2013State CompetitionCountdown Round
Problems 1–80
This section contains problems to be used in the Countdown Round.
MATHCOUNTS
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Copyright MATHCOUNTS, Inc. 2013. All rights reserved.
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Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Countdown Round
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AtRocketDogs,fourhotdogsandthreeordersoffriescost$9.Sixhotdogsandfiveordersoffriescost$14.Whatisthetotalcostofonehotdogandoneorderoffries?
Ifa+b−4z=500anda+b+6z=1200,whatisthevalueofa + b?
Whatisthevalueof10+20+30++20130−(1+2+3++2013)0?
Whatisthepositivedifferencebetweenthemedianandthemeanof18,21,24,27,31,35and40?
Ifonebarofchocolatesquaresis210calories,andthereare15squaresinabar,howmanycaloriesarein1chocolatesquare?
Howmanyminuteswillittaketodrive30milesatanaveragerateof40milesperhour?
Ifthreepainterscanpaintthreehousesinthreedays,howmanydaysdoesittakeonepaintertopainttwohouses?
Whatistheunitsdigitofthesum12+22+32++102?
EquilateraltrianglesABCand DEFareinscribedinacirclewiththeircorrespondingverticesatdiametricallyoppositepoints,asshown.WhatfractionoftheareaofABCisgray?Expressyouranswerasacommonfraction.
Whatisthevalueof(1−1+2−1)−1?Expressyouranswerasacommonfraction.
Theeighthtermofanarithmeticsequenceistentimesthefirstterm,andthethirdtermofthesequenceis50.Whatisthefirsttermofthesequence?
Howmanypositiveintegerfactorsdoes108have?
Twelvecoins,consistingofdimesandnickels,areworth85cents.Whatisthenumberofdimes?
(factors)
(calories)
($)
(days)
A B
C
DE
F
(minutes)
(dimes)
Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Countdown Round
If5m ≥
23 ,whatisthelargestpossiblevalueofm?Expressyouranswerasa
decimaltothenearesttenth.
Two30-60-90triangles,eachwithhypotenuse12inches,overlapsothattheyshareavertexandpartofaside,asshown.Whatisthetotalarea,insquareinches,oftheresultingfigure?Expressyouranswerinsimplestradicalform.Ifx2−45b2=4xb,whatisthelargestpossibleratioofbtox?Expressyouranswerasacommonfraction.
Howmanypositivetwo-digitintegersaremultiplesof6?
Bothpandqarepositiveintegers.Eachofthenumbers29,84and128leavesaremainderofpwhendividedbyq.Whatisthesumofp+q?
Abagcontainsonlyredandblackmarbles.Originallytheratiooftheblackmarblestoredmarbleswas4:5.After18blackmarbleswereadded,theratioofblackmarblestoredmarblesbecame5:4.Howmanymarbleswereinthebagoriginally?
IfConsuelahasameanscoreof84outof100pointsforfivetests,whatisthelowestpossibletestscoreshecouldhavereceived?
Inthesequence1,2,2,3,3,3,4,4,4,4,…,eachpositiveintegerkappearsktimes.Iftheinteger12firstappearsinthesequenceasthenthterm,whatisthevalueofn?
Thelegsofarighttrianglemeasure123 inchesand4inches.Ininches,whatis
thelengthofthehypotenuse?Expressyouranswerasamixednumber.
Whatisthegreatestsingle-digitpositiveintegernforwhichn!leavesnoremainderwhendividedby2n −1?
Whatisthevalueof13132–12122?
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(points)
(in2)
(marbles)
30˚
60˚
(inches)
(integers)
Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Countdown Round
Samglancesattheclockandnoticesthattheminutehandis55°behindthehourhand.Inhowmanyminuteswillthehandsoverlap?
Forwhatpositiveintegernis2n<100<2n+1?
Whatistheleastvalueofn,suchthatn>1andthevalueof 1 2 3 n+ + + + isaninteger?
Howmanyzerosaretotherightofthelastnonzerodigitof78!?
Atriangleisboundedbythelinesx=3,y=10andy=−x+11.Whatisitsarea,insquareunits?
Allbuttwostudentsinthealgebraclasstookatest,andthemeanscorewas84.Whenthescoresof100and94forthemissingtwostudentswereincluded,theaverageincreasedby1.Howmanystudentsareintheclass?
Inthefigure,eachofthefourlargecirclesistangenttotwooftheotherlargecircles,thesmallcircle,andtwosidesofthesquare.Iftheradiusofeachofthelargecirclesis1unit,whatistheradiusofthesmallcircle,inunits?Expressyouranswerinsimplestradicalform.
Forwhatpositiveintegernisthemeanofthefirstnpositiveintegersequalto2013?
Forpositiveintegersaandb,27 oftheproductabis60,and
35 ofthequotienta
b is914 .Whatisthevalueofa?
Anarithmeticprogressionhas20asits12thtermand12asits20thterm.Whatisthe2013thterm?
Whatnumberis10%ofone-thirdof750?
Duringthebasketballseason,Carascored25pointsmorethanAnnika,andMariascored37pointslessthanAnnika.WhatisthepositivedifferencebetweentheaveragenumberofpointsscoredbythethreeplayersandthenumberofpointsthatAnnikascored?
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(minutes)
(zeros)
(students)
(units)
(units2)
(points)
Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Countdown Round
Thequadraticequationx2+bx+c=0hasrealroots4and−6.Whatisthevalueofb+c?
IfarandomlythrowndarthitstheinteriorofABC,shownhere,whatistheprobabilitythatitalsohitstheinteriortrapezoidBDEC?Expressyouranswerasacommonfraction.
ThediagonalsofrhombusMNOP,havelengthsof12mmand16mm.Whatistheperimeter,inmillimeters,ofrhombusMNOP?
Whatisthesumofthepositivefactorsof45?
Anisoscelestrapezoidwithonebasetwiceaslongastheotherhasaheightof12cm.Theshorterbaseis18cm.Incentimeters,whatistheperimeterofthistrapezoid?
Whatisthelargestthree-digitpositiveintegerwithexactlythreepositivefactors?
Afueltankholds200gallonsofgasolinewhenfull.Thetankiscurrently45
full.Iffuelcosts$5.50pergallon,howmanydollarsworthoffuelareneededto
filltherestofthetank?Usingfourofthefivedigits1,5,6,8and9,howmanyfour-digitoddnumberscanbeformed,withnorepeateddigits?
Inarighttriangle,thelongerlegandthehypotenusehaveconsecutiveintegerlengthswhosesumis121inches.Ininches,whatistheperimeterofthetriangle?
Whatistheunitsdigitoftheproduct1×3×5×7×9×…×2013?
Inthefigure,theareaoftrapezoidDBCEis80cm2.TheratioofthebasesDEtoBCis3:5.WhatistheareaoftriangleADE,insquarecentimeters?
Whatistheunitsdigitoftheproduct2×4130?
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(mm)
A
D
B C
E
(cm)
(dollars)
(oddnumbers)
(inches)
(cm2)
A
B C
D E
Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Countdown Round
If2x =wandx ≠0,whatisthevalueof(wx)−3?Expressyouranswerasa
commonfraction.
Whatisthevalueof1332+(2)(67)(133)+672?
Thefigureshowniscomposedoftwosemi-circlesandonerighttrianglewithalegof10mmandahypotenuseof26mm.Whatistheperimeterofthefigure,inmillimeters?Expressyouranswerintermsofπ.
If25%of16isx,whatisx%of200?
Whatisthemeanofthedatarepresentedinthefrequencydistributionshown?
Dianna’sweeklypaycheckis15%lessthanBarbara’spaycheck.Dianna’sandBarbara’spaycheckshaveacombinedtotal$740.HowmuchisDianna’spaycheck,indollars?
Whatistheleastpositiveintegern,forwhichtherearepositiveintegersxandysuchthatx2+y3=nandn>60?
Onthenumberlineshown,therearedotsateveryintegerandthreedotsbetweeneachpairofconsecutiveintegers.Ifallofthedotsareevenlyspaced,howmanydotsliestrictlybetweena(notshown)and a ?
Asquarehasadiagonaloflengthacm.Insquarecentimeters,whatistwicetheareaofthesquareintermsofa?
Howmanypositiveintegersareafactorof7!butnotafactorof6!?
Ifk=12131415
120
+ + + + + ,anda<k<a+1whereaisaninteger,whatis
thevalueofa?
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1026
XX X XX X X XX X X X X X X X1 2 3 4 5 6 7 8 9
(dollars)
(mm)
1 2 3a
(dots)
(cm2)
(integers)
Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Countdown Round
Ifkisaconstant,and2x+5=3kx+5forallvaluesofx,whatisthevalueofk?Expressyouranswerasacommonfraction.
If12c=b2+4b+4,
wherebandcarepositive,thenwhatisthevalueofcin
termsofb?Expressyouranswerasacommonfraction.
Whatis23 of30%of362,970?
Iff(x)=xx+-3
3 1 andg(x)= 23x ,whatisthevalueof ( )1 2
2f g +
?
Whatisthemaximumintegervalueofxforwhich2xislessthaneachofthenumbers:413,89and167?
Annewasabsentforthelastalgebratest.Her19classmateshadaclassaverageofexactly77pointsonthistest.HowmanypointsmustAnnescoreonthistesttoraisetheclassaveragebyexactlyonepoint?
Iff(x)=x−11andg(x)=2x2+3x−2,whatisthevalueofg[f(10)]?
Threeanglesofaconvexpentagonmeasure88°,124°and92°.Theremainingtwoanglesarecongruent.Whatisthedegreemeasureofeachoftheremainingangles?
Thetopofapolerests8ftabovethegroundalongawall.Ifthebottomofthepoleismovedanother2ftfromthewall,theentirepolewilllieonthegroundwiththetopofthepoletouchingthewall.Infeet,howlongisthepole?
Howmanyintegersxsatisfy|13πx |<−π?
Whatistheclosestintegerto10.013?
Intheequation(x−8)(x−k)=x2−5kx+m,kandmarepositiveintegers.Iftheequationistrueforallvaluesofx,whatisthevalueofm?
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(integers)
(feet)
(points)
(degrees)
Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Countdown Round
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If3x+3x+3x+3x+3x+3x+3x+3x+3x=3n,whatisthevalueofn,intermsofx?
SquareWXYZislocatedinthefirstquadrantsothatsegmentsWXandYZareparalleltothey-axis.ThecoordinatesofWare(3,6),andthesquarehasarea16units2.IfsquareWXYZistranslated2unitsupand6unitstotheright,whatisthesumofthecoordinatesoftheimageofY?
If4+ b =7.2,whatisthevalueof4− b ?Expressyouranswerasadecimaltothenearesttenth.
Achefspendschourseachdayatherrestaurant,c3 hoursofwhichshespends
preparingdesserts.Atthisrate,afterhowmanydayswillthechefhavespent24hourspreparingdesserts?Expressyouranswerasacommonfractionintermsofc.
Thepoints(6,1)and(8,1)lieonthegraphofy=f(x),wheref(x)=|x−k |andkisaconstant.Whatisthevalueoff(0)?
Foursetsoflightssurroundabaseballfield.Thefieldissufficientlylitwhenatleasttwoofthesetsareon.Inhowmanydifferentwayscanlightsbeturnedonsothatthefieldissufficientlylit?
If(m+n)k−3=m6+6m5n+15m4n2+20m3n3+15m2n4+6mn5+n6,whatisthevalueofk?
Bartplayedagameinwhichhecouldscoreeither9or14pointsoneachturn.IfBartscoredexactly100points,onhowmanyturnsdidhescore9points?
InJanuary2013,bboysandggirlsbelongedtoamathclub.ThreegirlsandsomeboysjoinedthemathclubinFebruary2013andtheratioofboystogirlsremainedthesame.Intermsofbandg,howmanyboysjoinedthemathclubinFebruary2013?Expressyouranswerasacommonfraction.
(days)
W
YX
Z(3, 6)
y
x
(turns)
(boys)
(ways)