Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round

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Ifthesumoftwonumbersis4andtheirdifferenceis2,whatistheirproduct?

MaryandAnnliveatoppositeendsofthesameroad.Theyplantoleavehomeatthesametimeandridetheirbikestomeetsomewherebetweenthetwohouses.At11:00a.m.Maryhastraveledhalfofthedistancebetweentheirhouses.Annisridingmoreslowlyandhascoveredonly

38ofthedistance

betweenthehouses.Theyarestillonemileapart.Howmanymilesapartaretheirhouses?

Abagcontains8bluemarbles,4redmarblesand3greenmarbles.Inasingledraw,whatistheprobabilityofnotdrawingagreenmarble?Expressyouranswerasacommonfraction.

Thearithmeticmeanof11numbersis78.If1issubtractedfromthefirst,2issubtractedfromthesecond,3issubtractedfromthethird,andsoforth,until11issubtractedfromtheeleventh,whatisthearithmeticmeanofthe11resultingnumbers?

Anoptometristhasthislogoonhisstorefront.Thecentercirclehasarea36πin2,anditistangenttoeachcrescentatitswidestpoint(AandB).TheshortestdistancefromAtotheoutercircleis

13 thediameterofthesmaller

circle.Whatistheareaofthelargercircle?Expressyouranswerintermsofπ.

in2

miles

A

B

Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round

Excludingsalestax,howmuchwillDorissavewhenshebuysaDVDoriginallypricedat$12.00andnowonsalefor20%off ?

Whatisthevalueof2 2444 111

444 111−− ?

Theproductofthedigitsofpositiveintegern is20,andthesumofthedigitsis13.Whatisthesmallestpossiblevalueofn?

QuadrilateralABCDisasquarewithBC=12cm.BOC andDOC aresemicircles.Intermsofπ,whatistheareaoftheshadedregion?

Realnumbersaandbsatisfytheequation 2 4 3 15 5− +

+ =a a b .Whatisthe

valueofa−b?Expressyouranswerasacommonfraction.

Ifthepoint(x,x)isequidistantfrom(−2,5)and(3,−2),whatisthevalueofx?

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cm2

O

C

B

D

A

12

$

Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round

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Inabagofmarbles,25 ofthemarblesarered,

310 ofthemarblesare

whiteand110 ofthemarblesareblue.Iftheremaining10marbles

aregreen,howmanymarblesareinthebag?

Iftis40%greaterthanp,andpis40%lessthan600,whatisthevalueoft–p?

Howmanywayscanallsixnumbersintheset{4,3,2,12,1,6}beorderedsothatacomesbeforebwheneveraisadivisorofb?

Whatistheunitsdigitoftheproduct723×8105×318?

If4(a–3)–2(b+5)=14and5b–a=0,whatisthevalueofa+b?

ThetwoconesshownhaveparallelbasesandcommonapexT.TW=32m,WV=8mandZY=5m.WhatisthevolumeofthefrustumwithcircleWandcircleZasitsbases?Expressyouranswerintermsofπ.

Acoinisflippeduntilithaseitherlandedheadstwotimesortailstwotimes,notnecessarilyinarow.Ifthefirstfliplandsheads,whatistheprobabilitythatasecondheadoccursbeforetwotails?Expressyouranswerasacommonfraction.

marbles

m3

ways

T

YZX

VWU

Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round

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Theproductoftwoconsecutiveintegersisfivemorethantheirsum.Whatisthesmallestpossiblesumoftwosuchconsecutiveintegers?

Fournickels,onepennyandonedimeweredividedamongthreepiggybankssothateachbankreceivedtwocoins.Labelsindicatingtheamountineachbankweremade(6cents,10centsand15cents),butwhenthelabelswereputonthebanks,nobankhadthecorrectlabelattached.Sorayashookthepiggybanklabeledas15cents,andoutfellapenny.Whatwastheactualcombinedvalueofthetwocoinscontainedinthepiggybankthatwaslabeled6cents?

Supposethe9×9multiplicationgrid,shownhere,werefilledincompletely.Whatwouldbethesumofthe81products?

Insomelanguages,everyconsonantmustbefollowedbyavowel.Howmanyseven-letter‟words”canbemadefromtheHawaiianwordMAKAALAifeachconsonantmustbefollowedbyavowel?

Iff(x)=3x2,whatisthex-coordinateofthepointofintersectionofthegraphsofy=f(x)andy=f(x−4)?

InisoscelestrapezoidABCD,shownhere,AB=4unitsandCD=10units.PointsEandFareonCD with BE paralleltoAD andAF paralleltoBC .AF andBE intersectatpointG.WhatistheratiooftheareaoftriangleEFGtotheareaoftrapezoidABCD?Expressyouranswerasacommonfraction.

‟words”

A B

CDE F

G

cents

× 1 2 3 4 5 6 7 8 9123456789

14

12

40

6¢ 10¢

15¢

Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round

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Thesumoffiveconsecutive,positiveevenintegersisaperfectsquare.Whatisthesmallestpossibleintegerthatcouldbetheleastofthesefiveintegers?

If123+125+127+129+12x=1011102,whatisthevalueofx,thebaseofthefifthterm?

Aboxcontains r redballsandggreenballs.Whenrmoreredballsareaddedtothebox,theprobabilityofdrawingaredballatrandomfromtheboxincreasesby25%.Whatwastheprobabilityofrandomlydrawingaredballfromtheboxoriginally?Expressyouranswerasacommonfraction.

ThegameofConnexcontainsone4-unitpiece,twoidentical3-unitpieces,threeidentical2-unitpiecesandfouridentical1-unitpieces.Howmanydifferentarrangementsofpieceswillmakea10-unitsegment?The10-unitsegmentsconsistingofthepieces4-3-2-1and1-2-3-4aretwosucharrangementstoinclude.

Insquareunits,whatisthelargestpossibleareaarectangleinscribedinthetriangleshownherecanhave?

AlinesegmentwithendpointsA(3,1)andB(2,4)isrotatedaboutapointintheplanesothatitsendpointsaremovedtoA' (4,2)andB' (7,3),respectively.Whatarethecoordinatesofthecenterofrotation?Expressyouranswerasanorderedpair.

10 17

21

units2

(,)

arrange-ments