guardian:

OnFebruary9,2010and/orFebruary24,2010yoursonordaughterwillparticipateinthe61stannualAmericanMathematicsCompetitionscontest.Thiscontesthasgrownfromasinglecity-widecompetitioninNewYorkCityin1950,organizedbythelocalsectionofTheMathematicalAssociationofAmericatoasequenceofcontests.In2009,over220,000studentsfromover4,200USandinternationalschoolscompetedforschool,regional,andnationalawardsinthiscontestandfounditfunandrewarding.Top10,well-knownU.S.universitiesandcolleges,includinginternationallyrecognizedU.S.technicalinstitutions,askforAMCscoresontheirapplicationforms.Yourstudentsdeservethechancetolistthesescoresontheirapplications!

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.

Withthesecontests,thereareawardsineachschoolforthestudentwiththehighestscore,certificatesforhigh-scoringstudentsineachschool,state-wideawards,regionalawards,andevennationalawards.Thesecontestsleadtoothermoreselectivemathcontests,evenallthewaytotheUSAteamsenttotheInternationalMathematicalOlympiad,thepremierinternationalhighschoollevelproblemsolvingcontest.Buttherealrewardscomefromchallengingeachstudentwithmathematicsthatisnew,different,and“outsideofthebox.”Theproblemsonthecontestarehard,butdesignedtobewithinreach.Justbyparticipatinginthecontestyourstudentshouldstillfeelaccomplishment,becausetheseproblemsaremeanttobemorechallengingthanroutinelyencounteredinmathematicscourses.Mathematicsisincreasinglyimportantinourtechnologicalandscientificage.Takingenoughmathematicsinhighschoolisthegatewaytojobsandcareersofallkinds,eventhosethatarenotexplicitlymathematical,scientific,ortechnological.Wehopethatbyofferingthesecontests,wecanchallengeandinspirestudentstolearnmoremathematics.Wehopethatyoursonordaughterenjoyedthecontests,andwillcontinuetotakemathematicscoursesandcompetitionsinhighschoolandbeyond.

Sincerely,

Dr.StevenR.DunbarAMCDirector

Letter to Parents (for Reproduction)

MATHEMATICAL ASSOCIATION of AMERICAAmerican Mathematics Competitions



HeyMathStudents!Plantotakethe

The MaTheMaTical associaTion of aMerica American Mathematics Competitions

AMC 10AMC 12

onTuesday,February9,2010orWednesday,February24,2010

and/orthe

Practicematerialscanbefoundatwww.unl.edu/amc

For more information contact:


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











AMC 12 Participation Certificates

20

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

11th Annual

AMC 10 AAmerican Mathematics Contest 10A


©2010MathematicalAssociationofAmerica

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

61st Annual

AMC 12 AAmerican Mathematics Contest 12A


©2010MathematicalAssociationofAmerica

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.

22

AMC 10 Student Practice Questions

YouwillfindtheseandadditionalproblemsfortheAMC10andAMC12onAMC’swebsite:http://www.unl.edu/amc,availablefromthecurrentandpreviousAMC10/12TeacherManuals,(http://www.unl.edu/amc/e-exams/e6-amc12/archive12.shtml)orfromourProblemspagearchives(http://www.unl.edu/amc/a-activities/a7-problems/problem81012archive.shtml).

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 .



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%.


NCTM Standard: Algebra Standard: use mathematical models to represent and understandquantitative relationships.

Mathworld.com Classification: Number Theory > Arithmetic > Fractions > Percent

24


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.

Mathworld.com Classification: Geometry > Plane Geometry > Triangles > Special Triangles >Other Triangles > Isosceles Triangle

25


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


NCTM Standard: Geometry Standard: analyze properties and determine attributes of two- andthree-dimensional objects.

Mathworld.com Classification: Geometry > Plane Geometry > Triangles > Special Triangles >Other Triangles > 30-60-90 Triangle

26


Triangle ABC has a right angle at B, AB = 1, and

BC = 2. The bisector of ∠BAC meets BC at D.

What is BD ?

A

B CD

1

(A)

√3 − 1

2(B)

√5 − 1

2(C)

√5 + 1

2(D)

√6 +

√2

2(E) 2

√3 − 1

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

.



Mathworld.com Classification: Geometry > Trigonometry > Angles > Angle Bisector Theorem

27

AMC 12 Student Practice Questions

What number is one third of the way from 14 to 3

4 ?

(A)1

3(B)

5

12(C)

1

2(D)

7

12(E)

2

3

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


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.

Mathworld.com Classification: Algebra > Polynomials > Quadratic Polynomial

29


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


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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