2013 MATHCOUNTS School Sprint Round

2013School Competition

Sprint RoundProblems 1–30

Copyright MATHCOUNTS, Inc. 2012. All rights reserved.

MATHCOUNTS

®

Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation

Name

DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.

This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You are not allowed to use calculators, books or other aids during this round. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the left-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers.

In each written round of the competition, the required unit for the answer is included in the answer blank. The plural form of the unit is always used, even if the answer appears to require the singular form of the unit. The unit provided in the answer blank is the only form of the answer that will be accepted.

Total Correct Scorer’s Initials

National Sponsors

Raytheon Company * Northrop Grumman Foundation * U.S. Department of Defense * National Society of Professional Engineers * CNA Foundation * ConocoPhillips * Texas Instruments Incorporated * 3M Foundation * Art of Problem Solving * NextThought

Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 School Sprint Round

1. _____________

2. _____________

3. _____________

4. _____________

5. _____________

Whatisthevalueof2+3×4–5?

Barb,Dianna,Tina,PatandKeradecidetogotolunchtocelebrateKera’sbirthday.Theyagreetosplitthebillequally.Withtaxandtip,thebillis$71.20.However,atthelastminute,Kera’sfriendsdecidetopayhershareandsplitthebillevenlyamongthefourofthem.WhattotalamountdoeseachofKera'sfriendspay?

InrighttriangleABCwhatisthearithmeticmean(average),indegrees,ofthemeasuresofthetwosmallestangles?

An8-inchby10-inchphotoisplacedinsideaframewithinnerdimensionsof10inchesby12inches.Whatfractionoftheinteriorareaisoccupiedbythephoto?Expressyouranswerasacommonfraction.

Cleo’srecipeforadozencupcakescallsfor1cupofsugar.Howmanycupsofsugarareneededtomake30cupcakes?Expressyouranswerasamixednumber.

degrees

$

cups


If4birdseat4wormsin4minutes,howmanyminuteswillittake32birdstoeat32worms?

Gretchenistryingtorowherboat20milesupariverfromthestartinglinetothefinishlineofarace.Gretchenstartsouteachmorningat7a.m.androws10milesupstream.Eachnightshefloats5milesdownstream.HowmanydayswillittakeforGretchentogettothefinishline?

Whatnumber,whenaddedto1andthendividedby7,hasaresultof2?

Onehundredpeoplewereaskedwhichflavoroficecreamtheypreferoutofchocolate,strawberryandvanilla.Basedontheresultsshown,howmanymorepeoplesurveryedpreferchocolatethanpreferstrawberry?

Amachineneedstobeprogrammedtolayasingle,continuouslineofgluecoveringallofthesegmentsofthepatternshown.Thelettersidentifythepointswherethegluepathcanchangedirection.ThereisonlyoneendingpointforanypaththatstartsatG,coversallsegments,andneverretracesanysegment.Atwhatpointwilleverysuchpathend?

6. _____________

7. _____________

8. _____________

9. _____________

10._____________

days

minutes

45%25%

30%

ChocolateStrawberry

Vanilla

Preferred Ice Cream Flavorpeople

Point A B C

DEF

GH

I


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12._____________

13._____________

14._____________

15._____________

Whatcommonfractionisequivalentto0.6%?

Attheannualmusicfestival,thenumberofBayviewMiddleSchoolstudentscompetingintheband,chorusandorchestracompetitionsis14,13and17students,respectively.Fourstudentswillcompeteinallthreecompetitions.Fivestudentswillcompeteinthebandandchoruscompetitions.Sixstudentswillcompeteinthebandandorchestracompetitions.Sevenstudentswillcompeteinthechorusandorchestracompetitions.HowmanystudentsfromBayviewMiddleSchoolwillcompeteinthismusicfestival?

Marielikestojumpupsteps.Shecanjumpuponestepatatime,twostepsatatime,oracombinationofboth.Howmanydifferentwayscanshejumpupasetoffivesteps?

TheWritingEmporiumsellspencilsandpensindividually.Alanabuysapencilandtwopensfor$1.10.Jeffreybuystwopencilsandapenfor$1.00.HowmuchdoesEdwardspendtobuysixpencilsandsixpensfromtheWritingEmporium?

UsingtheMayancalendar,auinalis20days,andatunis18uinals.Howmanydaysarein2tuns?

days

students

ways

$

Band Chorus

Orchestra


16._____________

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20._____________

ThestudentbodyatCentralMiddleSchoolconsistsof303eighthgraders,312seventhgradersand285sixthgraders.TenpercentofthestudentsrespondedtoapollthataskedwhenFieldDayshouldtakeplace.Ofthosewhoresponded,30%wereinfavorofhavingFieldDaytakeplaceonthelastdayofschool.HowmanystudentswhorespondedtothepollwerenotinfavorofFieldDaytakingplaceonthelastdayofschool?

LastseasontheDescartesMiddleSchoolfootballteamscored36times,earningatotalof212points.Theteamscoredonlyfieldgoals,eachworth3points,andtouchdowns,eachworth7points(includingallextrapoints).Howmanyfieldgoalsdidtheteamscorelastseason?

Fatimaplanstoresumehersummerjobatthelocalsupermarket,whichrequirestheuseofherpersonalvehicletodelivergroceries.Includingtips,Fatimaearnsanaverageof$9.00perdelivery,andsheanticipatesspending$298eachmonthongasandvehiclemaintenance.IfFatimawillworkforthreemonths,whatistheminimumnumberofdeliveriesshemustmaketoearn$3286fortuitionandcoverhergasandvehiclemaintenanceexpenses?

Whatexpressionrepresents5x(1+7y)+x(y–5)whenitiswritteninsimplestformintermsofxandy?

Estefanbringstwobagsofsandwichestoaparty.Onebagcontains6peanutbuttersandwiches,4cheesesandwichesand2hamsandwiches.Theotherbagcontains4peanutbuttersandwiches,2cheesesandwiches,4hamsandwichesand2chickensaladsandwiches.IfJohnpicksonesandwichfromeachbagatrandom,whatistheprobabilitybothwillbepeanutbuttersandwiches?Expressyouranswerasacommonfraction.

students

deliveries

fieldgoals


21._____________

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25._____________

CirclePistangenttocircleOatpointAandtodiameterBCatpointO,asshown.Withlength4units,OA isadiameterofcirclePandaradiusofcircleO.Whatistheareaoftheshadedregion?Expressyouranswerintermsofπ.

Timpurchasedapairofrunningshoesfor$79.20,whichincludeda10%salestax.Iftheshoeswereonsalefor40%offtheoriginalprice,whatwastheoriginalpriceoftherunningshoes?

Abankrobberhopsintohisgetawaycarandtravelsatanaveragespeedof60mi/hintothecountryside.Exactlytwominutesaftertherobberleavesthebank,apoliceofficerfollowshimfromthebankalongthesameroute.Iftheofficeristravelingatanaveragespeedof75mi/h,howfarfromthebankwilltherobberbeintercepted?

Ofthefivecoplanarlinesshown,onlymandnareparallel.Whatisthedegreemeasureof∠1?

PaulisthreetimesasoldasManny.AlexistwiceasoldasManny.Timisone-fourththesumofPaul’sandAlex’sages.ThesumoftheagesofAlex,TimandPaulis75.HowoldisManny?

miles

yearsold

60˚

50˚

1

m

n

units2A

B CO

P

$

degrees


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30._____________

If2n–2(3m–4)=12andn–6=5m,whatisthenumericalvalueof3m+n?

Duringhismorningcommutetoworkinrushhourtraffic,Justin’saveragespeedwas30mi/h.Duringhisafternooncommutebackhomealongthesameroute,hisaveragespeedwas60mi/h.Whatwashisaveragespeedfortheentireroundtrip?

Abagcontains20redmarblesand12bluemarbles.Whatistheprobabilitythattwomarblesdrawnatrandom,withoutreplacement,willbethesamecolor?Expressyouranswerasacommonfraction.

ApointPislocatedintheinteriorofrectangleABCDsuchthatPB=25cm.ApointQislocatedonAB suchthatPQ isperpendiculartoAB .Ifalllengthsaremeasuredincentimeters,AQ=PQ,PQ hasthesmallestintegerlengthpossibleandBQ isofintegerlength,whatisthelengthofAB ?

Aregularhexagonwitharea54 3 in2isinscribedinacircle.Whatisthecircumferenceofthecircle?Expressyouranswerintermsofπ.

mi/h

A B

CD

Q

P

cm

in

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2013 MATHCOUNTS School Sprint Round