MATHCOUNTS - tarantamath [licensed for non-commercial...
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2012 State Competition Sprint Round Problems 1–30 Total Correct Scorer’s Initials Copyright MATHCOUNTS, Inc. 2011. All rights reserved. MATHCOUNTS ® Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO. This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You are not allowed to use calculators, books or other aids during this round. If you are wearing a calculator wrist watch, please give it to your proctor now. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the left-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers. In each written round of the competition, the required unit for the answer is included in the answer blank. The plural form of the unit is always used, even if the answer appears to require the singular form of the unit. The unit provided in the answer blank is the only form of the answer that will be accepted. 03-S12SPR 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 HONOR PLEDGE I pledge to uphold the highest principles of honesty and integrity as a Mathlete ® . I will neither give nor accept unauthorized assistance of any kind. I will not copy another’s work and submit it as my own. I understand that any competitor found to be in violation of this honor pledge is subject to disqualification. Signature _____________________________________________ Date ___________ Printed Name __________________________________________________________ School _______________________________________________________________ Chapter ______________________________________________________________
Transcript of MATHCOUNTS - tarantamath [licensed for non-commercial...
2012State Competition
Sprint RoundProblems 1–30
Total Correct Scorer’s Initials
Copyright MATHCOUNTS, Inc. 2011. All rights reserved.
MATHCOUNTS
®
Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation
DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.
This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You are not allowed to use calculators, books or other aids during this round. If you are wearing a calculator wrist watch, please give it to your proctor now. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the left-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers.
In each written round of the competition, the required unit for the answer is included in the answer blank. The plural form of the unit is always used, even if the answer appears to require the singular form of the unit. The unit provided in the answer blank is the only form of the answer that will be accepted.
03-S12SPR
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
HONOR PLEDGE I pledge to uphold the highest principles of honesty and integrity as a Mathlete®. I will neither give nor accept unauthorized assistance of any kind. I will not copy another’s work and submit it as my own. I understand that any competitor found to be in violation of this honor pledge is subject to disqualification.
Signature _____________________________________________ Date ___________
Printed Name __________________________________________________________
School _______________________________________________________________
Chapter ______________________________________________________________
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Sprint Round
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Abucketisfilledwith10gallonsofwater.Aholeiscreatedinthebucketsothatitloses2gallonsofwaterevery10minutes.Afterhowmanyminuteswillthebucketbecompletelyempty?
Thesumofthedigitsofapositive,two-digitintegeris9.Thepositivedifferencebetweentheintegercreatedwhenthetwodigitsarereversedandtheoriginalintegeris27.Whatistheproductofthesetwodigits?
Howmanydiagonalsdoesaconvexoctagonhave?
Jeanistwiceaslikelytomakeafreethrowassheistomissit.Whatistheprobabilitythatshewillmissthreefreethrowsinarow?Expressyouranswerasacommonfraction.
Apyramidhas6verticesand6faces.Howmanyedgesdoesithave?
Theproductoftheintegersfrom1through7isequalto2 j⋅3k⋅ 5 ⋅ 7.Whatisthevalueofj–k ?
Astandardsix-sideddiewasrolledrepeatedly.Thefrequencytablebelowshowshowmanytimeseachnumberwasrolled.Whatisthemeanofthe20numbersrolled?Expressyouranswerasadecimaltothenearesthundredth.
diagonals
edges
minutes
1 2 3 4 5 64 4 3 4 2 3
Number Times rolled
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Sprint Round
Thecoordinatesoftheverticesofatrapezoidare(1, 7),(1, 11),(8, 4)and(4, 4).Whatistheareaofthetrapezoid?
Malikaran3miles.Sheranthefirstmilein6minutes,45seconds.Hertimetocompleteeachmilewas
19 longerthanhertimeto
completethepreviousmile.Inseconds,howlongdidittakeMalikatorunall3miles?
Fourconsecutiveintegersaresubstitutedineverypossiblewayfordistinctvaluesa,b,candd.Whatisthepositivedifferencebetweenthesmallestandlargestpossiblevaluesof(ab+cd )?
TriangleMNOisanisoscelestrianglewithMN=NO=25cm.Alinesegment,drawnfromthemidpointofMOperpendiculartoMN,intersectsMN atpointPwithNP:PM=4:1.WhatisthelengthofthealtitudedrawnfrompointNtoMO ?Expressyouranswerinsimplestradicalform.
Let{a,b,c,d}beasetofnumberschosenfromthefirstninepositiveintegers.Ifyouaddeverypossiblepairofthesefournumbersyougetthesesums:7,9,10,12,13and15.Whatisthesmallestpossibleproductofthesefournumbers?
Inasequenceofpositiveintegers,everytermafterthefirsttwotermsisthesumofthetwoprevioustermsinthesequence.Ifthefifthtermis2012,whatisthemaximumpossiblevalueofthefirstterm?
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cm
seconds
units2
M
N
O
P
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Sprint Round
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Thefigureshowsthefirstthreestagesofafractal.Ifthepatterncontinues,howmanycircleswillStage5ofthefractalcontain?
Productsarefoundbymultiplyingthreedifferentnumbersfromtheset{1,2,3,4,5}.Amongthoseproducts,howmanypairsofrelativelyprimenumbersarethere?
InrectangleABCD,AB=6units,themeasureof∠DBCis30°,MisthemidpointofsegmentADandsegmentsCMandBDintersectatpointK.WhatisthelengthofsegmentMK?Expressyouranswerissimplestradicalform.
JackandJilldroveinseparatecarstotheirfavoritehill,leavingfromthesameplaceatthesametime.Jilldrove20%fasterthanJackandarrivedhalfanhourearlier.HowmanyhoursdidJackdrive?
Whatisthelargestfive-digitintegersuchthattheproductofthedigitsis2,520?
Arectangularprismiscomposedofunitcubes.Theoutsidefacesoftheprismarepaintedblueandthesevenunitcubesintheinteriorremainentirelyunpainted.Howmanyunitcubeshaveexactlyonepaintedface?
pairs
circles
units
Stage 1 Stage 2 Stage 3
B
A
C
DM
K
hours
unitcubes
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Sprint Round
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Iff (x)=x2+5andg (x)=2 (f (x)),whatisthegreatestpossiblevalueoff (x +1)wheng(x)=108?
Arighttrianglehassideswithlengths8cm,15cmand17cm.Acircleisinscribedinthetriangle.Incentimeters,whatistheradiusofthecircle?
Giventhatx≠0,whatquantitycanbeaddedtoxx+1
ormultipliedbyxx+1
togivethesameresult?Expressyouranswerintermsofx.
IntrapezoidABCDsegmentsABandCDareparallel.PointPistheintersectionofdiagonalsACandBD.Theareaof∆PABis16squareunits,andtheareaofΔPCDis25squareunits.WhatistheareaoftrapezoidABCD?
Twopositivenumbershavethepropertythatthesumoftheirsquaresis20andthesumoftheirreciprocalsis2.Whatistheirproduct?Expressyouranswerasacommonfraction.
Atrianglehasanglesmeasuring15°,45°and120°.Thesideoppositethe45°angleis20unitsinlength,andtheareaofthetrianglecanbeexpressedinsquareunitsas(m−n q ),whereqisaprimenumber.Whatisthevalueofthesumm+n +q?
cm
A B
CD
P
units2
Copyright MATHCOUNTS, Inc. 2011. All rights reserved. 2012 State Sprint Round
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Asilo-shapedplanefigureisformedbypositioningasemicircleaboveasquare.Thediameterofthesemicircleis2unitslongandcoincideswiththetopofthesquare.Whatistheradius,r,ofthesmallestcirclethatcontainsthisfigure?Expressyouranswerasacommonfraction.
Inhowmanywayscansixdifferentgiftsbegiventofivedifferentchildrenwitheachchildreceivingatleastonegiftandeachgiftbeinggiventoexactlyonechild?
Ifthecostofadozeneggsisreducedbyxcents,abuyerwillpayonecentlessforx+1eggsthanifthecostofadozeneggsisincreasedbyxcents.Whatisthevalueofx?
Forhowmanytwo-elementsubsets{a,b}oftheset{1,2,3,...,36}istheproductabaperfectsquare?
InrectangleABCD,shownhere,pointMisthemidpointofsideBC,andpointNliesonCD suchthatDN:NC=1:4.SegmentBNintersectsAM andAC atpointsRandS,respectively.IfNS:SR:RB=x:y:z, wherex,yandzarepositiveintegers,whatistheminimumpossiblevalueofx+y+z?
ways
units
r
subsets
A B
CD N
MR
S
