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Transcript of MATHCOUNTS - Eat Pie Institute of State Countdown.pdf · PDF fileCopyright MATHCOUNTS,...

  • 2013State CompetitionCountdown Round

    Problems 180

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

    MATHCOUNTS

    Copyright MATHCOUNTS, Inc. 2013. All rights reserved.

    Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation

    03-S13CDR

    National Sponsors Raytheon Company * Northrop Grumman Foundation * U.S. Department of Defense * National Society of Professional Engineers * CNA Foundation * ConocoPhillips * Texas Instruments Incorporated * 3M Foundation * Art of Problem Solving * NextThought

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

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    AtRocketDogs,fourhotdogsandthreeordersoffriescost$9.Sixhotdogsandfiveordersoffriescost$14.Whatisthetotalcostofonehotdogandoneorderoffries?

    Ifa+b4z=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(11+21)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.Ifx245b2=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!leavesnoremainderwhendividedby2n1?

    Whatisthevalueof1313212122?

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    (points)

    (in2)

    (marbles)

    30

    60

    (inches)

    (integers)

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

    Samglancesattheclockandnoticesthattheminutehandis55behindthehourhand.Inhowmanyminuteswillthehandsoverlap?

    Forwhatpositiveintegernis2n60?

    Onthenumberlineshown,therearedotsateveryintegerandthreedotsbetweeneachpairofconsecutiveintegers.Ifallofthedotsareevenlyspaced,howmanydotsliestrictlybetweena(notshown)and a ?

    Asquarehasadiagonaloflengthacm.Insquarecentimeters,whatistwicetheareaofthesquareintermsofa?

    Howmanypositiveintegersareafactorof7!butnotafactorof6!?

    Ifk=12131415

    120

    + + + + + ,anda