2012National Competition

Sprint Round with ANSWERSProblems 1–30

Copyright MATHCOUNTS, Inc. 2012. All rights reserved.

MATHCOUNTS

®

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04-N12SPR-A

Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2012 National Sprint Round

1. _____________

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Hyramishelpinghislittlesisterwriteherintegersfrom1to50.Hesuggeststhattheytakeabreakaftershehaswrittenher33rddigit.Whatwasthelasttwo-digitnumberthatHyram’slittlesisterwrotebeforetheytookthebreak?

The660studentsatMandelbrotMiddleSchoolvotedontheirchoiceforfavoriteamongsixmathematicians.Thetableshowstheresultsofthevote.Finnmadeacirclegraphtorepresentthedatainthetable.Indegrees,whatisthemeasureofthecentralangleofthesectorthatrepresentsvotesforGauss?

Whatisthesumofallpositiveintegersnsuchthat 3 ≤n≤ 99 ?

Tenitemshavethefollowingweights,inpounds:2,3,5,5,6,6,8,8,8,9.Theitemsarethendividedintothreesetsof20poundseach.Howmanysetscontainan8poundweight?

Thesumoffivedifferentpositiveintegersis80.Whatisthelargestpossiblevalueofthesecondlargestnumber?

Fromthetimeashopopensat10a.m.,onecustomerenterstheshopevery15minutes,untiltheshopclosesat7p.m.Thereisa 13 chancethatthesalesmanwillconvincethecustomertobuyawidget.Theshopownermakesaprofitof$6oneachwidgetsold.Whatisthemosttheshopownercanpaythesalesman(indollarsperhour)toexactlybreakeven?Thatis,whathourlyratewillmaketheamountpaidtothesalesmanequalthetotalamountofincomefromwidgetsales?

degrees

sets

$

Mathematicians Number of Votes Euclid 100 Gauss 121

Germain 200 Hypatia 48 Pascal 66

Pythagoras 125

21

66

44

3

36

8.00or8

perhr

Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2012 National Sprint Round

TheoddsagainsteventAoccurringare3:1andtheoddsinfavorofeventBoccurringare3:5.Whatistheprobabilitythatatleastoneoftheseindependenteventsoccurs?Expressyouranswerasacommonfraction.

ThepopulationofExpotowniscurrently54,000people.Ifthepopulationhasincreasedby200%inthepast30years,whatwasthepopulation30yearsago?

Ifa+2b +3c =30anda +3b +5c =8,whatisthevalueofa +b +c?

Abagcontainstenidenticalbluemarblesandtenidenticalgreenmarbles.Inhowmanydistinguishablewayscanfiveofthesemarblesbeputinarowifthereareatleasttwobluemarblesintherowandeverybluemarbleisnexttoatleastoneotherbluemarble?

Forhowmanyintegers,n,in{1,2,…,20}isthetensdigitofn2odd?

LineLcontainingthepoint(−11, −4)hasthesamey-interceptasy=−3x −2.Whatistheproductoftheslopeandy-interceptoflineL?Expressyouranswerasacommonfraction.

7. _____________

8. _____________

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11._____________

12._____________

people

ways

integers

18,000

52

11

4

-411

1732

Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2012 National Sprint Round

13._____________

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TriangleABCisarighttrianglewithAB=8units,

BC=10unitsandAC=6units.TrianglesBDC,AFB

andCEAareisosceleswithvertexanglesof90°,120°

and120°,respectively.IfRdenotestheareaofΔCEA,

SdenotestheareaofΔAFBandTdenotestheareaof

ΔBDC,whatisthevalueof R ST+ ?Expressyouranswer

asacommonfractioninsimplestradicalform.

Inthefigure,acircleislocatedinsideatrapezoidwithtworightanglessothatapointoftangencyofthecircleisthemidpointofthesideperpendiculartothetwobases.Thecirclealsohaspointsoftangencyoneachbaseofthetrapezoid.Thediameterofthecircleis 23 thelengthofsegmentEF,asshown.Iftheareaofthecircleis9πunits2,whatistheareaofthetrapezoid?

Thesumofthereciprocalsofthreeconsecutivepositiveintegersisequalto47dividedbytheproductoftheintegers.Whatisthesmallestofthethreeintegers?

Thefirstthreetermsofageometricsequenceare3,6+pand30–p,andeachtermisapositivenumber.Whatisthefifthtermofthissequence?

Astandarddiewhosefacesarenumbered1,2,3,4,5,6isrolledthreetimes.Whatistheprobabilitythatthesumofthenumbersrolledis8?Expressyouranswerasacommonfraction.

units2

E F

E

A B

C

D

F

54

3

243

772

33

Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2012 National Sprint Round

18._____________

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Ifx,yandz satisfyxy =1,yz =2andxz =3,whatisthevalueofx2+y2+z2?Expressyouranswerasacommonfraction.

Acubicalroomhasedgelength10feetwithAandBdenotingtwocornersthatarefarthestapart.AcaterpillarcrawlsfromAtoBalongthewalls.Infeet,whatisthelengthoftheshortestpathfromAtoBthatthecaterpillarmayhavetaken?Expressyouranswerinsimplestradicalform.

Whatisthemaximumvalueof xyif x+ 1y =72 andy+

1x =

75 ?Expressyour

answerasadecimaltothenearesttenth.

Asphereofradius4inchesisinscribedinaconewithabaseofradius6inches.Ininches,whatistheheightofthecone?Expressyouranswerasadecimaltothenearesttenth.

IncircleO,shown,OP=2units,PL=8units,PK=9unitsandNK=18units.PointsK,PandMarecollinear,asarepointsL,P,OandN.WhatisthelengthofsegmentMN?

Considerthesequence1,2,3,4,5,6,7,8,9,19,29,…,wherethesumofthedigitsofan,thenthterm,isequalton,andan+1isthesmallestpositiveintegersuchthatan +1>an.Whatisthevalueofthe25thtermofthissequence?

feet

N

M

K

L

O

P

units

inches

10 5

2.5

14.4

12

799

496

Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2012 National Sprint Round

24._____________

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Fourequilateraltriangles,ABG,BCH,CDEandDAF,areconstructedinsidesquareABCD,asshown.PointsE,F,GandHaretheverticesofthetrianglesthatliewithinsquareABCD.WhatistheratiooftheareaofsquareEFGHtotheareaofsquareABCD?Expressyouranswerinsimplestradicalform.

Inaround-robinchesstournamenteveryplayerplaysonegamewitheveryotherplayer.Fiveparticipantswithdrewafterplayingtwogameseach.Noneoftheseplayersplayedagameagainsteachother.Atotalof220gameswereplayedinthetournament.Includingthosewhowithdrew,howmanyplayersparticipated?

Thebase-threerepeatingdecimal012. isequivalenttowhatbase-tencommonfraction?

Anarithmeticseriesofpositiveintegershas8termsandasumof2008.Whatisthesmallestpossiblevalueofanymemberoftheseries?

Howmanywholenumbersn,suchthat100≤n ≤1000,havethesamenumberofoddfactorsasevenfactors?

Ifxandy satisfy(x−3)2+(y−4)2=49,whatistheminimumpossiblevalueofx2+y2 ?

Eightidenticalunitcubesarestackedtoforma222cube,asshown.A“shortpath”fromvertexAtovertexBisdefinedasonethatconsistsofsixone-unitmoveseitherright,uporbackalonganyofthesixfacesofthe2-unitcube.Howmany“shortpaths”arepossible?

shortpaths

numbers

A B

CD

E

F

G

H

A

B

players

2− 3

26

58

6

225

4

54