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703-299-9006 info@mathcounts.org www.mathcounts.org

Unauthorized reproduction of the contents of this publication is a violation of applicable laws.Materials may be duplicated for use by U.S. schools.

MATHCOUNTS® and Mathlete® are registered trademarks of the MATHCOUNTS Foundation.

2010–2011

School Handbook:Volume I

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 more than 100 creative math problemsthat meet NCTM standards for grades 6-8.

MATHCOUNTS 2010–2011 19

FORMS OF ANSWERSThe following list explains acceptable forms for answers. Coaches should ensure that Mathletes are familiar with these rules prior to participating at any level of competition. Judges will score competition answers in compliance with these rules for forms of answers.

All answers must be expressed in simplest form. A “common fraction” is to be considered a fraction in the form ± a

b , where a and b are natural numbers and GCF(a, b) = 1. In some cases the term “common fraction” is to be considered a fraction in the form AB , where A and B are algebraic expressions and A and B do not share a common factor. A simplified “mixed number” (“mixed numeral,” “mixed fraction”) is tobe considered a fraction in the form ± N a

b , where N, a and b are natural numbers, a < b and GCF(a, b) = 1. Examples:Problem: Express 8 divided by 12 as a common fraction. Answer: 2

3 Unacceptable: 46

Problem: Express 12 divided by 8 as a common fraction. Answer: 32 Unacceptable: 12 1

8 2, 1Problem: Express the sum of the lengths of the radius and the circumference of a circle with a diameter of 1

4 as a common fraction in terms of π. Answer: 1 28

+ π

Problem: Express 20 divided by 12 as a mixed number. Answer: 23 Unacceptable: 8 5

12 3,

Ratios should be expressed as simplified common fractions unless otherwise specified. Examples:Simplified,AcceptableForms: 7 3 4

2 6, , −ππ Unacceptable:

141

2 3, , 3.5, 2:1

Radicals must be simplified. A simplified radical must satisfy: 1) no radicands have a factor which possesses the root indicated by the index; 2) no radicands contain fractions; and 3) no radicals appear in the denominator of a fraction. Numbers with fractional exponents are not in radical form. Examples:Problem: Evaluate 15 5× . Answer: 5 3 Unacceptable: 75 Answers to problems asking for a response in the form of a dollar amount or an unspecified monetary unit (e.g., “How many dollars...,” “How much will it cost...,” “What is the amount of interest...”) should be expressed in the form ($) a.bc, where a is an integer and b and c are digits. The only exceptions to this rule are when a is zero, in which case it may be omitted, or when b and c are both zero, in which case they may both be omitted. Examples:Acceptable: 2.35, 0.38, .38, 5.00, 5 Unacceptable: 4.9, 8.0

Units of measurement are not required in answers, but they must be correct if given. When a problem asks for an answer expressed in a specific unit of measure or when a unit of measure is provided in the answer blank, equivalent answers expressed in other units are not acceptable. For example, if a problem asks for the number of ounces and 36 oz is the correct answer, 2 lbs 4 oz will not be accepted. If a problem asks for the number of cents and 25 cents is the correct answer, $0.25 will not be accepted.

Do not make approximations for numbers (e.g., π, 23 , 5 3 ) in the data given or in solutions unless

the problem says to do so.

Do not do any intermediate rounding (other than the “rounding” a calculator performs) when calculating solutions. All rounding should be done at the end of the calculation process.

Scientific notation should be expressed in the form a × 10n where a is a decimal, 1 < |a| < 10, and n is an integer. Examples:Problem: Write 6895 in scientific notation. Answer: 6.895 × 103

Problem: Write 40,000 in scientific notation. Answer: 4 × 104 or 4.0 × 104

An answer expressed to a greater or lesser degree of accuracy than called for in the problem will not be accepted. Whole number answers should be expressed in their whole number form. Thus, 25.0 will not be accepted for 25 nor vice versa.

The plural form of the units will always be provided in the answer blank, even if the answer appears to require the singular form of the units.

1 1

3

20 MATHCOUNTS 2010-2011

VOCABULARY AND FORMULASThe following list is representative of terminology used in the problems but should not be viewed as all‑inclusive. It is recommended that coaches review this list with their Mathletes.

interior angle of a polygonintersectioninverse variationirrational numberisosceleslateral surface arealateral edgelattice point(s)LCM linear equationmeanmedian of a set of datamedian of a trianglemidpointmixed numbermode(s) of a set of datamultiplemultiplicative inverse

(reciprocal)natural numbernumeratorobtuse angleoctagonoctahedronodds (probability) opposite of a number (additive

inverse)ordered pairordinateoriginpalindromeparallelparallelogramPascal’s trianglepentagonpercent increase/decreaseperimeterpermutationperpendicularplanarpolygonpolyhedronprime factorizationprime numberprincipal square root

degree measuredenominatordiagonal of a polygondiagonal of a polyhedrondiameterdifferencedigitdigit‑sumdirect variationdividenddivisibledivisoredgeendpointequationequiangularequidistantequilateralevaluateexpected valueexponentexpressionexterior angle of a polygonfactorfactorialFibonacci sequencefiniteformulafrequency distributionfrustumfunctionGCFgeometric meangeometric sequenceheight (altitude)hemispherehexagonhypotenuseimage of a point (under a

transformation)improper fractioninequalityinfinite seriesinscribeinteger

abscissaabsolute valueacute angleadditive inverse (opposite)adjacent anglesalgorithmalternate interior anglesalternate exterior anglesaltitude (height)areaarithmetic meanarithmetic sequencebase 10binarybisectbox‑and‑whisker plotcenterchordcirclecircumferencecircumscribe coefficientcollinearcombinationcommon denominatorcommon divisor common factorcommon fractioncommon multiplecomplementary anglescomposite numbercompound interestconcentricconecongruentconvexcoordinate plane/systemcoordinates of a pointcorresponding anglescounting numberscounting principlecubecylinderdatadecimal

MATHCOUNTS 2010–2011 21

prismprobabilityproductproper divisorproper factorproper fractionproportionpyramidPythagorean Triplequadrantquadrilateralquotientradiusrandomrange of a data setrateratiorational numberrayreal numberreciprocal (multiplicative

inverse)rectanglereflectionregular polygonrelatively prime

remainderrepeating decimalrevolutionrhombusright angleright circular coneright circular cylinderright polyhedronright trianglerotationscalene trianglescientific notationsegment of a linesemicirclesequencesetsimilar figuressimple interestslopeslope‑intercept formsolution setspheresquaresquare rootstem‑and‑leaf plotsumsupplementary angles

system of equations/inequalities

tangent figurestangent linetermterminating decimaltetrahedrontotal surface areatransformationtranslationtrapezoidtriangletriangular numberstrisectunionunit fractionvariablevertex vertical anglesvolumewhole numberx‑axisx‑coordinatex‑intercepty‑axisy‑coordinatey‑intercept

The list of formulas below is representative of those needed to solve MATHCOUNTS problems but should not be viewed as the only formulas that may be used. Many other formulas that are useful in problem solving should be discovered and derived by Mathletes.

CIRCUMFERENCECircle C=2 × � × r=� × d AREASquare A= s 2

Rectangle A = l × w=b × hParallelogram A= b × hTrapezoid A = 1

2 (b1 + b2) × hCircle A = π × r2

Triangle A = 12 × b × h

Triangle ( )( )( )= − − −A s s a s b s c

Equilateral triangle 2 34

=sA

Rhombus A = 12 × d1 × d2

SURFACE AREA & VOLUMESphere SA= 4 × π × r2

Sphere V = 43 × π × r3

Rectangular prism V = l × w × hCircular cylinder V = π × r2 × hCircular cone V = 1

3 × π × r2× hPyramid V = 1

3 × B × h

Pythagorean Theorem c2 = a2 + b2

Counting/ Combinations

!( !)(( )!)

=−n r

nCr n r

MATHCOUNTS 2010-2011 27

Warm-Up 11. ��������� Ifatotaloftwentymilliondollarsistobedividedevenlyamongfourmillionpeople,

howmuchmoney,indollars,willeachpersonreceive?

2.��������� TriangleXYZhassideXZ=6cm.SegmentYAisperpendiculartoXZ.IfAYis4cm,whatistheareaoftriangleXYZ?

3.��������� Wilhelminawenttothestoretobuyafewgroceries.Whenshepaidforthegrocerieswitha$20bill,shecorrectlyreceived$4.63backinchange.Howmuchdidthegroceriescost?

4.��������� Howmanymultiplesof8arebetween100and175?

5.��������� Athree-digitintegeristoberandomlycreatedusingeachofthedigits2,3and6once.Whatistheprobabilitythatthenumbercreatediseven?Expressyouranswerasacommonfraction.

6.��������� Inthefollowingarithmeticsequence,whatisthevalueofm?-2,4,m,16,...

7.��������� Aparticularfractionisequivalentto23.Thesumofitsnumeratoranddenominator

is105.Whatisthenumeratorofthefraction?

8.��������� Whatistheleastnaturalnumberthathasfourdistinctprimefactors?

9.��������� Theperimeterofaparticularrectangleis24inches.Ifitslengthandwidtharepositiveintegers,howmanydistinctareascouldtherectanglehave?

10. �������� TrapezoidABCDandtrapezoidEFGHarecongruent.WhatisthedifferencebetweentheareaoftriangleABCandtheareaoftriangleEFJ?

$

multiples

$

sqcm

areas

squnits

A B

CD

E F

GJH

X

Y

ZA

MATHCOUNTS 2010-201128

Warm-Up 21. ��������� Thebargraphshowsthenumberofcartonsofmilksoldper

daylastweekatJonesJuniorHigh.Whatisthepositivedifferencebetweenthemeanandmediannumberofcartonsofmilksoldperdayoverthefive-dayperiod?

2.��������� AnequilateraltrianglePBJthatmeasures2inchesoneachsideiscutfromalargerequilateraltriangleABCthatmeasures5inchesoneachside.WhatistheperimeteroftrapezoidPJCA?

3.��������� Currently, 14 ofthemembersofalocalclubareboys,and

thereare80members.Ifnoonewithdrawsfromtheclub,whatistheminimumnumberofboysthatwouldneedtojointomaketheclub 1

3 boys?

4.��������� IfWilliamhas3pairsofpantsand4shirts,andanoutfitconsistsof1pairofpantsand1shirt,howmanydistinctoutfitscanWilliamcreate?

5.��������� Jessicareads30pagesofherbookonthefirstday.Thenextday,shereadsanother36pages.Onthethirdday,shereadsanother42pages.Ifshecontinuestoincreasethenumberofpagesshereadseachdayby6,howmanydayswillittakehertoreadabookthathas270pages?

6.��������� Abasketoffruitcontains4oranges,5applesand6bananas.Ifyouchooseapieceoffruitatrandomfromthebasket,whatistheprobabilitythatitwillbeabanana?Expressyouranswerasacommonfraction.

7.��������� Whatnumberishalfwaybetween 58 and

1116 ?Expressyouranswerasacommonfraction.

8.��������� FarmerFredreadthathiscropneedsafertilizerthatis8%nitrateforoptimalyield.Heneedstoapply4poundsofnitrateperacre.Ifhisfieldhas188acres,howmanypoundsoffertilizerwillFreduse?

9.��������� Foursquaresarecutfromthecornersofarectangularsheetofcardboard.Itisthenfoldedasshowntomakeaboxthatis15incheslong,8incheswideand2inchestall.Whatwastheareaoftheoriginalpieceofcardboard?

10. �������� Twoconsecutiveanglesofaregularoctagonarebisected.Whatisthedegreemeasureofeachoftheacuteanglesformedbytheintersectionofthetwoanglebisectors?

inches

days

0

10

20

30

40

Day

Mon Tues Wed Thurs Fri

Num

ber

Sold

Cartons of Milk Sold per Day

pounds

A

B

C

P J

sqin

cartons

outfits

boys

degrees

MATHCOUNTS 2010-2011 29

Workout 1 1. ��������� Sweaterscost$49atalocaldepartmentstore,andthecatalogdepartmentof

thesamestoresellsthemfor$44online.Thelocaldepartmentstorecharges7%tax,andthecatalogdepartmentcharges$3.50persweaterforshipping(butdoesnotchargetax).Howmuchwouldapersonsavebybuying6sweatersthroughthecatalogdepartmentratherthanatthedepartmentstore?

2.��������� IfJordanpassedmilemarker138at5:08pmandthenpassedmilemarker216at6:20pm,whatwasheraveragespeedinmilesperhour?

3.��������� Agroupof5womenand8menweighatotalof1921pounds.Ifthewomenofthegrouphaveanaverageweightof109pounds,whatistheaverageweightofthemeninthegroup?

4.��������� OnplanetWobble,1wompisequalinvalueto3wamps,and2wampsareequalinvalueto5wemps.Howmanywompsareequalinvalueto15wemps?

5.��������� Howmanypositivethree-digitperfectcubesareeven?

6.��������� Billywasplayingatriviagame.Accordingtotherules,Billywouldreceive25pointsforeachquestionheansweredcorrectlybutwouldlose50pointsforeachquestionheansweredincorrectly.Attheendofthegame,Billyhadatotalof450points.Hehad5timesasmanyquestionscorrectasincorrect.IfBillyansweredeveryquestion,howmanyquestionswereaskedinthegame?

7.��������� Whatisthevalueofthefollowingexpression,expressedasarepeatingdecimal?

( )1 1 29 7 9+ −

8.��������� Garychoosesatwo-digitpositiveinteger,addsitto200,andsquarestheresult.WhatisthelargestnumberGarycanget?

9.��������� ThesalepriceofKara’sdresswas$43.20aftera40%discount.Whatwastheoriginalpriceofthedress?

10. �������� Thesupplementofanangleis5.5timesthecomplementoftheangle.Whatisthemeasureoftheangle?

mph

cubes

pounds

questions

degrees

$

$

womps

MATHCOUNTS 2010-201130

ways

$

(,)

Warm-Up 3 1. ��������� Thirtypercentoftwiceanumberis30.Whatwastheoriginalnumber?

2.��������� Abouncyballbounces16feethighonitsfirstbounceandexactlyhalfashighonthenextbounce.Foreachlaterbounce,theballbounceshalfashighasonthepreviousbounce.Howmanyincheshighdoestheballbounceonitsseventhbounce?

3.��������� Tamika’sdadboughtaMustangconvertiblefor$12,000tocelebratehisfirstjobin1978,whichpaidhim$15,000ayear.IfTamikaistousethesamefractionofhersalaryasherdadtobuyaMustangconvertible,whichnowsellsfor$32,000,whatwillherfirstjobhavetopay?

4.��������� Arectangularprismhasdimensions4by6byx.Ifthetotalsurfaceareaoftheprismis248squareunits,whatisthevalueofx?

5.��������� HowmanywayscanthelettersinFACTORberearrangedsothatthefirstandlastlettersarevowels?

6.��������� Ifa#b=(a+b)2,whatisthevalueof3#1?

7.��������� Thetriangulararrayofpositiveintegersshowncontinuesindefinitely,witheachrowcontainingoneentrymorethantherowaboveit.Whatisthesumofthetwointegersdirectlyabove100?4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

8.��������� Inaclassof20students,theaverage(arithmeticmean)scoreonatestis84points.If6studentseachscored100pointsand4studentseachscored50points,whatistheaverageofthescoresoftheremainingstudents?

9.��������� If 23 4 5a ba b−−

= ,whatistheratioofatob?Expressyouranswerasacommonfraction.

10. �������� IftriangleWINisreflectedoverthex-axistocreatetriangleW’I’N’,whatwillbethecoordinatesofpointI’?Expressyouranswerasanorderedpair.

points

inches

2 4 6 8 10-2

4

6

-2

W (-1, 2)

I (3, 5)

N (11, 2)

x

y

MATHCOUNTS 2010-2011 31

Warm-Up 41. ��������� Asurveyofthecostofone-poundbagsofpotatochips

atfivelocalsupermarketsisshowninthestem-and-leafplot.Howmanybagsofchipscostlessthanthemedianpriceofabagofchips?

2.��������� Vivianhas45songsonherMP3-player,15ofwhicharealternativerockandtherestofwhicharerap.Malikhas60songsonhisMP3-playerinthesameratioasVivian’ssongs.HowmanymorerapsongsdoesMalikhaveonhisMP3-playerthanVivianhasonhers?

3.��������� Lashaewillplantpansiesandvioletsaroundtheedgeofhergarden.Shemustusethesamenumberofeachplant,andshemustusealloftheflowersshebuys.Ifpansiesaresoldinpackagesof6andvioletsaresoldinpackagesof10,whatistheleastnumberofpackagesofvioletsshecouldbuy,giventhatshebuysatleast1package?

4.��������� ThefirstfourrowsofPascal’striangleareshowntotheleft.IfthefirsttwonumbersinaparticularrowofPascal’striangleare1and10,whatisthefourthnumberofthenextrow?

5.��������� Inthefigure,thegraysmallsquaresandthewhitesmallsquaresareallthesamesize,andtheoutercornersoftheoutergraysquareslayontheedgeofthelargestsquare.Whatfractionofthelargestsquareisshaded?Expressyouranswerasacommonfraction.

6.��������� LastJulytherewere56dogsforsaleinthenewspaperads,andthisJulytherewere84dogsforsale.Whatisthepercentincreaseinthe

numberofdogsforsale?

7.��������� Whatisthevalueoftheexpression3x+4y3ifx=3andy=−2?

8.��������� PointsA,BandCarecollinear,withBbetweenAandC.IfthedistancefrompointAtopointBis8morethantwicethedistancefrompointBtopointC,andthelengthofsegmentACis5timesthelengthofsegmentBC,howlongissegmentAC?

9.��������� Thesumoftwonumbersis3timestheirpositivedifference.Whatistheratioofthesmallernumbertothelargernumber?Expressyouranswerasacommonfraction.

10. �������� Thenetshowniscomposedof6congruentsquaresandhasaperimeterof56units.Whenthenetisfoldedintoacube,whatisthevolumeofthecube?

%

units

morerapsongs

pckgs

bags

Potato Chip Prices

15

16

17

18

19

20

21

22

23

24

25

9

9

5

8

9

9

5

5

9

9

9

9

9 9 9 9

5 8 9

9 9

Key

15 9 means $1.59|

11 1

1211 3 3 1

units3

MATHCOUNTS 2010-201132

Workout 21. ��������� Sarahas30poundsofdriedbananasthatshewantstodivideintosmallbagsand

sell.Ifeachsmallbagholds 58ofapoundandwillsellfor$1.58,howmuch

willSarabringinfromsalesifshesellsallthebags?

2.��������� Howmanyfive-digitintegersareperfectsquares?

3.�������� Fourfriends(Amanda,Barbara,CharlesandDylan)invested$3000,$5000,$7000and$9000,respectively,inacertainstart-upcompany.Profitswillbedividedinthesameratioastheinvestment.Ifthebusinessmadeaprofitof$72,000thisyear,howmuchoftheprofitwillthelargestinvestorreceive?

4.��������� ThemeanscoreonMr.Pascal’sfirstsemesterexamforhis28studentsis 73points.Ifheremovesthelowestscoreof19points,whatistheadjustedclassaverage(arithmeticmean)?

5.��������� Thesumoffourconsecutiveevenintegersis596.Whatistheproductoftheleastandgreatestoftheseintegers?

6.��������� InthisNumberWall,youaddthenumbersnexttoeachotherandwritethesumintheblockdirectlyabovethetwonumbers.WhatnumberwillbeintheblocklabeledN?

7.��������� ChordABofcircleOis6unitslongand4unitsfromcenterO.WhatistheareaofcircleO?Expressyouranswerintermsof�.

8.��������� Threefairsix-sidednumbercubes,eachwithfaceslabeled0,2,4,6,8,10,arerolled.Whatistheprobabilitythatthesumofthenumbersrolledisgreaterthan20?Expressyouranswerasacommonfraction.

9.��������� InthepatternsofdotswithP1,P2,P3andP4,eachsuccessivepatternofdotshasonemorerowandonemorecolumn.WhatistheratioofthenumberofdotsinP50tothenumberofdotsinP25?Expressyouranswerasacommonfraction.

10. �������� ThepopulationofCircletownhasbeengrowingatanannualrateof5%forthelast5years.Thepopulationisnow3105people.WhatwasthepopulationofCircletown5yearsago?Expressyouranswertothenearestwholenumber.

points

$

integers

$

peopleP1 P2 P3 P4

welcome

to

Circletown

N

65

34

9 7

15

4

O{

A B

squnits

MATHCOUNTS 2010-2011 33

Warm-Up 51. ��������� IfMinhpicksaletterofthealphabetatrandom,whatistheprobabilityitoccurs

inthename“Hattiesburg,Mississippi”?Expressyouranswerasacommonfraction.

2.��������� ThegraphshowstheamountofsalesmadebyeachgradeatMemorialMiddleSchoolintheannualfundraiser.WhatpercentofthetotalsaleswasmadebytheGrade7class?Expressyouranswertothenearesttenth.

3.��������� Whatistheintegervalueof0−5+10−15+20−25+30−...+240?

4.��������� Inthisadditivemagicsquarethesumofthethreenumbersineachrow,ineachcolumnandalongeachdiagonalisthesame.Whatisthevalueofz?

5.��������� Whatistheareaofthequadrilateralwithverticesat(1,1),(4,1),(7,5)and(4,5)?

6.��������� Ifa#b=ab-b

a,whatisthevalueof3#5?

7.��������� CoachKennedyhadthestartersofhisbasketballteampracticefreethrows.Eachplayerhadtoshoot20freethrows.OnWednesday,4ofthe5startersdidthefreethrowdrillandaveragedmaking15outof20attemptedbaskets.Whenthefifthstarterdidhis

drill,theteamaveragedroppedto14madeoutof20attempted.Whatpercentofhisfreethrowsdidthefifthstartermake?

8.��������� Williamistravelingatarateof75milesperhour.HowfastisWilliamtravelinginfeetpersecond?

9.��������� Tocreateamoreaffordablelineofhergourmetchocolates,Kindramixes10poundsofchocolateworth$1.50perpoundwithanothertypeofchocolate,worth$9.00perpound.Shewishestomakeamixtureofchocolateworth$4.00perpound.Howmanypoundsofthe$9.00chocolatemustsheaddtocreatehernewproduct?

10. �������� Asolidisformedbyplacingasquarepyramidwithabaseof3inchesby3inchesontopofacubewithedgesof3inches.Thevolumeofthiscombinedsolidis54cubicinches.Whatistheheightofthepyramid?

%

ftpersec

z

6

10

11

inches

Grade 8:

1250

Grade 5:

750

Grade 7:

500

Grade 6:

1000

Total Sales By Grade

%

pounds

squnits

MATHCOUNTS 2010-201134

Warm-Up 61. ��������� Whatisthesmallestpositiveintegerthatisdivisibleby2,5,6and9?

2.��������� Whatisthevalueof(0.07)3?Expressyouranswerasadecimaltothenearestmillionth.

3.��������� Ifx= ab andy= ba ,thenwhatisthesquareoftheproductofxandy?

4.��������� Whatistheonlynumberthatwhenaddedtoitsreciprocalisequalto2?

5.��������� Twosquareshavecentersattheoriginofacoordinateplaneandsidesparalleltotheaxes.Thesmallersquarehasanareaof18squareunits,andthelargersquarehasanareaof50squareunits.Howmanypointswithonlyintegercoordinatesareoutsidethesmallersquareregionandinsidethelargersquareregion?

6.��������� Pedroearns$7.50anhourathisjobatMatrixCinemas.Ifheworks20hoursthisweekandhisemployerwithholdsatotalof22%ofhisweeklypayfortaxesandSocialSecurity,whatishistake-homepayfortheweek?

7.��������� MaryleavesNewYorkCityat9:00am,travelingtoCharlotte,NC,atanaveragerateof55milesperhour.SimbaleavesonehourlaterthanMaryandfollowsMary’srouteatanaveragerateof65milesperhour.AtwhattimewillSimbacatchuptoMary?

8.��������� Whatorderedpairofpositiveintegers(r,s)satisfiestheequation5r+6s=47,suchthatr>s?

9.��������� Whatistheminimumnumberof3-inchby5-inchindexcardsneededtocompletelycovera3-footby4-footrectangulardesktopwithoutcuttingtheindexcards?

10. �������� PointsA,B,CandDarethecentersoffourcircles,andtheyarealsointersectionpointsofthesecircles,asshown.Eachcirclehasaradiusof6feetandistangenttotwosidesofsquareEFGH.WhatistheareaofsquareEFGH?

(,)

indexcards

points

$

A B

CD

E F

H G

sqft

pm

MATHCOUNTS 2010-2011 35

Workout 31. ��������� Whentherepeatingdecimal 0.38 isexpressedasacommonfraction,whatisthe

sumofthenumeratoranddenominator?

2.��������� Thesumofthreedifferenttwo-digitprimenumbersis79.Thelargestofthenumbersis43.Thedifferenceoftheothertwois10.Whatistheproductofthethreenumbers?

3.��������� Fivewoodendisksarenumbered-3,-2,1,4and7.Iftwodisksarechosenatrandom,withoutreplacement,whatistheprobabilitythattheirproductisnegative?Expressyouranswerasacommonfraction.

4.��������� Thedigits1,3,5,6,7and9areeachusedonceandonlyoncetoformthreetwo-digitprimenumbers.Whatisthelargestpossiblesumthatcanbeformed?

5.��������� Acylinderandarectangularprismhavethesamevolume.Ifthelength,widthandheightoftherectangularprismaredoubledandtheradiusofthecylinderisdoubled,whatistheratioofthevolumeofthenewcylindertothevolumeofthenewrectangularprism?Expressyouranswerasacommonfraction.

6.��������� AquadrilateralisformedbyconnectingtheverticesA(5,0),B(2,6),C(–9,0)andD(2,–6).WhatwouldbethecoordinatesofpointCifyoumovedittotherightjustenoughtotransformquadrilateralABCDintorhombusABCD?Expressyouranswerasanorderedpair.

7.��������� Threestoplightsondifferentstreetseachoperateontheirownindependentschedules,asfollows:thefirststoplightisred1minuteoutofevery2minutes(1minutered,then1minutegreen),thesecondisred2minutesoutofevery3minutes(2minutesred,then1minutegreen)andthethirdisred3minutesoutofevery5minutes(3minutesred,then2minutesgreen.)At9:00ameachstoplightturnsred.Thelightsareeitherredorgreen(don’tworryaboutyellow).Whattimeisitwhenthenext1-minutesegmentoftimeinwhichallthreestoplightsareredbegins?

8.��������� IfSaramakes80%ofthefreethrowssheattempts,whatistheprobabilitythatshemissesexactlytwoofhernextthreefreethrows?Expressyouranswerasacommonfraction.

9.��������� A4-unitby4-unitgridhasthispatternshadedinthedesignofacompany’slogo.Whatistheareaoftheshadedportionofthelogo?

10. �������� Whatisthe2011thtermofthearithmeticsequence−4,−1,2,5,...,whereeachtermafterthefirstis3morethantheprecedingterm?

am

(,)

squnits

A

B

C

D

y

x

MATHCOUNTS 2010-201136

Systems of Equations Stretch

Forproblems1-3,solveeachofthefollowingsystemsofequations.Describethemethodyouused.Whatothermethodscouldyouhaveusedtosolveeachsystem?Expresseachanswerasanorderedpair.Expressanynon-integervalueasacommonfraction.

1. 2x+3y=11 2. 32 11x y+ = 3. 2x2+3y

2=11 3x-y=11 3 1 11x y− = 3x

2-y2=11

Inproblems4and5,forwhatvalueofkdoesthelinearsystemhavenosolution?Expressanynon-integervalueasacommonfraction.

4. kx+3y=11 5. 2x+ky=11 3x-y=11 3x-y=11

6.Giventhat2a+b=19,2c+d=37andb+d=24,whatisthevalueofa+b+c+d?

7.Giventhata+b=29andab=204,whatisthevalueofa2+b

2?

8.Solveeachofthefollowinglinearsystems.Expressyouranswerasanorderedpair.

a. 5x+6y=7 b. x+2y=3 8x+9y=10 4x+5y=6

Manyproblemscanbesolvedusingalinearsystemofequations.Forproblems9and10,uselinearsystemsofequationstohelpyousolve.

9a.Foreachequationbelow(i,ii,iiiandiv),doesthereexistasolution(x,y)withpositiveintegersxandy?Ineachcase,eitherfindallpairsofintegerssatisfyingtheequationorexplainwhynoneexist.Hintforthefirstequation:x

2–y2=(x+y)(x–y),sox+y=12andx–y=4isonecaseto

check.

i.x2–y

2=48 ii.x2–y

2=23 iii.x2–y

2=45 iv.x2–y

2=90

b.Ingeneral,forwhattypeofintegers,n,doesx2–y

2=nhaveatleastonesolution?

10.Ashippingclerkwishestodeterminetheweightsofeachoffiveboxes.Eachboxweighsadifferentintegeramountlessthan100kg.Unfortunatelytheonlyscalesavailablemeasureweightsinexcessof100kg.Theclerkthereforedecidestoweightheboxesinpairssothateachboxisweighedwitheveryotherbox.Theweightsforthe10pairsofboxesare(inkilograms)110,112,113,114,115,116,117,118,120and121.Fromthisinformationtheclerkcandeterminetheweightofeachbox.Whataretheweightsofeachofthefiveboxes?

MATHCOUNTS 2010–2011 37

What About Math?In an effort to bridge the gap between the knowledge students gain from MATHCOUNTS and the important applications of this knowledge, MATHCOUNTS is providing a What About Math? section in Volume I of the MATHCOUNTS School Handbook. This year the problems are from the fields of engineering and statistics. Written by actual professionals in these fields, these problems illustrate how students can put the skills they are learning today into practice in the near future.

This year’s problems and solutions were written for MATHCOUNTS by members of the Association for Unmanned Vehicle Systems International (AUVSI), The Actuarial Foundation (TAF) and the Consumer Electronics Association (CEA).

AUVSI Foundation Writers/Editors: Wendy Amai, Sandia National Laboratories; Dr. Kenneth Berry, Assistant Director of the Science and Engineering Education Center, University of Texas at Dallas; Angela Carr, Web Services Manager, AUVSI; Dr. Deborah A. Furey, DARPA Program Manager; Lisa Marron, Sandia National Laboratories and Dr. Dave Novick, Sandia National Laboratories

TAF Writers/Editors: Charles Cicci, ACAS, MAAA; Kevin Cormier, FCAS, MAAA; Jeremy Fogg; Mubeen Hussain; Jeremy Scharnick, FCAS, MAAA; John Stokesbury, FSA, FCA, EA, MAAA; Pete Rossi, FSA, FCA, CERA, MAAA and Patricia Teufel, FCAS, MAAA. CEA Writers/Editors: Bill Belt; Deepak Joseph; Brian Markwalter, PE and Dave Wilson

ASSOCIATION FOR UNMANNED VEHICLE SYSTEMS INTERNATIONAL FOUNDATION (WWW.AUVSIFOUNDATION.ORG)

The AUVSI Foundation is a nonprofit, charitable organization that was established to support the educational initiatives of the Association for Unmanned Vehicle Systems International (AUVSI). The AUVSI Foundation’s primary objective is to develop educational programs that attract and equip students for careers in robotics. To do this, the AUVSI Foundation provides K-12 and university level students with hands-on educational activities that highlight the world of robotics while emphasizing STEM curriculum–science, technology, engineering and math. Currently, the AUVSI Foundation directly sponsors five annual advanced robotics competitions that attract nearly 200 student teams, representing more than 2000 students from schools around the world. Last year, nearly $100,000 in prize money was awarded through these student competitions. For more information on the AUVSI Foundation and its educational programs, please visit www.auvsifoundation.org.

THE ACTUARIAL FOUNDATION (WWW.ACTUARIALFOUNDATION.ORG)

The Actuarial Foundation (TAF) was founded in 1994 as the charitable arm of the actuarial profession. TAF’s mission is to develop, fund and execute education and research programs that serve the public by harnessing the talents of actuaries. Through the Foundation’s youth education math activities, actuaries assist with the development of math materials that help students develop a love for math and embrace mathematical concepts in a fun and engaging way in order for them to understand how math is used in the “real world.” If you like math – consider becoming an actuary. “Actuary” was rated the number two profession in the most recent Jobs Rated Almanac. The actuarial profession’s ability to continue providing competent service in the future requires supporting mathematics achievement among today’s students. The Actuarial Foundation is proud to support and sponsor MATHCOUNTS!

CONSUMER ELECTRONICS ASSOCIATION (WWW.CE.ORG)

The Consumer Electronics Association (CEA) is the preeminent trade association promoting growth in the $165 billion U.S. consumer electronics industry. More than 2000 companies enjoy the benefits of CEA membership, including legislative advocacy, market research, technical training and education, industry promotion, standards development and the fostering of business and strategic relationships. CEA also sponsors and manages the International CES–The Global Stage for Innovation. All profits from CES are reinvested into CEA’s industry services.

MATHCOUNTS 2010-201138

What About Math?AUVSI Foundation

1a. �������� Robots often have two treads (like tank treads) instead of four wheels. You can calculate the amount of weight a tread or wheel is carrying per square inch of tread or wheel by dividing the weight of the robot by the total area of the tread or wheel on the ground. If a robot weighs 120 kg and each of the robot’s 2 treads touches an area of ground that is 30 cm long and 5 cm wide, how much weight is carried over each square centimeter of tread? Express your answer as a decimal to the nearest tenth.

1b. �������� Compare this to the same robot with 4 wheels (instead of tread) that each touch a section of ground that is 3 cm × 5 cm. How much weight is on each square centimeter of the wheels?

1c. �������� A particular robot wheel is 6 cm in diameter. The motor attached to the wheel runs at a speed of 120 rpm (rotations per minute). How many centimeters will the robot travel in 30 seconds? Express your answer as a decimal to the nearest tenth.

2. ��������� The buoyancy of an object is defined as the mass of the displaced fluid. A particular underwater robot that has a mass of 10 kg has a volume equivalent to the right circular cylinder shown. The density of water is 1000 kg/m3, and the volume of a right circular cylinder is defined as V = πr2h. For d = 9 cm and h = 100 cm (use 3.14 for π), determine if this robot will sink or float. How much flotation will need to be added (or mass will need to be added/removed) to get this robot to be neutrally buoyant in water? Express your answer as a decimal to the nearest ten thousandth. Note: For an object to be neutrally buoyant, the mass of the object must be equal to the mass of an identical volume of water.

3a. �������� An unmanned aerial vehicle (UAV) carries 45 gallons of fuel. It burns fuel at a rate of 3 gph (gallons/hour) and flies at a speed of 160 mph (miles/hour). We want to fly it to a location X, fly it in the area for 1 hour, and then fly it back to its launching point. What is the greatest number of miles away from the UAV’s launching point can location X be?

3b. �������� How far away can location X be if we want the UAV to arrive back home with a fuel reserve of 0.5 hour?

kg

cm

kg

miles

miles

kg

h

d

MATHCOUNTS 2010-2011 39

4a.�������� Acommunicationnetworkcancarry128,000,000bitspersecond.Adigitalvideocameraonthenetworkmakespicturesthatare640pixelswideby480pixelshigh.Eachpixeliscomposedof3bytes.Abyteis8bits.Apictureiscalleda“frame.”Howmanyframespersecondfromthecameracanthisnetworkcarry?Expressyouranswertothenearestwholenumber.

4b.�������� Wecanincreasetheframerateifwedecreasethesizeofthepicture.Ifwewantedtosendbacksquareframesatarateof30frames/second,howmanypixelswidewouldasquareframe(samesizehighaswide)be?Expressyouranswerasadecimaltothenearesttenth.

5a.�������� Obstaclessuchasfallentrees,fencesandditchescanbedifficultforwheeledroboticvehiclestoovercome.Imaginethataspeciallyequippedrobotcouldjumpoversuchthings.Atthepeakofthejump,therobot’sverticalvelocityis0m/s.StandardgravityonEarth,g,is9.8m/s2.Ifthetimeittakestofallfromthepeakis1.7s,howhighdidtherobotjump?Heightisequalto½(g)(t)2,whereg=gravityandt=timetofalltotheground,inseconds.Expressyouranswerasadecimaltothenearesthundredth.

5b.�������� Howfast,inmeterspersecond,istherobottravelingwhenithitstheground?Velocity=(g)(t).Expressyouranswerasadecimaltothenearesthundredth.

6.��������� In2003,NASAlaunchedtheroverSpiritontothesurfaceofMarstoexploreourplanetaryneighbor.NeartheendofJanuary2010,Marswas99.33millionkmawayfromEarth.Assumethespeedofcommunicationisatthespeedoflight,whichis300,000,000m/s.IfaresearcheronEarthrequestedinformationfromSpiritaboutitstemperature,howlongdidtheresearcherhavetowaitforareplyaftersendingtherequest?Expressyouranswerintheformxminutesyseconds,wherexandyarewholenumbers.

7.��������� Arobothasabatterylifeof3hourswhilecarrying0pounds.Therobot’smaximumvelocityis20feetperminute.Foreverypoundthattherobotneedstocarry,thetotalamountofbatterytimetherobotwilloperatewhiletravelingatmaximumvelocitydecreasesby8minutes.Startingwithafullychargedbattery,howmanyfeetcantherobottravelatmaximumvelocitywhilecarrying10pounds?

8.��������� Asmallgroundrobothasatelescopingliftingarmwithamotorthatcanpickupa20-poundweightwhenitis3feetaway.Themotorcansupportamaximummoment,ortorque,of60foot-pounds(moment=weight×distance).Howmuchweight,inpounds,couldthearmliftat5feetaway?

feet

pounds

pixels

minsec

meters

20 lb

3 ft

? lb

5 ft

meterspersecond

framespersec

MATHCOUNTS 2010-201140

9.��������� Torqueisameasureof“rotationalforce,”anditisequaltoforcetimesdistance.Amotorattheendofan8-inchrobotarmisliftinga3-poundweight.Howmuchtorqueisthemotorgenerating?Note:Torqueismeasuredinfoot-pounds.

10a. ������� Gearsperformtwoimportantduties.First,theycanincreaseordecreasethespeedofamotor.Second,theycandecreaseorincreasethepowertransferred.Gearswiththesamesizeteethareusuallydescribednotbytheirphysicalsizebutbythenumberofteetharoundthecircumference.Thegearratioistherelationshipbetweenthenumbersofteethontwogearsthataremeshed.Gearratio=(numberofteethonoutput)/(numberofteethoninput).Whenmultiplegearsareinvolved,thegearratiosfromeachpairofmeshedgearsaremultipliedtocalculatethegeartrain’sgearratio.

Whatisthegearratioofthegeartrainshowniftherightmostgearistheinput?Expressyouranswerasacommonfraction.

10b. ������� Thespeedofanoutputgearisequalto(inputspeed)×(1÷gearratio).Ifamotorwith120rpm(revolutionsperminute)isattachedtotheleftmostgear(input),whatisthespeedofthefinalrightmostgearthatistheoutput?

20 teeth

120 rpm

60 teeth

30 teeth

25 teeth

40 teeth

foot-pounds

rpm

MATHCOUNTS 2010-2011 41

What About Math?The Actuarial Foundation

1. ��������� Theprobabilitythataparticularstatewillhaveneitherahailstormnoratornadoinagivenmonthis55%.Inthesameperiod,theprobabilityofahailstormis35%andtheprobabilityofatornadois25%.Iftheprobabilityofahailstormandtheprobablityofatornadoareindependentbutnotmutuallyexclusive,whatistheprobabilityofbothahailstormandatornadooccurringinagivenmonth?Expressyouranswerasadecimaltothenearesthundredth.

Usethefollowinginformationforproblems2and3.

Theamountofdamage(inmillionsofdollars)thatcouldbecausedbyanearthquakeinthefictitiouscountryofShakalotisshowninthetablebelow.

2.��������� Whatistheexpected(mean)damageamount,inmillionsofdollars,foranearthquakeinShakalot?Expressyouranswerasadecimaltothenearesttenth.

3.��������� Whatistheinterquartilerange,inmillionsofdollars,forthisdata?

4.��������� Aninsurancecompanyhassold10,000policies.Thepolicyholdersareclassifiedusinggender(MaleorFemale)andage(YoungorOld).Ofthesepolicyholders,3000areOld,4000areYoungMalesand4000areFemale.Howmanyofthecompany’spolicyholdersareOldFemales?

5.��������� SneakyJoehasaloadeddiewiththenumbers1,2and3eachhavingaprobabilityof1/4ofbeingrolledandthenumbers4,5and6eachhavingaprobabilityof1/12ofbeingrolled.NoonebutJoeknowsabouttheloadeddie.SneakyJoeofferstotakethefollowingbetwithyou:Yourollthedie.Iftheresultisanevennumber,youwin$x;ifit’sanoddnumber,youloseandpayJoe$5.HowmuchshouldSneakyJoepayyouforawintomakethisafairbet?

$

$

policy-holders

$

Dollar Valueof Damage (millions)

Probability

5 10%10 15%25 20%50 30%75 20%100 4%150 1%

million

million

MATHCOUNTS 2010-201142

6.��������� Acertaindiseaseisexpectedtoinfect1outofevery10,000individualsinacountry.Atestforthediseaseis99.5%accurate.Itnevergivesafalseindicationwhenitisnegative,so0.5%ofthepeoplewhotakethetestwillgetinaccuratereadings,allofwhichwillbefalsepositives(meaningthatthepeopletestpositivebutdonothavethedisease).Letussupposeyoutestpositive;whatistheprobabilitythatyouactuallyhavethedisease?Expressyouranswerasapercenttothenearestwholenumber.

Usethefollowinginformationforproblems7–9.Frequency=theratioofthenumberofclaimstothenumberofpoliciesSeverity=themeansizeofagroupofclaimsPurePremium=Frequency×SeverityChargedPremium=PurePremium+Expenses+Profit

Supposethereare50policiesthatgeneratethefollowingfiveclaimsin2009:$7500$5000$3000$11,500$8000

7.��������� Whatisthefrequency?Expressyouranswerasadecimaltothenearesthundredth.

8.��������� Whatistheseverityofthefiveclaims?

9.��������� Howmuchpremiumshouldbechargedif15%ofthechargedpremiumistobeusedforexpensesand5%ofthechargedpremiumisforprofit?

10. �������� AutoMakers,Inc.isacarmanufacturerandcurrentlyhasthreeoperationalplantscleverlynamedPlantA,PlantBandPlantC.PlantAcanproduce100carsaday.PlantBcanproduce80carsaday.PlantCcanproduceonly70carsaday.Ifallthreeplantsarerunningatfullproductioncapacity,howmanyfulldaysareneededtoproduce1600cars?

claims/policies

$

$

days

%

MATHCOUNTS 2010-2011 43

What About Math?Consumer Electronics Association

AproviderofwirelessInternetservicewillofferservicetoaneighborhoodthatisshapedlikeasquareandis1km(1000m)longoneachside.Theserviceproviderwillbeusingatowerthatislocatedpreciselyinthemiddleofthesquare.Thesignalfromthetowerwilltravelanequaldistanceinalldirections,producingacircularcoveragearea.Tofunctioncorrectly,aconsumer’swirelessInternetdevicemustreceiveasignalfromthewirelessInternetserviceproviderthatisatleast1microwatt(1x10-6watt)instrength.Radiofrequencysignalslosetheirstrengthastheytraveloveradistanceaccordingtothefollowingformula:Receivedsignalstrengthinwatts=Transmittedsignalstrengthinwatts÷[4πd/λ]2,whereλisthewavelengthofthewirelessInternetsignalinmetersanddisthedistancefromthetransmittertothereceiverinmeters.Thewavelength(λ)ofaradiofrequencysignalinmetersisequivalenttothespeedoflightinmetersperseconddividedbythefrequencyofthesignalinhertz(Hz).Thespeedoflightisapproximately3×108meterspersecond.ThewirelessInternetserviceproviderwillbeusingthefrequency900megahertz(MHz),whichisequivalentto900,000,000Hz.

1. ��������� Howstrongdoesthesignalsentbythetransmitterhavetobeinordertoprovideatleastonemicrowatttoeveryreceiverwithinthe1-square-kilometerneighborhood?Expressyouranswertothenearestwholenumber.Note:Disregardtheheightofthetransmitters.

Anelectriccircuitinahousegoesfromthecircuitbreakerboxinthebasementtothekitchen.Inthekitchentherearefouroutletsconnectedtothiscircuit.Thesearetheonlythingsinthiscircuit.Thewireinthewallfromthebasementtothekitchenisabletohandle15ampsofcurrentwithoutbecomingtoohot.Thecircuitbreakerforthiscircuitinthebasementprotectsthewiringinthewallfromdamage.Theciruitbreakerwilltripifmorethan15ampsofcurrentflowsthroughthecircuit.Thehousehasstandard120-voltelectricservice.Inthekitchenthereisonethingconnectedtotheoutletsinthecircuit—an1100-watttoaster.Thepowerrequiredforoperatingthetoasterwillbepulledfromthetotalpoweravailabletothecircuit.Thehomeownerwantstopurchaseacoffeemakerandconnectittoanotheroutletinthesamecircuit.

2.��������� Whatisthemaximumpower,inwatts,thatthecoffeemakercanconsumewithoutcausingthecircuitbreakerinthebasementtotripwhenboththecoffeemakerandthetoasterareoperatingsimultaneously?Note:Power=voltagexcurrent(thatis,watts=volts×amperes).

Signal must reach

at least this far.

Neighborhood

1 km

1 km

watts

watts

MATHCOUNTS 2010-201144

Plug-inhybridelectricvehicles(PHEVs)combineoneormoreelectricmotorsandagasordieselengineforpropulsion.Energyisstoredinabatterypackfortheelectricmotorandinthegastankforthecombustionengine.Thecombinationallowsshorttripstobemadeentirelyonelectricpowerandraisestheeffectivefuelefficiencyofthecar.ThebatteryinatypicalPHEVstores9000watt-hours(W-h)ofenergy.Someadditionalinformationyouwillneed:Ahousehasstandard120-voltelectricservice.power=voltage×current(thatis,watts=volts×amperes)energy=power×time(thatis,watt-hours=watts×hours)Astandardwalloutletwitha15-amperecircuitbreakercansupply12amperesforcontinuouscharging,whichavoidstrippingthecircuitbreaker(circuitbreakersareset25%abovethemaximumcontinuouscurrentacircuitisdesignedtocarry).ElectricvehicleengineerscallchargingfromastandardoutletLevel1charging.

3.��������� HowlongwillittaketochargeaPHEVwitha9000-Wh-batteryfromastandardwalloutletinyourhome?Expressyouranswerasadecimaltothenearesthundredth.

Level2chargingusesa240-voltoutletthattypicallysupplies32amperesofcurrent.

4.��������� Howlongwillittaketochargethesame9000-Wh-PHEV(discussedinquestion3)withaLevel2chargingoutlet?Expressyouranswerasadecimaltothenearesthundredth.

Televisionsuseelectricpowerwhileturnedoffandwhileturnedon.ThesmallamountofpowerusedwhenaTVisoffallowstheTVtorespondtoacommandfromaremotecontrol.MostofthepowerusedwhentheTVisturnedongoestowardproducingthelightforthepictureweseeonthescreen.TheaverageTVspends5hoursadayturnedonand19hoursadayturnedoff.A55-inchTVuses197watts/hourwhenitisonand0.4watt/hourwhenitisoff.Thus,a55-inchTVuses992.6wattsperday,whichcanbecalculatedbyaddingtogetherthepowerusedwhiletheTVisonandthepowerusedwhiletheTVisoffeachday.ThepowerusedwhiletheTVisonisfoundbymultiplyingthenumberofhourstheTVisontimesthepowerusedperhourwhileitison.AcorrespondingformulaappliesforthepowerusedwhiletheTVisoff.

5.��������� HowmanywattsperdaydoesasmallTVuseifituses33watts/hourwhileonand0.4watt/hourwhileoff?AssumethattheTVisonfor5hoursandofffor19hours.Expressyouranswerasadecimaltothenearesttenth.

hours

hours

watts

MATHCOUNTS 2010-2011 45

Digital displays (such as TVs and computer monitors) are built according to standard dimensions (width and height) and resolutions (pixels). The width and height dimensions of a display can also be expressed as a ratio called the aspect ratio. The aspect ratio of a display is the ratio of the width of the image to its height, expressed as two numbers separated by a colon. The resolution of a display is the number of distinct pixels (smallest elements of a screen) that can be displayed in each dimension. For example, a display that has resolution of 800 × 600 pixels, or 800 pixels in the width dimension and 600 pixels in the height dimension has an aspect ratio of 4:3. This is because the aspect ratio 800:600 can be reduced to 4:3.

6. ��������� Ifahigh-definitiondigitalTVhasaresolutionof1920×1080pixels,whatisitsaspect ratio? Express your answer in the form a:b, where a and b are positive integersandhavenocommonfactorsotherthan1.

Internetspeedtoahomeis25megabitspersecond(Mbps)peakthroughput.Viewingorusingcertaincommonwebsitestakesupvaryingbandwidthasshowninthelistbelow: (a)ViewingMySpacetakesup1Mbps (b)UsingPandoratakesup5Mbps (c)ViewingFacebooktakesup2Mbps (d)ViewingAmazonOnDemandtakesup24Mbps (e)ViewingYouTube-HDtakesup12Mbps Forexample:ViewingMyspace(1Mbps)andlisteningtoPandora(5Mbps)atthesametimeuses1Mbps+5Mbps=6Mbpsofbandwidthofthe25Mbpsavailable.Thisisbecausebandwidthusage is additive. If a group of websites exeeds the available bandwidth, they cannot all be used simultaneously.

7. ��������� What is the maximum number of distinct websites from the above list that can be simultaneously used at the home?

websites

MATHCOUNTS 2010-201146