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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 ?

(,)

(,,)