volume 30 MATHEMATICS COMPETITIONS - …

1 MATHEMATICS COMPETITIONSj o u r n a l o f t h e WORLD FEDERATION OF NATIONAL MATHEMATICS COMPETITIONSA M T Pu b l i s h i n gv o l u m e 3 0 n u m b e r 1 2 01 7 MATHEMATICS COMPETITIONSJOURNAL OF THE WORLD FEDERATION OF NATIONAL MATHEMATICS COMPETITIONSA M T PUBLISHINGVOLUME 30 NUMBER 1 2017 MATHEMATICSCOMPETITIONSMATHEMATICS COMPETITIONS JournAl of The World federATion of nATionAl MATHEMATICS COMPETITIONS (ISSN 1031 7503)Published biannually byA M T Pu b l i s h i n g Au sT r A l i A n MA Th eM A Ti c s Tr u sTun i v e r s iT y o f cA n b e r rA lo c k e d bA g 1cA n b e r rA g P o A c T 2 6 01Au sT r A l i AWith significant support from the UK MATHEMATICS (in English) are send articles to:The EditorMathematics CompetitionsWorld Federation of National MATHEMATICS CompetitionsUniversity of Canberra Locked Bag 1 Canberra GPO ACT 2601 AustraliaFax:+61-2-6201-5052orDr Jaroslav vr cekDept.

2 Of Algebra and GeometryPalack University of Olomouc 17. listopadu 1192/12771 46 OlomoucCzech RepublicEmail: 2017 AMT Publishing, AMTT Limited ACN 083 950 341 MATHEMATICS COMPETITIONS volume 30 nuMber 1 2017 CONTENTS PAGEWFNMC Committee 1 From the President 4 From the Editor 6 Call for nominations 8 How Does Age Factor Matter - a Case Study 10 Borislav Lazarov & Albena Vassileva A Fraction Problem 25 Arthur Holshouser & Harold Reiter Scanning Cats 45 Yen-Kang FuThe 58th International Mathematical Olympiad, Rio de Janeiro, Brazil, 2017 50 International MATHEMATICS Tournament of Towns Selected Problems from Spring 2017 Tournament 60 Andy LiuMathematicsCompetitionsVol30No12017 agerBRGK eplerKeplerstrasse18020 GrazAUSTRIAP rofessorAliRejaliIsfahanUniversityofTech nology8415683111 Svr yUniversity,OlomoucCZECHREPUBLIC1 MathematicsCompetitionsVol30No12017 Secretary:SergeyDorichenkoSchool179 MoscowRUSSIAI mmediatePastPresident&Chairman,AwardsCom mittee:ProfessorMar aFalkdeLosadaUniversidadAntonioNarinoCar rera55#45-45 Bogot aCOLOMBIAT reasurer:EmeritusProfessorPeterTaylorPOB ox6165O ConnorACT2601 AUSTRALIAR egionalRepresentativesAfrica.

3 ProfessorJohnWebbDepartmentofMathematics UniversityofCapeTownRondebosch7700 SOUTHAFRICAAsia:vacantEurope:ProfessorNi kolayKonstantinovPOBox68 Moscow121108 RUSSIAP rofessorFranciscoBellot-RosadoRoyalSpani shMathematicalSocietyDosDeMayo16-8#DCHAE -47004 ValladolidSPAIN2 MathematicsCompetitionsVol30No12017 NorthAmerica:ProfessorIanVanderBurghUniv ersityofWaterlooWaterlooCANADAO ceania:ProfessorDerekHolton605/228 TheAvenueParkvilleVic3052 AUSTRALIAS outhAmerica:ProfessorRadmilaBulajichFacu ltaddeCiencias, , ,Morelos62209 ,andresearchassociatedwith,mathematicsed ucationthroughtheuseofschoolmath-ematics COMPETITIONS ; ; ,no-tablythroughtheJournaloftheFederatio n; ; ; ,Iamhappytoinviteyoutothe8thInternationa lCongressoftheWorldFederationofNationalM athematicsCompetitions(WFNMC).

4 ItwilltakeplaceonJuly18 24,2018nearGraz,Austria,andtheChiefOr-ga nizerisourVicePresidentRobertGeretschl :ThejournalMathematicsCompetitionsisinte restedinreceivingarticlesdealingwithmath ematicscompetitions,notonlyatnationaland inter-nationallevel, Thejournaltraditionallycontainsmanydiffe rentkindsofarti-cles,includingreports, Tomaintainandimprovethequalityofthejourn alanditsuse-fulnesstothoseinvolvedinmath ematicscompetitions, ,butcanbesummarisedbysayingthatanarticle acceptedmustbecorrectandappropriate,thec ontentaccurateandinteresting,and,whereth efocusismathematical, ,themostimportantthingisthatifyouhaveany thingtocontributeonanyaspectofmathematic scompetitionsatanylevel,local,regionalor national, , , , ,andcorrespondence,canalsobeforwardedtot heeditorbymailtoTheEditor.

5 MathematicsCompetitionsAustralianMathema ticsTrustUniversityofCanberraLockedBag1 CanberraGPOACT2601 AUSTRALIA ortoDrJaroslav Svr Svr cekOctober20177 MathematicsCompetitionsVol30No12017 CallfornominationsPaulErd osAward2018 TheAwardsCommitteeoftheWorldFederationof MathematicalCom-petitionsherebylaunchest heprocessofselectingawardeesofthePaulErd , osAwardThefollowingdocumentsandadditiona linformationmustbesubmittedinEnglish: Aone-totwo-pagestatementwhichincludesach ievementsofthenomineeandadescriptionofth econtributionsbythenomineewhichreflectst heobjectivesofWFNMC. Nomineespresentcontactinformation,homeor businessaddress,telephone,e-mailaddresse s. ,whereasalltheworkattributabletothecandi datemaybeentirelywithinonesowncountry, : MembersoftheWorldFederationofNationalMat hematicsCom-petitionsExecutiveCommittee RegionalRepresentativesofWFNMCTheFederat ionencouragesthesubmissionofsuchnominati onsfromDi-rectorsorPresidentsofInstituti onsandOrganisations,fromChancellorsorPre sidentsofCollegesandUniversities, ,BulgarianAcademyofSci-ences, , , (ChH)startedin1992adopt-ingtheformatofth eAustralianMathematicsCompetition(AMC)[6 ].

6 Inthebeginningtherewasjustonecompetition paperforallparticipantswhowereadvanced9t h 11thgradestudents[3].ThepresentformatofC hHincludes5divisionsforstudentsoftwocons ecutivegrades,startingwith3rd&4thandfini shingwith11th& , [9].Figure1 performance, (whichwasthecaseinthe2015issueofChH).Sim ilaristhecasewithAMC1990 IntermediateDivisionwherethe9thand10thgr adersgraphsalmostcoincides,butasignifica ntshiftofthegraphsappearsagainintheSenio rDivision[9]. [4],[5]. , (takeintoaccountthatablankresponsesheeti sworth60points). ,weconsiderthescoresbetween80and85points asaverage, (VR)isnormal-distribution-likefortheenti repopulationandthereisashiftoftheVRshape of6thgradewithrespecttotheVRof5thgrade,w hichissimilartotheoneinFigure1[8].

7 :5thgradestudents 72in2014and56in2015;6thgraders (item)willbe(yyyy,n) (item)nthedifficultyofthecorrespondingte stitemforthen-thgradestratum(n=5,6).Thev aluesforT(item)narecalculatedas(100minus thepercentageofthenumberofcorrectrespons es), (item) :Distributionof itemfor2014and2015competitionpapersWedef inetheage-factorindex(AFI)ofthetestitema sthedifference (item)=T(item)5 T(item) (item) (item).(2015,10)Whichisthearithmeticoper ationreplacedbythesymbol intheequation(555+5 55) (5 5:5) 5=2015?(A)addition(B)subtraction(C)divis ion(D)multiplication(E) 4 5=2015;505 4=2020 505 4= (D). (2015,10)5= ,T(2015,10)6= (2015,10)= ;D(2015,10)5= ,D(2015,10)6= (2015,10) (2015,10)5,6 (2015,4) ,1943, (A)43(B)16(C)27(D)20(E) ,1916,andthereforehewas43 16= (C).

8 (2015,4)5= ,T(2015,4)6= (2015,4)= 2;D(2015,3)5= ,D(2015,4)6= , , ,despitetheyarerelatedtosometopicsinclud edintheschoolcurriculum.(2014,9)Whatisth eanglebetweentheclockhandsat10:10o clock?(A)120 (B)118 (C)115 (D)112 (E) .In10minutesthehourhandtravels16ofthearc between10and11marks, .Thus,theanglebetweenthehourhandandthe12 hourmarkis55 .Answer(C). (2014,9)5= ,T(2014,9)6= (2014,9)= ;D(2014,9)5= ,D(2014,9)6= (2014,9) (2014,9) (2014,17)TriangleABCisdividedintofivesma llertriangleswithequalareas:AMQ,PMQ,MNP, ,howlong(incm)isMN?..ABCQPMN(A)12(B)15(C )16(D)18(E) [ANC]=4[BNC]followsthatAN=4BN,whichyield sBN=9,AN= [APM]=2[MNP]followsthatAM=2MN,whichyield sMN= (A). (2014,17)5= ,T(2014,17)6= (2014,17)= ;D(2014,17)5= ,D(2014,17)6= [PKL][PMN]=KLMN forcollinearpointsK,L,M,Nisimplicitlyinc ludedintheschoolcurriculum (2014,17) (2014,17) (2014,17) experience , (2014,19)Aboxis14cmlongand1cmwideandisdi videdinto14squarecellsof1 (A)6(B)10(C)12(D)14(E) (otherwisethefleawilllandoutsidethebox).

9 Thefleacouldmake13jumpslikethis:1+2 3+4+5 6+7 8+9 10+11 12+ (E). (2014,19)5= ,T(2014,19)6= (2014,19)= ;D(2014,19)5= ,D(2014,19)6= ,theyjusttestedoneoranotherroadofF :practicallysameresultsinbothagestrata.( 2014,20)Inhowmanywayscanthenumber5berepr esentedasasumofsmallerpositiveintegersif theorderofsummandsmatters?Forexample,3=1 +2=2+1=1+1+1, (A)5(B)10(C)15(D)30(E) ,forexample,if5=1+2+2,weconsiderthisrepr esentation:5=1+(1+1)+(1+1).Itisequivalen ttothefollowingbreakingoffiveoneswiththe divider*:1*11* ,wecaneasilyestablishthatthenumberofways wearelookingforisequaltothenumberofwayst o17 MathematicsCompetitionsVol30No12017break down11111withatleastone*.Butthisnumberis equaltothenumberofwaystoputornottoputadi vider*in4placeswithatleastonedividerpres ent, (C).

10 (2014,20)5= ,T(2014,20)6= (2014,20)= ;D(2014,20)5= ,D(2014,20)6= (2014,19) ( ) ,thereisconsiderabledifferencebetween(20 14,19)and(2014,20) (2015,3)Letusdefinetwonewoperationsforth enumbersaandb:a b=ab a banda b=ab+a+ ,5 2=5 2 5 2=3,5 2=5 2+5+2= 62 3 + x(A)20(B)12(C)22(D)18(E) 6=9,2 3=11,x=9+ (A). (2015,3)5= ,T(2015,3)6= (2015,3)= ;D(2015,3)5= ,D(2015,3)6= [2].However,thelargevalueof (2015,3)indicatesthatsuchcombinationofid easisinsidethe6thgraderszoneofproximalde velopment(ZPD)butfortheonethirdofthefive gradersitisbeyondtheirZPD[10].(2015,12)A lex, ,thenadded3totheresult,thendividedthenew resultto3, ,multipliedthere-sultto3,subtracted3from theproduct, ,thenmultipliedtheresultby3,added3tothep roduct, ,BandC?