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