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2010–2011
School Handbook:Volume II
For questions about your local MATHCOUNTS program,please contact your chapter (local) coordinator. Coordinator contact information is available through the Find a Coordinator option of the Competition Program
link on www.mathcounts.org.
Contains 200 creative math problemsthat meet NCTM standards for grades 6-8.
MATHCOUNTS 2010-2011 9
Warm-Up 71. ��������� Melinda chooses a three-digit positive integer, subtracts it from 3000, and triples
the result. What is the largest integer Melinda can get?
2. ��������� Eachtermofasequence,afterthefirstterm,isthreelessthanthesquareoftheprecedingterm.Ifthefirsttermofthesequenceis2,whatisthe2011thterm?
3. ��������� Susanreadsatarateof240wordsperminute.Howmanyhourswillittakehertoreada480-pagebookthataverages600wordsper page?
4. ��������� Afaircoinistobeflipped5times.Whatistheprobabilitythattheresultwillnotbe5headsinarow?Expressyouranswerasacommonfraction.
5. ��������� Whatisthedegreemeasureoftheonlyanglethatiscongruenttoitscomplement?
6. ��������� If a + 2b = 11 and a – b=-4,whatisthevalueof4a – b?
7. ��������� What is the value of 10113expressedasanumeralinbase2?
8. ��������� Fromthesetofdigits{1,2,3,4,5}three-digit,positiveintegersareformed,noneofwhichhasrepeatingdigits.Howmanysuchoddintegerscanbemade?
9. ��������� TriangleABChasverticeswithcoordinates(1,0),(5,0),and(3,7).Kadimaccidentallyswitchedthex- and y-coordinatesofeverysinglevertexwhencalculatingtheareaofthetriangle.WhatisthepercentofchangebetweentheareaoftheoriginaltriangleABCandtheareaofthenewtriangleKadimmade?
10. �������� EightstudentsmetforthefirsttimeataMATHCOUNTScompetition.Everyoneofthemexchangedcellphonenumberswitheachoftheothersevenstudents,andtheyallenteredthephonenumbersintotheirphones.Howmanytimeswerecellphonenumbersenteredintophonesamongthegroupof8students?
integers
%
2
degrees
times
hours
MATHCOUNTS 2010-201110
Warm-Up 81. ��������� Attheschoolstorepencilsandpenshavedifferentprices.Sixpencils
andfourpenscost$4.30.However,fourpencilsandsixpenscost$5.20.Whatisthecostoffivepencilsandfivepens?
2. ��������� Whattermisthesmallestfive-digitpalindromeinthearithmeticsequence 2, 7, 12, 17, … ?
3. ��������� Achevronisinscribedinasquareofside12yardswherepointXisatthe center of the square and the upper corners of the chevron touch the upper corners of the square. What is the area of the chevron?
4. ��������� Onanumberline,whatisthepositivedifferencebetweenthetwonumbersthatarethetrisectionpointsofthelinesegmentwithendpointsat 1
8 and 34 ?Express
youranswerasacommonfraction.
5. ��������� Mannycanmowhisone-acreyardin1.5hoursonhisridinglawnmower.TimmytakesfivetimesaslongtomowManny'syardwithhispushmower.Howmanyminutes
willittakethemworkingtogether?
6. ��������� Whatistheunitsdigitoftheproduct(375)(2113)?
7. ��������� Whatisthevalueoftheexpression1 1 13 4 51 1 12 5 6
+ ++ +
?Expressyouranswerasacommonfraction.
8. ��������� Theproductofthreedifferentpositiveintegersis2010.Whatisthemaximumpossible sum of the three integers?
9. ��������� Youhaveasupplyofboxesofvolumes1,3,9,27and81cubicmeters.Giventhattheboxesmustbefilledcompletely,whatistheleastnumberofboxesthatwillholdexactly300cubicmeters of sand?
10. �������� If F (n)=3n–5,G(n)=n 2 + 3n−2andH(n)=15−0.3n,whatisthevalueof× −−
(4) ( 10)( 3)
F HG
?
$
boxes
sqyds
minutes
X
MATHCOUNTS 2010-2011 11
Workout 41. ��������� Whenwaterfreezes,itsvolumeisincreasedbyone-eleventh.Inotherwords,the
volumeoficeequalsthesumofthevolumeoftheoriginalamountofwaterandtheproductofone-eleventhandthevolumeofthewater.If979cubiccentimetersofwateristobefrozen,whatwillbethevolumeoftheicethatwillbeformed?
2. ��������� Arighttrianglehasanareaof84m2 and has integral side lengths. What is the perimeter of this triangle?
3. ��������� Whatistheleastpositiveintegerthathasaremainderof0whendividedby3,aremainderof1whendividedby4,andaremainderof3whendividedby7?
4. ��������� If u + w = x and w + x = y and x + y = z,whatistheintegervalueofthesumu + w + x + y + z, given that u=8andz = 10?
5. ��������� What is the smallest counting number nsuchthattheproduct245n is a perfect square?
6. ��������� Akiteisgraphedinthecoordinateplaneasshown.Ifthiskiteisrotated180°clockwiseaboutpointC(0,1),thentranslateddown4unitsandtotheleft2units,whatarethecoordinatesofthefinalimageofpointB(2,3)?
7. ��������� In1973,agallonofgasolinesoldfor40cents.In2008,agallonofgasolinesoldfor$3.75.Bywhatpercentdidthecostpergallonofgasolineincrease?Expressyouranswertothenearesttenth.
8. ��������� Jasonwasdescribingaspinnertohisfriend.Hesaidthespinnerwasdividedintosixequalsections,eachcontaininganintegerthatwasnotnecessarilydistinctfromtheotherintegersonthespinner.Hetoldhisfriendthatonthespinnertheprobabilityoflandingon2is 1
6,theprobabilityoflandingon6is16 ,theprobability
of landing on a factor of 27 is 23, and the sum of the odd numbers on the spinner is
16.Whatistheproductofthesixnumbersonthespinner?
9. ��������� Usingonlylinesthatarealreadydrawn,howmanytrianglesareintheregularhexagonshown?
10. �������� Whatisthelength,ininches,ofarectanglethathasanareaof18.75sqinandalength that happens to be 3
8 oftheperimeter?Expressyouranswerasadecimaltothe nearest tenth.
triangles
cm3
meters
(,)
A
B
C
D
inches
%
MATHCOUNTS 2010-201112
Warm-Up 91. ��������� Acertainobjecthasavolumeof216cubiccentimetersandasurfaceareaof
240 square centimeters. Another object is a smaller-scale replica of the original object.Ifthesurfaceareaofthesmallerobjectis60squarecentimeters,whatisits volume?
2. ��������� FivethousandrunnerswillparticipateintheMountainMarathon.Eachrunnerwillrandomlybegivenadifferentnumberfrom1to5000towearduringtherace.Whatistheprobabilitythatthenumberofthesecond-placefinisherwillbebothgreaterthanthenumberofthethird-placefinisherandlessthanthenumberofthefirst-placefinisher?(Assumenoties.)Expressyouranswerasacommonfraction.
3. ��������� Tricia has a bag of marbles. She gave one-third of them to Barbara and one-fourth oftheremainingmarblestoSam.Iftherearenow24marblesinthebag,howmanymarbles did Tricia give to Barbara?
4. ��������� Acubehasasidelengthof50inches.Whatisthepositivedifferencebetweenthenumerical values of the surface area of the cube in square inches and the volume of the cube in cubic inches?
5. ��������� ThepointP(1,4)isreflectedovertheliney = xtopointP’.WhatisthedistancebetweenPandP’?Expressyouranswerinsimplestradicalform.
6. ��������� Thinkofthesetofpositivetwo-digitintegershavingtwodifferentdigitsthatareselectedfromthedigits2,3,4and7.Inthisset,whatistheratioofprimenumberstocompositenumbers?Expressyouranswerasacommonfraction.
7. ��������� ThestudentsatValleyMiddleSchoolheldaWalk-a-Thontoraisemoneyforalocalcharity.Thebargraphshowshowmuchmoneywascollectedbyeachgrade. If the data had been graphed in a pie chart instead,howmanydegreeswouldbeinthecentralangle of the sector representing grade 7?
8. ��������� IfthesevenlettersofALABAMAaretobearrangedatrandom,whatistheprobabilitythatallfourAswillbetogether?Expressyouranswerasacommonfraction.
9. ��������� Atrapezoidwithverticesat(–1,2),(1,2),(3,–2)and(–3,–2)isgraphedinthecoordinateplane.Whatistheareaoftheregionofthetrapezoidwherex≥0?
10. �������� What is the product of the roots of 2 3 464 16 4x x x x+ − − = 0?
degrees
units
units2
cm3
marbles
500
1000
1500
2000
2500
16001500
900
2000
Walk-a-Thon Fundraiser
Money
Rais
ed
(in
dollars)
5 6 7 8
Grade
MATHCOUNTS 2010-2011 13
Warm-Up 101. ��������� Kevincanmowalawnin40minutes.WhenJoshhelpshim,theycanfinish
thelawnin15minutes.HowlongwouldittakeJoshtomowthelawnbyhimself?
2. ��������� Intheadditionproblemshown,A,B,CandDrepresentdistinctdigits. ABC +DBBB 2011 WhatisthevalueofA+B+C+D?
3. ��������� Whatistheprobabilitythatapositivethree-digitintegerthathasa3intheunitsplaceisdivisibleby3?Expressyouranswerasacommonfraction.
4. ��������� A96-inchpieceofwireiscutintopieces,andthenthepiecesaretobegluedtogethertoformtheedgesofabox.Whatisthelargestpossiblevolumeofaboxthatcouldhavethosepiecesforedges?
5. ��������� Ajuicedrinkcontainsonly10%juice.Howmanyquartsofpurejuicemustbeaddedto1gallonofthejuicedrinktocreateadrinkthatis60%juice?
6. ��������� Howmanyfactorsdoes1800havethataregreaterthan15?
7. ��������� Twocubes,eachwithfaceswith0through5,aretoberolled.Whatistheprobabilitythatthesumwillbelessthan6?Expressyouranswerasacommonfraction.
8. ��������� The mean, median and range are all the same for a particular set of four numbers. What is the largest number in the set if the smallest number is 17?
9. ��������� Thediagonalsofasquareintersectatthepoint(3,2).Thecoordinatesofoneoftheverticesofthesquareare(5,4).Whatistheareaofthissquare?
10. �������� Inthemultiplicativemagicsquareshown,theproductofthethreenumbersineachrow,ineachcolumnandalongeachdiagonalis1.Whatisthevalueofr+s?Expressyouranswerasacommonfraction.
quarts
cu in
minutes
factors
units2
qp r
1s t
4u 1
8–
MATHCOUNTS 2010-201114
Workout 51. ��������� Apaintertakestwodaystopaintaroom(allfourwallsandtheceiling).Ifheworks
atthesamepace,howmanydayswillittakehimtopaintaroomthatistwiceaswide,twiceaslong,andtwiceashigh?
2. ��������� Thegraphshowstheamountoffuelusedbyfivedifferentvehiclesthateachdrove200milesduring a road test. What is the average miles per gallonforthetest?Expressyouranswerasadecimalto the nearest tenth.
3. ��������� Whatisthelengthofthesideoftheonlyequilateraltrianglewhoseperimeterhasthesamenumericalvalueinunitsasitsareainsquareunits?Expressyouranswerasa decimal to the nearest tenth.
4. ��������� ThreebooksandeightCDscosttwiceasmuchasonebookandfiveCDs.IfonebookandoneCDcost$45,whatisthecostofsixbooks?
5. ��������� Ifthelargesttriangleinthisfigureisarightisoscelestrianglewithanareaof1squareunit,andifallofthetrianglesaresimilar,whatisthecombinedareaoftheshadedregions?Expressyouranswerasacommon fraction.
6. ��������� What is the sum of the integer values of x,wherex<8,thatproduceanintegervaluefortheexpression 2 4 6 x+ + + ?
7. ��������� Assumeadiamond’svalueisdirectlyproportionaltothesquareofitsweightincarats.Ahugediamondweighing10caratsisbrokenintoa4-caratpieceanda6-caratpiece.Thesumofthevaluesofthetwopiecesiswhatpercentoftheoriginalvalue?
8. ��������� Atwo-yearstudywasjustcompleted.Theprojectbeganwith40participantsandendedwith90participants.Afterthefirstyear,Pparticipantswereinthestudy.Ifthepercentincreaseofthenumberofparticipantswasthesameforbothyears,whatisthevalueofP?
9. ��������� Whenadarthitswithinthecentercircularregionofthiscirculartarget,thescoreis50,andwhenadarthitswithintheouterring,thescoreis7.Ifthreedartsrandomlyhitthetarget,whatistheprobabilitythatthetotalscoreis64?Expressyouranswerasacommon fraction.
10. �������� Asquarewithadiagonaloflength7unitsisinscribedinacircle.Whatisthepositivedifferencebetweentheareaofthesquareandtheareaofthecircle?Expressyouranswerasadecimaltothenearesthundredth.
%
units
4
8
12
16
0FuelU
sed
(gallons/2
00
miles)
U V W X Y
Vehicle
Comparison of Fuel Usage
1 1
50 pts
7 pts
days
mpg
$
units2
sq units
MATHCOUNTS 2010-2011 15
Warm-Up 111. ��������� Inbasketball,aplayercanscorevia3-pointshots,2-pointshotsand
1-pointfreethrows.IfShakeelmadeeight2-pointshotsandscored30pointsinall,whatistheminimumnumberoffreethrowshecouldhave made?
2. ��������� Theperimeterofarectangleis62meters.Thelengthis1metermorethanfourtimesthewidth.Whatistheareaoftherectangle,insquaremeters?
3. ��������� AlinesegmentABhasendpointsatA(5,9)andB(-7,-15).Whatisthesumofthecoordinates of its midpoint?
4. ��������� IfPisdividedbyR,theresultis 23 .IfRisdividedbyS,theresultis 3
4 . What is theresultifPisdividedbyS?Expressyouranswerasacommonfraction.
5. ��������� Theregionshowniscomposedofarighttriangleandthreesemicircles. What is the area of the entire enclosed region if the trianglehaslegsof6and8units?Expressyouranswerintermsofπ.
6. ��������� Juliecollectstoyfrogs.Shedisplaysthemonashelfandrearrangesthemoften.Ifthereare3greenfrogsand2yellowfrogslineduponhershelf,whatistheprobabilitythatthefrogsonbothendswillbegreen?Expressyouranswerasacommonfraction.
7. ��������� What is the product of all integer values of xforwhich −2 9x is a prime number?
8. ��������� ConvexoctagonABCDEFGHhascongruentanglesA,B,EandFandcongruentanglesC,D,GandH.If m∠A=(2x)°,m∠B = y°andm∠C=(x+50)°,whatisthe value x + y?
9. ��������� Asquareofwidth100unitscontainsacircletangenttoallfoursidesofthesquare.Twosmallersquares,congruenttoeachotherandeach containing a circle tangent to all four sides of the square, rest atopthelargesquare,withthethreesquarestogetherformingarectangle.Whatisthedifferencebetweentheareaoftheshadedsemicircleandthecombinedareaofthetwosmallershadedcircles?
10. �������� TwosubsetsofthesetS={a, b, c, d, e}aretobechosensothattheirunionisSandtheirintersectioncontainsexactlythreeelements.EachoftheelementsofSisdistinct.Inhowmanywayscanthisbedone,assumingthattheorderinwhichthesubsets are chosen is irrelevant?
free throws
units2
m2
ways
6
8
A B
C
D
F E
H
G
100
units2
MATHCOUNTS 2010-201116
Warm-Up 121. ��������� Thenumeratorofafractionis8lessthanthedenominator.Ifthenumeratoris
doubledandthedenominatorisdecreasedby1,thevalueoftheresultingfractionis1.Whatistheoriginalfraction?Expressyouranswerasacommonfraction.
2. ��������� PointAhascoordinates(-2,5)andpointBhascoordinates(10,8).PointPisonsegmentAB,withAP=2PB.WhatarethecoordinatesofpointP?Expressyouranswerasanorderedpair.
3. ��������� What is the value of xif52(5)+52(11)+52(17)+52(78)+52(14)=5x?
4. ��������� Theproductoftwoconsecutive,positive,oddintegersis783.Whatisthesumofthetwointegers?
5. ��������� Theaspectratioofarectangleisitslengthdividedbyitswidth.Thefigureshowsanimage(thewhiterectangle)withaspectratio16:9displayedonascreen(thelargestrectangle)withaspectratio4:3.Whatpercentoftheareaofthescreenisoccupiedbytheimage?
6. ��������� Themeanofthesquaresofthreenon-zeronumbersa, b and c is three times the square of their mean. If b = 2a,whatisthevalueof ca ?Expressyouranswerasacommon fraction.
7. ��������� When creating a particular pattern, the designer beginswithalinesegment1unitlong.Eachsuccessivestageiscreatedbyusingcongruentsegments that are half as long as those in the previousstage.IfyoutracethedesignatStage4,howmanyunitslongerisitthanthelength of the design at Stage 1?
8. ��������� Twodifferentunitsquaresarerandomlyselectedfromthe16unitsquares in the 4 ×4gridshown.Whatistheprobabilitythattheydonothaveavertexincommon?Expressyouranswerasacommonfraction.
9. ��������� The graphs of y = x2–8x–35andy = –2x2+16x+3intersectintwopoints.Whatis the sum of the x-coordinatesofthetwopointsofintersection?
10. �������� Inthefigureshown,AC is the diameter of the circle. If the measureofarcBCis60°,AC BE⊥ ,andBE=12units,whatisthe length of AB?
%
unitslonger
1 unit
Stage 0 Stage 1
Stage 2
A
B
E
CD
units
(,)
MATHCOUNTS 2010-2011 17
Workout 61. ��������� Twoapples,fivebananasandonecarrotcostatotalof$2.05.Three
apples,onebananaandfourcarrotscostatotalof$1.89.Threeapples,twobananasandthreecarrotscostatotalof$1.98.Whatisthetotal cost, in cents, of one apple, one banana and one carrot?
2. ��������� Inmathclass,themeantestscoreforthegirlswas92pointsandthemeantestscorefortheboyswas87points.If12girlsand10boystookthetest,whatwasthemeantestscoreforthewholeclass?Expressyouranswerasadecimaltothenearesthundredth.
3. ��������� Triapaid$11,500foranewcaraftertheoriginalpricewasreducedbyarebateof$4500.WhatpercentageoftheoriginalpricedidTriapayforthecar?Expressyouranswertothenearesttenth.
4. ��������� HowmanymilescanScottyrideonabike,goingattherateof 8milesperhour,ifhemustwalkbacktothestartingpointatarateof3milesperhour(followingthesameroutehetraveledonhisbike)andheistobegoneatotalof11hours?
5. ��������� InordertosendhisDVDplayerbacktothemanufacturerforrepair,Jeremymustpackitsecurely.Hehasapackingboxwithinternaldimensions22”×18”×6”.TheDVDplayerisarectangularprismwithdimensions15”×12”×2”.Hehasaboxwithexternaldimensions3”×3”×8”forthepowercord,whichalsowillgoinsidethepackingbox.Howmanycubicinchesofpackingmaterialmustheputinthepackingboxtofillitcompletely?
6. ��������� Stella sells 30 gismos a month at a price of $20 per gismo. She notices that each time sheraisesthepriceofagismoby$4,shesells2fewergismospermonth.Whatisthemaximumamountofrevenue,indollars,thatStellacanmakeinamonthofsellinggismos?
7. ��������� IsoscelestrapezoidABCDhasbaseswithlengths12and24.PointsEandFarethemidpointsoflegsABandCD,respectively.PointsGandHarethemidpointsofbasesADandBC,respectively.Iftheheightofthetrapezoidis6units,whatisthedifferencebetweentheareasoftrapezoidABCDandparallelogramHFGE?
8. ��������� Mr.Kellyisbuildinga60-foot-longrampthatwillbe24feethighatthehighestpoint.Hewantstoplaceaverticalsupportbeamundertherampwithitstop15feetdownthelengthoftheramp.Howlongdoesheneedtomakethesupportbeam?
9. ��������� A circular garden is placed inside a rectangular patio that measures20feetby14feet.Thecirculargardenistangenttothreesidesoftherectangularpatio,asshown.Whatpercentofthepatioisoutsideofthegarden?Expressyouranswertothenearestwholenumber.Use 22
7 asanapproximationforπ.
10. �������� Abagcontainsfiveredmarbles,threebluemarblesandtwogreenmarbles.Ifsixmarblesaredrawnwithoutreplacementfromthebag,whatistheprobabilitythattwomarblesofeachcoloraredrawn?Expressyouranswerasacommonfraction.
inches3
cents
%
miles
$
feet
points
%
A
E
B CH
G D
F
units2
MATHCOUNTS 2010-201118
Warm-Up 131. ��������� Sixpointsareequallyspacedaroundacircle.Ifthreeofthepoints
areselectedatrandom,whatistheprobabilitythattheyformascalenetriangle?Expressyouranswerasacommonfraction.
2. ��������� Point P is on the y-axisandisequidistantfromthepoints(1,3)and(7,5).Whatisthe y-coordinate of point P?
3. ��������� Amanufacturingplantperiodicallyopensalargevalveatthetopofatanktopumpwaterintothetankforuseinamanufacturingprocess.Whenthevalveisopened,waterispumpedintothetankatarateof75gallonsperminute.Anothervalve,atthebottomofthetank,sendswateroutofthetanktoanassemblylineatarateof24gallonsperminute.Bothvalvesareopenedsimultaneouslyat7:30amandbeginfillingtheemptytank.Atwhattime,tothenearestminute,willthetankbefilledtoits600-galloncapacity?
4. ��������� Asetof5positiveintegershasameanof11,amedianof10andauniquemodeof7.What is the greatest possible range of the set of integers?
5. ��������� Ajarcontains$5.00inquarters,$5.00indimes,$5.00innickelsand$5.00inpennies.IfConnierandomlychoosesonecoinfromthejar,whatistheprobabilitythatthecoinchosenisadime?Expressyouranswerasacommonfraction.
6. ��������� MarketinganalystsareinvestigatingcylindricalpackagingforanewbeveragecalledStarJuice.Onememberofthedesignteamproposedacanthathasadiameterof6cmandaheightof20cm.Anothersuggestedacanthatis17cmtallandhasadiameterof8cm.Whatistheratioofthesmallervolumetothelargervolume?Expressyouranswerasacommonfraction.
7. ��������� Forhowmanyintegervaluesofxisthefollowinginequalitytrue?-5≤2x–7≤21
8. ��������� Thefigureshownisanetofacube.Whenfoldedtoformacube,whatisthelargestsumoftwonumbersonopposite faces?
9. ��������� YesterdayAlicedroveonehourlongerthanBobatanaveragespeedthatwasfivemilesperhourfasterthanBob'sspeed.ClarkdrovetwohourslongerthanBobatanaveragespeedthatwastenmilesperhourfasterthanBob'sspeed.Alicedrove 50milesmorethanBob.HowmanymoremilesthanBobdidClarkdrive?
10. �������� InrighttriangleABC,themediantothehypotenusehaslength15unitsandthealtitudetothehypotenusehaslength12units.WhatisthelengthoftheshorterlegoftriangleABC?Expressyouranswerinsimplestradicalform.
units
values
am
3 2 5
4 7 6
miles
MATHCOUNTS 2010-2011 19
Warm-Up 141. ��������� In base b,321–123=154.Whatisthevalueofb?
2. ��������� What is the value of nif4(3n + 2)+45(3n)=3?
3. ��������� Inthefigureshown,AD=4cm,BC=8cmandCD=20cm.Also, AD ABandBC AB⊥ ⊥ . What is the length of AP? Expressyouranswerasacommonfraction.
4. ��������� Arectanglewithsidesoflength5unitsand6unitsthatareparalleltothex-axisandy-axishasonevertexat(1,3).Whatisthelargestpossiblesumofthecoordinatesofanyoftheotherverticesofthisrectangle?
5. ��������� What is the units digit of Ninthefollowingequation?× × × ×
= × ×3 3 3 3 3 1 1 1
2 3 3N
6. ��������� Thecycleshopsellsonlybicyclesandtricycles.Allthebicycleshavetwowheelsandallthetricycleshavethreewheels.Ifyouaddupthetotalwheelsonthe67cyclesintheshop,youget157wheels.Howmanytricyclesareintheshop?
7. ��������� IfABCDEFGHisaregularoctagonwithsidesoflength2cm,whatisthearea,insquarecentimeters,oftriangleABC?Expressyouranswerinsimplestradicalform.
8. ��������� Runningat12milesperhourburnsabout1200caloriesperhour.Runningat5milesperhourburnsabout450caloriesinanhour.Howmanyminuteslongerdoesonehavetoruntoburn300caloriesat5milesperhourthanat 12 miles per hour?
9. ��������� Ofthe2000studentsinHoustonHighSchool,70%aremale.Howmanymalestudentsmustbemovedtoanotherschoolsothat60%oftheHoustonHighstudentbodywillbemales?
10. �������� Supposethatsixbrothers(George,John,Thomas,Fred,Andrew,Martin)eachrollafair6-sideddie.Atleastthreeofthemhaverolledanevennumber.WhatistheprobabilitythatJohn,FredandMartinhaveeachrolledanevennumber?Expressyouranswerasacommonfraction.
cm
tricycles
cm2
minutes
P
D
A
B
C
students
MATHCOUNTS 2010-201120
Workout 71. ��������� AtMarshMelLowMiddleSchool34studentsplaysoccer,36studentsplayfootball
and29studentsplaybasketball.Ofthesestudents15playbothsoccerandfootball,18playbothbasketballandfootballand13playbothsoccerandbasketball.Whatisthesmallestpossiblenumberofstudentsthatplayallthreesports?
2. ��������� If x and yaretwotriangularnumberslessthan100thatwhenadded,produceasumthatisasquarenumber,whatisthelargestpossiblevalueofx or y?
3. ��������� Tria’snewcarcanget50%moremilespergallonofgasolinethanheroldcarcould.Ifhernewcargets24milespergallon,howmanygallonsofgasolinewillshesaveusinghernewcartotravel480milestovisitrelativesincomparisonwithdrivingtheoldcarthesamedistance?Expressyouranswertothenearestwholenumber.
4. ��������� Mr.Davidsonlives30milesfromhisoffice.DrivingtoworkonMonday,heaveraged50mph.However,duringhisreturntripfromwork,therewasconstructionthatslowedtraffic. Mr.Davidsonaveragedonly20mphgoinghome.HowmanyhoursdidMr.DavidsonspendtravelingtoandfromworkonMonday?Expressyouranswerasadecimaltothenearesttenth.
5. ��������� Exactlyoneofthefollowingstatementsistrue.Whichone?A.Exactlyoneofthesestatementsisfalse. B.Exactlytwoofthesestatementsarefalse. C.Exactlythreeofthesestatementsarefalse. D.Exactlyfourofthesestatementsarefalse. E.Exactlyfiveofthesestatementsarefalse.
6. ��������� InregularoctagonABCDEFGH,whatistheratiooftheareaoftriangleADFtotheareaoftriangleAHF?Expressyouranswerasadecimaltothe nearest hundredth.
7. ��������� If$1000isinvestedat6%annualinterestcompoundedsemiannually,howmuchinteresthasbeenearnedattheendofoneyear?
8. ��������� Howmanyrectanglesarethereinthefollowingfigure?
9. ��������� Fourchildren(Alice,Brad,CathyandDan)arearrangedinaline.IfBradandCathycannotbenexttoeachother,inhowmanywayscanthefourchildrenbearranged?
10. �������� Inthecoordinateplane,whatistheareaoftheregionthatisabovethex-axis,betweenx = –3 and x=3,andbelow = − −1 2y x ?
students
gallons
hrs
$
rect
ways
units2
A B
F E
H
G
C
D
MATHCOUNTS 2010-2011 21
Warm-Up 151. ��������� Fantasygumissoldin3-ounceand10-ouncepackages.Onepound
isequalto16ounces.If76packagesweresoldatB-RiteMartlastmonthandthetotalweightsoldwaslessthan16pounds,whatisthegreatestnumberof10-ouncepackagesB-Ritecouldhavesold?
2. ��������� Threestandard,six-sideddicearerolled.Thesumofthethreenumbersontopislessthan6.Whatistheprobabilitythatallthreediceshowa1?Expressyouranswerasacommonfraction.
3. ��������� AlinethroughpointP(4,3)isparalleltothelinex – 2y=8.WhatistheequationofthelinethroughpointP?Expressyouranswerinslope-interceptformwheretheslope is a common fraction.
4. ��������� HowmanydifferentwayscanthelettersinthenameHANKbescrambledsothatthenewlycreated“words”startandendwithaconsonantandarenotHANK?
5. ��������� Eightsophomoresareonastudentcouncilcommittee.Apartyplanning subcommittee must be selected from these eight student councilcommitteemembers.Howmanydifferentpartyplanningsubcommittees are possible if it must have more than 1 but less than5members?
6. ��������� Aregularhexagonandaregularoctagonhavethesameperimeter.Ifthesidelengthsofthesepolygonsarewholenumbersandthesumoftheperimetersislessthan200,howmanydifferentpairsofhexagonsandoctagonscanbemade?
7. ��������� What is the integer closest to +150 75 ?
8. ��������� Arectanglehasbeenpartitionedintoninesquares,asshown.Ifthe area of each of the three medium squares is 100 square units, whatistheareaofthelargestsquare?
9. ��������� Acirclewitharadiusof 4 3 cmisinscribedinsidearegularhexagon.Whatistheareaofthehexagon?Expressyouranswerinsimplestradicalform.
10. �������� Inthefigure,l m and p r .Whatisthevalueofx?
pairs
packages
cm2
130º
xºl
m
p r
ways
sub-committees
units2
MATHCOUNTS 2010-201122
sides
Warm-Up 161. ��������� Inthisfigure,sideABofparallelogramABCDis10units
long.Acircleofradius3unitsisdrawnintheinteriorsuchthatsidesABandCDoftheparallelogramaretangenttoit.Whatistheareaoftheshadedregion?Expressyouranswerintermsofπ.
2. ��������� If f (x)=5x, and g (x)=f (x)–3x–7,whatisthevalueofg (8)?
3. ��������� Thesumofthedigitsofatwo-digitpositiveintegeris8.Whenthedigitsarereversed,thenewintegeris3morethan4timestheoriginalinteger.Whatisthenewinteger?
4. ��������� The largest angle of a quadrilateral is four times its smallest angle. Another angle is10degreesmorethantwicethesmallest.Thefourthangleis60degreeslessthan 3 times the smallest. What is the measure of the largest angle?
5. ��������� AtMelvilleMiddleSchool73studentsareintheband,65studentsareinthechorusand114studentsplaysports.Thirty-twoarebothinthebandandplaysports,12playsportsandsinginthechorus,butnooneisinboththebandandthechorus.Fivehundredfifty-eightstudentsdonotparticipateinanyoftheseactivities.Howmanystudentsarethere at Melville Middle School?
6. ��������� Howmanysidesdoesaconvexpolygonwith20diagonalshave?
7. ��������� Onthenumberline,whatpercentoftheinterval–10≤x≤10satisfiestheinequality52
2x
x+ <
−?
8. ��������� If aΔb = −
=+
2 6a ba bb ,whatisthevalueof3Δ(5Δ3)?Expressyouranswerasa
common fraction.
9. ��������� Howmanyintegersnsatisfying3000≤n≤4000havetheproductoftheirdigitsequaltozero?
10. �������� LetPQRSbeanisoscelestrapezoidwithbasesPQ=100unitsandRS=26units.SupposePS=QR=xunitsandacirclewithcenteronbasePQistangenttobothsegmentsPSandQR.Ifxisthesmallestpossiblevalue,thenwhatisthevalueofx2 ?
degrees
units2
integers
students
A B
CD
%
MATHCOUNTS 2010-2011 23
Workout 81. ��������� EachofJonathan’sbicycletiresrevolved2327timesasherodethreemiles.What
isthediameterofoneofJonathan’sbicycletires,ininches?Expressyouranswertothenearestwholenumber.
2. ��������� SixfriendsA,B,C,D,EandFeatdinneratalocalrestaurant.Aeatsthereeveryday.Beatsthereeveryotherday.Ceatsthereeverythirdday.Deatsthereeveryfourthday.Eeatsthereeveryfifthday.Featsthereeverysixthday.IfallsixofthemeattogetherattherestaurantonJanuary15,howmanymoretimeswilltheyeattogetherinthatcalendaryear?
3. ��������� ThekettleinKerry’skitchenis80%fullofwater.After25%ofthewaterinithasbeenpouredout,thereare1200mlofwaterleft.HowmanylitersofwaterdoesKerry’skettleholdwhenitisfull?
4. ��������� Asquareisinscribedinacirclewithradius6inches.Whatistheareainsidethecirclebutoutsidethesquare?Expressyouranswertothenearestwholenumber.
5. ��������� Nextweek,tocelebrate50yearsinbusiness,thegrocerystoreisgivingawayabagoffreegroceriestoevery150thcustomer,startingwiththe150thcustomer.If2000customersshopatthestorenextweek,whatisthepercentageofcustomerswhowillreceiveafreebagofgroceries?Expressyouranswertothenearest hundredth.
6. ��������� Tombuystwoshirtsandonepairofsocksfor$15.Sambuysoneshirtandtwopairsofsocksfor$12.HowmuchwillJimpayfor15shirtsand10pairsofsocks?
7. ��������� InisoscelestriangleABC,AB=AC=13ft,andBC=10ft.SquareRSTUisinscribedinthetriangle,withRandSonBC,withTonACandwithUonAB. What isthelengthofasideofthesquare?Expressyouranswerasacommonfraction.
8. ��������� Alinewithslopeequalto1andalinewithslopeequalto2intersectatthepointP(4,6),asshown.PointsQandRareboth on the x-axis.WhatistheareaoftrianglePQR?
9. ��������� WhenAndy,Becky,Chad,Dave,EleanorandFredlineupinarow,whatistheprobabilitythatAndyandBeckyarenotstandingnexttoeachother?Expressyouranswerasacommonfraction.
10. �������� InconvexpentagonJKLMN,KL=LM=MN=4unitsandJK=JN=8units.Ifm∠J=60°andm∠L = m∠M,whatistheareaofthepentagon,expressedinsimplest radical form?
liters
inches
times
%
P (4, 6)
R
Q
inches2
$
feet
units2
units2
MATHCOUNTS 2010-201124
Warm-Up 171. ��������� Foreverydayhishomeworkislate,Sam’sgradeisreducedby10%.
Forexample,ifSam'shomeworkis2dayslate,hisscorewillbemultipliedby0.8.Samwillget4moreproblemscorrectforeachextradayheworksonhishomeworkpasttheduedate.IfSamhas14outof30problemscorrectthedayhisassignmentisdue,howmanydayslateshouldheturnitintogetthemaximumgrade?
2. ��������� Eightidenticalmarblesaretobeplacedinfiveboxesnumbered1through5sothateachboxcontainsatleastonemarble.Inhowmanywayscanthisbedone?
3. ��������� Twonaturalnumbershaveagreatestcommonfactorof3andaleastcommonmultipleof216.Ifthedifferencebetweenthetwonumbersis3,whatisthesumofthetwonumbers?
4. ��������� Thetablelistspoints(x, y)thatsatisfythelinearrelationy = mx + b. What is the product of m and b ?Expressyouranswerasacommonfraction.x 1 5 9 13y 2 5 8 11
5. ��������� SquareABCDhassidesoflength4units.MisthemidpointofAB ,andKisonCDsothatDK:KC=3:1.DM and AK intersect at J. MC and BK intersect at L. What is the area of quadrilateral JKLM?Expressyouranswerasacommonfraction.
6. ��������� Giventhatn>1,whatisthesmallestpositiveintegernwhosedivisorshaveaproduct of n 8?
7. ��������� IntriangleRST,JisonlinesegmentRT,withRJ:JT=2:1.Also,KisonlinesegmentST,withTK:KS=2:1.LinesegmentsSJandRKintersectatpointP.IftheareaoftriangleSPKis7units2,whatistheareaoftriangleRPS?
8. ��������� At3:00,thehourhandandminutehandofa12-hourclockareperpendicular.Whatistheleastnumberofminutesthatmustelapseforthistobetrueagain?Expressyouranswerasamixednumber.
9. ��������� Onehundredredballsarelinedupinarow.Startingfromtheleftend,everyfourthballisreplacedwithagreenball.Then,startingfromtherightend,everyfifthballisreplacedwithawhiteball.Finally,startingfromtheleftend,everysixthballisreplacedwithayellowball.Howmanyredballsremainintherow?
10. �������� IfABCDEFisaregularhexagon,whatistheratiooftheareaoftriangleAHDtotheareaofhexagonABCDEF?Expressyouransweras a common fraction.
days
units2
ways
minutes
A
B C
D
EF
H
A B
CD K
L
M
J
units2
balls
MATHCOUNTS 2010-2011 25
Warm-Up 181. ��������� What is the product of x and y if x2 + y2=36–2xy and x2 – y2 = 12?
2. ��������� Foreachpositivetwo-digitinteger,Johnaddsthetwodigits.Forexample,34gives3 + 4 = 7. What is the sum of all of his results?
3. ��������� Thefigureshows∆PTUinscribedinsquarePQRS.IfPQ=12unitsandsegmentsPTandPUtrisect∠QPS,whatistheareaof∆PTU?
4. ��������� Themean,medianandrangeareall6foracollectionoffivepositiveintegers.Whatis the sum of all possible distinct values that could be the greatest number in all the possible collections?
5. ��������� What is the sum of the reciprocals of the roots of 2x 2 + 3x–8=0?Expressyouranswerasacommonfraction.
6. ��������� ThisSudoku-likefigureisa4×4gridtobefilledsothateachofthedigits1,2,3and4appearsineachrowandineachcolumn.The 4 × 4 grid is divided into four 2 × 2 squares. Each of these 2 × 2 squares also is to contain each of the digits 1, 2, 3 and 4. What digit replaces P?
7. ��������� A small square is inscribed in a circle that has another, larger square circumscribed about it. What is the ratio of the area of the small square to the area of the large square?Expressyouranswerasacommonfraction.
8. ��������� Three rectangles have the same area. Their dimensions are mbyn,(m+2)by(n–2)and(m-2)by(n+10).Whatistheareaofeachrectangle?
9. ��������� Bencameupwithanewideaforaclockandtellingtime.Hewantstomeasureadayastwo10-"hour"cyclesratherthantwo12-hourcycles.However,Bendoesnotwanttochangethe
durationofaminute,soadaywouldstillhavethesamenumberofminutes.Howmanyminutesarethereina“Ben-clockhour”?
10. �������� Twored,twoyellowandtwogreenfaces,allunitsquares,areavailableforbuildingacube.Howmanydistinctcubescanbebuilt?
P S
U
RTQ
1
2
3
4
P
101
2
3
4
56
7
8
9
units2
units2
minutes
cubes
MATHCOUNTS 2010-201126
Workout 91. ��������� Each of the integers 1 through 9 is to be placed in one of the nine
given circles so that the sum, S, of the four numbers along each sideofthetriangleisthesame.Twoofthenumbershavebeenentered. What is the value of S?
2. ��������� IntrapezoidABCD,basesABandCDhavelengths10and6,respectively.LinesegmentPQisdrawnparalleltothebases,withPonAD and Q on BC so that the areaofABQPisthree-fourthsthatofABCD.WhatisthelengthofPQ?Expressyouranswerinsimplestradicalform.
3. ��������� Thenumber20canbeexpressedasasumofthreenaturalnumbersinmanyways.Threedistinctexamplesare1+12+7,3+14+3and14+3+3.Includingtheexamplesshown,inhowmanydistinctwayscan20beexpressedasasumofthreenatural numbers?
4. ��������� BaseBCDofrighttetrahedronABCDisanequilateraltrianglewithsidesoflength6mm.Eachofthelateralsidesofthetetrahedronisanisoscelesrighttriangle.Whatisthevolumeofthetetrahedron?Expressyouranswerinsimplestradicalform.
5. ��������� Ofthefirst2011naturalnumbers,howmanyhaveexactlythreedigitsof1whenwritteninbase2form?
6. ��������� EachsideofequilateraltriangleABChaslength60cm.PointUisonAB,withAU:UB=1:2;pointVisonBC withBV:VC=1:3;andpointWisonAC ,withCW:WA=1:4.WhatistheareaoftriangleUVW?Expressyouranswerinsimplestradical form.
7. ��������� OnSunday,Johndrovefromhishousetohisuncle’shouseforavisit.Ifhisaveragespeedhadbeen10milesperhourslower,thetripwouldhavetaken2hourslonger.Ifhisaveragespeedhadbeen20milesperhourfaster,thetripwouldhavetaken2hoursless.HowmanymilesisitfromJohn’shousetohisuncle’shouse?
8. ��������� Ifapositivetwo-digitintegerisc times the sum of its digits, the number formed byinterchangingthedigitsisthesumofthedigitsmultipliedbywhatexpressionthat involves c?
9. ��������� Howmanyorderedpairsofintegers(x, y)satisfyx2 + y2=652?
10. �������� There are several natural numbers, n, such that B = n 2+2008n and B has a units digit of 4. What is the largest possible digit that can be in the tens place of B?
6
9
units
ways
mm3
numbers
cm2
miles
ordered pairs
MATHCOUNTS 2010-2011 27
Solids Stretch IfyourevolverectangleABCDaboutsideAB,youwillgenerateacylinder.LineABistheaxisofrevolutionforthissolidofrevolution. Forquestions1and2,describethesolidofrevolutiongeneratedbyrevolvingthegivenfigure360°aboutsegment AB.
1. ������������ 2. ������������
Forquestions3and4,whatplanefigurewillgeneratethegivensolidwhenrevolved360°aboutanaxisofrevolution?Drawordescribethefigureandindicatetheaxisofrevolution. 3. ������������ 4. ������������ 5. IfyourevolvethisrectangleaboutsideABorsideBC,youformacylinder. a.Revolvingaroundwhichofthesetwosegmentsproducesacylinderwitha larger volume? ��������� b.Revolvingaroundwhichofthesetwosegmentsproducesacylinderwitha larger total surface area? ��������
6. ��������� Considerthisshadedrectanglewithb>a.LetVa denote the volume of the cylinderthatiscreatedwhentheshadedrectangleisrevolvedaroundsidea, and let Vbdenotethevolumeofthecylinderthatiscreatedwhentheshaded rectangle is revolved around side b. What is the value of the ratio
b
a
VV ?Expressyouranswerasacommonfractionintermsofaandb.
7. ��������� Let Sadenotethetotalsurfaceareaofthecylinderthatiscreatedwhentheshadedrectangleusedinquestion6isrevolvedaroundsidea,andletSb denote the total surface areaofthecylinderthatiscreatedwhentheshadedrectangleisrevolvedaroundsideb.What is the value of the ratio b
a
SS ?Expressyouranswerasacommonfractionintermsof
a and b.
8. ��������� Theseconescanbeformedbyrevolvingarighttriangleaboutitstwolegs.Whatistheratioofthe volume of cone A to the volume of cone B? Expressyouranswerasacommonfraction.
9. Thesidelengthsinthefigurearegiven.Whatisthevolumeofthesolidthatresultsfromrevolvingthetriangle360°around... a)sideAC?Expressyouranswerintermsofπ.__________ b)sideBC?Expressyouranswerintermsofπ.__________ c)sideAB?Expressyouranswerasacommonfractionintermsofπ.__________
10.Anicecreamconeispackedfulloficecream,andahemisphereoficecreamisplacedontop.
a.Whatplanefigurewillgeneratethissolidwhenrevolved360°aboutanaxisofrevolution?Drawordescribethefigureandindicatetheaxisofrevolution.______
b. If the volume of ice cream inside the cone is the same as the volume of ice cream outsidethecone,howmanycentimetersistheheightofthecone?___________
A
B
A
B
D
C
A
B
A
B
A B
CD
12
8
b
a
2425
7C
A
B
8 cm
? cm
2c
c
cone A2c
c
cone B
units3
units3
units3
cm
MATHCOUNTS 2010-201128
Sum and Product SUPER Stretch
1. ��������� Whatisthesumofthesolutionsof6x2+5x −4=0?Expressyouranswerasacommon fraction.
2. ��������� A quadratic equation of the form x2 + kx + m = 0 has solutions 3 2 2x = + and 3 2 2x = − . What is the value of k + m ?
3. ��������� What is the sum of the reciprocals of the solutions of 4x2 − 13x+3=0?Expressyouranswerasacommonfraction.
4. ��������� If r and s are the solutions of 2x2 + 9x+3=0,whatisthevalueofr 2 + s 2? Expressyouranswerasacommonfraction.
5. ��������� If r and s are the solutions of x2+6x −2=0,whatisthevalueofr 3 + s 3?
6. ��������� The solutions of x2 + bx + c=0areeach5morethanthesolutionsof x2 + 7x + 3 = 0. What are the values of b and c ?Expressyouranswerasanorderedpair(b, c ).
7. ��������� A cubic equation of the form x3 + bx2 + cx + d = 0 has solutions x = 3, x = 4 and x=5.Whatarethevaluesofb, c and d ?Expressyouranswerasanorderedtriple(b, c, d ).
8. ��������� What is the sum of the reciprocals of the solutions of x3 − 3x2 − 13x+15=0?Expressyouranswerasacommonfraction.
9. ��������� What is the sum of the squares of the solutions of x3 −15x2+66x −80=0?
10. �������� The solutions of x3 −63x2 + cx −1728=0formageometricsequence.Whatisthevalue of c ?
(,)
(,,)
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