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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.

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03-S12CDR

National Sponsors

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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)


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)


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)


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)


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)


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


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

MATHCOUNTS - WikispacesState+Coun… · Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012...

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