Download - MATHCOUNTS - Eat Pie Institute of Mathematics State Countdown.pdf · Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Countdown Round x + 3 = m

2013State CompetitionCountdown Round

Problems 1–80

This section contains problems to be used in the Countdown Round.

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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)


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)


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)


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


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)


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)


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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)