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2012State CompetitionCountdown Round
Problems 1–80
This section contains problems to be used in the Countdown Round.
MATHCOUNTS
®
Copyright MATHCOUNTS, Inc. 2011. All rights reserved.
Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation
03-S12CDR
National Sponsors
Raytheon Company * U.S. Department of Defense * Northrop Grumman Foundation * National Society of Professional Engineers * Bezos Family Foundation * ConocoPhillips * CNA Foundation * Texas Instruments Incorporated * ThinkFun * 3M Foundation
2012 MATHCOUNTS National Competition Sponsor
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Countdown Round
1. _____________
2. _____________
3. _____________
4. _____________
5. _____________
6. _____________
7. _____________
8. _____________
9. _____________
10._____________
11._____________
12._____________
13._____________
If6andxhavethesamemeanastheset{2,4,24},whatisthevalueofx?
RightXYZhassideswithlengthsx,yandzunits,suchthatx>y>z.If
P=15x2+5y2+5z2andQ=y2+z2,whatisthevalueofPQ ?
Forallsubsetsoffivedistinctintegersfromthesetofconsecutiveintegers{1,2,3,...,99,100},whatisthelargestpossibledifferencebetweenthemeanandmedianofasubset?
Whatisthevalueoftheproduct15 ×0.1×
12 ×5×
12 ×8×5?
EachwidgetproducedbytheWidgetCompanyistobeassignedauniqueproductcodeconsistingofaletterfollowedbytwodigits.Whatisthenumberofallpossibleproductcodes?TwoexamplestoincludeareG55andR92.
Theareaofaregularhexagonis96 3 cm2.Whatisitsperimeter,incentimeters?
Ifn= 27 ,whatisthevalueofn23 ?
Ifaϒb=b a∙a (b − a),whatisthevalueof(2ϒ5)?
Whatisthearea,insquareunits,oftheregioninthexy-planeboundedbyy=x,y=2andx=6?
Iff(x)=x 2−2x−5andg(x)=x 3−11,whatisthevalueoff (4)+g (5)?
Whatistheunitsdigitoftheproduct242∙319?
Theprimefactorizationof720is2a∙3b∙5c.Whatisthevalueofa+b+c?
Inright∆ABC,shownhere,AB=15units,AC=24unitsandpointsD,EandFarethemidpointsofAC ,AB andBC,respectively.Insquareunits,whatistheareaofDEF?
(cm)
(codes)
(units2)
C
D F
A E B
(units2)
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Countdown Round
Inhislastgameofbowling,Robertoearnedascoreof241,andhisaveragescoreincreasedfrom205to209.Whatscorewouldhehaveneededinthelastgametoincreasehisaverageto211?
Whatdigitisinthehundred-thousandthsplacewhen(0.5)5issimplifiedandwrittenasadecimal?
If110objectsareseparatedintotwogroupssothatonegroupcontains150%ofthenumberofobjectsintheothergroup,howmanyobjectsareinthelargergroup?
Inagroupoffivefriends,thesumsoftheagesofeachgroupoffourofthemis58,59,61,62and64.Whatistheageoftheoldestofthefivefriends?
InsquarePQRS,pointTisthemidpointofsegmentQRandTU ⊥SQ. Whatistheratiooftheareaof∆QUTtotheareaofsquarePQRS?Expressyouranswerasacommonfraction.
If16×2x=2y+2y + 2y + 2y,whatisthepositivedifferencebetweenxandy?
Whentwoverticesofaconvexoctagonareselectedatrandom,whatistheprobabilitythatthesegmentconnectingthosetwopointsisadiagonal?Expressyouranswerasacommonfraction.
Whatisthesumofallofthepositivedivisorsof50?
IfAis(0,y),Bis(4,4)andthelengthofABis5units,whatisthesumofallpossiblevaluesofy?
Twoprimenumbershaveasumof50.Whatisthemaximumpossiblevalueoftheproductoftwosuchprimes?
If30%ofxis12,345,whatis40%ofx?
Thelinesy=−0.4x+9andy=mx−2areperpendicular.Whatisthevalueofm?Expressyouranswerasadecimaltothenearesttenth.
14._____________
15._____________
16._____________
17._____________
18._____________
19._____________
20._____________
21._____________
22._____________
23._____________
24._____________
25._____________
Q
T
R
UP
S
(years)
(objects)
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Countdown Round
Sphericalweights,eachwithavolumeof0.003m3,willbedroppedintoarectangulartankhalf-filledwithwater.Thelength,widthandheightofthetankare6m,4mand2m,respectively.Howmanyweightsmustbedroppedintothetankinorderforthewaterleveltorise0.5m?
Whatisthesumofthefirst10termsofthearithmeticsequencethatbegins−12,−8,−4,…?
Howmanydistinctarrangementsoffourdifferentbowlsandonecupcanbemadeifthefiveitemsaretobeinalineonashelfandthreebowlscannotbenexttoeachother?
Howmanynon-congruentisoscelestriangleshaveatleastoneangleofmeasure70°andatleastonesideoflength70units?
ThesetScontains19differentintegerswithameanof101.ThemeanofthetensmallestintegersinSis50,andthemeanofthetenlargestintegersinSis150.WhatisthemedianofalltheintegersinS?
Eightswimmers,numberedfrom1to8,standinascendingorderclockwisearoundtheedgeofacircularswimmingpool.Goingaroundclockwise,everysecondswimmerstillstandingwilljumpintothepool.Swimmer#2isthefirsttojumpintothepool.Whatisthenumberoftheswimmerwhoistheseventhtojumpintothepool?
Whatisthesmallestoffourconsecutiveprimenumberswhosesumis220?
When3x+5yissubtractedfromx+7y,theresultis3x.Whatistheratioofxtoy?Expressyouranswerasacommonfraction.
SquareABCDhassidesoflength22cm.Whatisthearea,insquarecentimeters,oftheregionshadedgray?
Katiewritesdownsixdifferentprimenumbers,p,q,r,s,tandu,eachlessthan20.Ifp+q=r+s=t+u,whatisthesumofalltheprimenumberslessthan20thatKatiedidnotwritedown?
26._____________
27._____________
28._____________
29._____________
30._____________
31._____________
32._____________
33._____________
34._____________
35._____________
(weights)
(triangles)
(Number)
(arrange-ments)
A BE
CD
(cm2)
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Countdown Round
Whatisthevalueoftheproduct11 11 115 4
− − ?Expressyouranswerasacommonfraction.
Sixnumbershaveameanof12.Twelvenumbershaveameanof6.Whatisthemeanofthe18numberswhenbothsetsofnumbersarecombined?
SegmentsABandPQshareamidpoint.PointAisat(3,2)andtheendpointsofsegmentPQareP(7,8)andQ(9,6).WhatisthesumofthecoordinatesofpointB?
Thesumofthreepositiveintegersis2012.Thesumoftheleastandgreatestofthethreeintegersis1350,andthesumofthetwogreatestofthesethreeintegersis1362.Whatisthepositivedifferencebetweenthegreatestintegerandtheleastinteger?
Howmanypositive3-digitintegerscontainbothevenandodddigits?
Inthefigureshown,ABCDisasquare,∆CDEisequilateralandthetwoshapesarecoplanar.Whatisthemeasureof∠CBE,indegrees?
Whatisthevalueof3+6–9+12+15–18+21+24–27+…+84+87–90?
Theproductofthepositiveintegersa,bandcis72,anda>b>c>1.Whatisthegreatestpossiblevalueofa?
Whatisthedegreemeasureoftheindicatedexteriorangleformedwhentworegulardodecagonsshareacommonsideandtwoverticesbutnocommoninteriorpoints,asshown?
If2x∙42x= 23 ∙ 42 ∙ 8 ,whatisthevalueofx?
Giventhat10k isafactorof25!,whatisthegreatestpossiblevalueofk?
Insideasquarewithsidesoflength5units,howmanylocationsarepossibleforapointPwhosedistancesfromeachofthefoursidesare1unit,2units,3unitsand4units?Onesuchlocationtoincludeisshown.
36._____________
37._____________
38._____________
39._____________
40._____________
41._____________
42._____________
43._____________
44._____________
45._____________
46._____________
47._____________
(degrees)
(locations)
A B
CD
E
P
5
4
53
12
θ
(degrees)
(integers)
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Countdown Round
Therearetobeninepositiveoddintegersinthegridshown.Eachnumberisunique,andthenumbersineachrow,columnanddiagonalhavethesamesum.Whatisthevalueofx?
Whatistheperimeter,inunits,ofthisfigurecomposedof12unitsquares?
Whatarethecoordinatesoftheimageof(5, −2)afteratranslationof3unitsrightand7unitsdown?Expressyouranswerasanorderedpair.
Whatpercentoftheprimenumberslessthanorequalto30arefactorsof30?
If3brickmasonscancompletethefrontofonehousein2days,howmanydayswillittakeadozenbrickmasons,workingatthesamerate,tocompletethefrontsof50identicalhouses?
TheamountofalgaecoveringtheSmith’sbackyardponddoubledeverydayuntilitwascompletelycoveredinalgaeonday30ofthemonth.Onwhatdayofthatmonthwas75%ofthepondalgae-free?
Ifb(b4∙b3)2=b3x,whatisthevalueofx?
Fromtheset{1,2,3,4,5},DougwillchoosethreedistinctnumbersandReginawillchoosethreedistinctnumbers.WhatistheprobabilitythatDougandReginawillchooseexactlytwoofthesamenumbersaseachother?Expressyouranswerasacommonfraction.
Marissagives15 ofherpotatochipstoCraig.Shethengives
12 oftheremaining
chipstoPatandthengives14 ofthechipsthatremaintoZeke.IfMarissanow
has12potatochipsleft,howmanychipsdidshehavebeforegivinganyaway?
Theliney= 34 x+2isreflectedoverthey-axis,andtheresultinglineisthenreflectedoverthex-axis.Whatistheslopeofthisthirdline?Expressyouranswerasacommonfraction.
Whatisthe101stoddpositiveinteger?
48._____________
49._____________
50._____________
51._____________
52._____________
53._____________
54._____________
55._____________
56._____________
57._____________
58._____________
(units)
(,)
3 7
13
15x
(percent)
(days)
(Day)
(chips)
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Countdown Round
Inaboxcontainingadozencookiesthereare4chocolatechipcookies,4oatmealraisincookiesand4macadamianutcookies.If3cookiesarechosenatrandom,withoutreplacement,whatistheprobabilitythatonecookieofeachtypeisselected?Expressyouranswerasacommonfraction.
Anoperation × isdefinedas a × b= ( )3 2 2a bab+ .Finda × a2ifa=−3.
Whatisthe6thtermofthegeometricsequencewithafirsttermof3andasecondtermof−6?
Fortheset{2,5,x,9,16},writteninascendingorder,themeanandmedianareequal.Whatisthevalueofx?
Whatisthesmallestpositiveintegerwithexactlyeightpositivefactors?
Alexiohasninecards:threeareredonbothsides,threeareblueonbothsides,andtheotherthreeareredononesideandblueontheotherside.Acardischosenatrandom,andthecolorofonesideisobserved.Whatistheprobabilitythattheothersideofthechosencardisthesamecolor?Expressyouranswerasacommonfraction.
Thesquareofapositiveintegeristripled.Whentheresultisdividedbyfivethequotientis15.Whatwastheoriginalpositiveinteger?
Howmanydifferentstraightlinesegmentscontainingexactlysixdotscouldbedrawninthissquarearrayof36dots?
Whatisthesumofallvaluesofxsatisfying|x+1|=7?
TheareaoftriangleABC,shownhere,is36a8b4units2,anditsheight,h,is8a2b2units.Inunits,whatisthelengthofsideAC,intermsofaandb?
Twopoints,AandB,arerandomlyselectedfromtheinteriorofasquare.IfpointAandpointBarenotthesamedistancefromtheintersectionofthediagonals,whatistheprobabilitythatpointAisclosertotheintersectionthanpointB?Expressyouranswerasacommonfraction.
59._____________
60._____________
61._____________
62._____________
63._____________
64._____________
65._____________
66._____________
67._____________
68._____________
69._____________
(linesegments)
(units)
A
B
C
h
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Countdown Round
70._____________
71._____________
72._____________
73._____________
74._____________
75._____________
76._____________
77._____________
78._____________
79._____________
80._____________
Theproductofthreeprimenumbersis285.Whatistheirsum?
TheGleeClubissellingraffleticketsfor$2each,or3ticketsfor$5.Theprizeisabigscreentelevision,whichtheypurchasedfor$500.Iftheysell700tickets,whatisthesmallestprofittheywillmake,indollars?
AwormcrawlsfrompointA,duenorthalongastraightpathatarateof3cmpersecond.Aminutelater,anantleavesfrompointA,crawlingduesouthat5cmpersecond.Fiveminutesaftertheworm’sdeparturehowmanycentimetersapartfromeachotherarethewormandtheant?
Inonetossoftwofairdicewhatistheprobabilitythattheproductofthetwonumbersrolledisaprimenumber?Expressyouranswerasacommonfraction.
Whatpositiveintegerisclosesttothedifference 98 − 65 ?
Whatpercentof500is4?Expressyouranswertothenearesttenth.
Ifeachedgeofacubeistripledinlengthtoproducealargercube,thelargercube’ssurfaceareaishowmanytimestheoriginalcube’ssurfacearea?
Malcolmgot50%ofthequestionsonatestcorrect.Ifhehad10questions
correctoutofthefirst12,and 14 oftheremainingquestionscorrect,howmany
questionswereonthetest?
Twinprimesareprimepairsthatdifferby2.Whatisthesumofthelargestpairoftwinprimeslessthan50?
IntrapezoidABCD,shownhere,thenumberofunitsintheperimeterisbetweentwoconsecutive,positiveintegers.Whatisthesumofthosetwointegers?
Whatisthevalueof( ) ( )( ) ( )2014 20062011 2009
2 2
2 2--
?
(dollars)
(cm)
(questions)
(percent)
(times)
8
8
5
A
B C
D
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