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2012National Competition
Sprint Round with ANSWERSProblems 1–30
Copyright MATHCOUNTS, Inc. 2012. All rights reserved.
MATHCOUNTS
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04-N12SPR-A
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Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2012 National Sprint Round
1. _____________
2. _____________
3. _____________
4. _____________
5. _____________
6. _____________
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
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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. _____________
9. _____________
10._____________
11._____________
12._____________
people
ways
integers
18,000
52
11
4
-411
1732
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Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2012 National Sprint Round
13._____________
14._____________
15._____________
16._____________
17._____________
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
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Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2012 National Sprint Round
18._____________
19._____________
20._____________
21._____________
22._____________
23._____________
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
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Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2012 National Sprint Round
24._____________
25._____________
26._____________
27._____________
28._____________
29._____________
30._____________
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