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SpeedQuestions(Grade6)

2022MathLeagueInternationalSummerChallenge(Unofficialversion,forreferenceonly)

Note:

ListedbelowarethefourtypesofcoinscurrentlybeingmintedinUnitedStates.

Penny:1￠

Nickel:5￠

Dime:10￠

Quarter:25￠

1.

(-1)100=

-1

1

C)-100

D)100

Answer:B

2.

(x+2)+(2x+4)+(3x+6)+(4x+8)=

A)x+20

B)9x+20

C)10x+20

D)20x+20

Answer:C

3.

(y+2)(y–2)=

A)y2–4

B)y2–2y+4

C)y2+2y+4

D)y2+4Answer:A

4.

123...100

246...200

1200

1100

12

D)100

Answer:C

5.(x2)(x3)(x4)=

x5

x9

x24

x234

Answer:B

6.

x2–10x+24=A)(x–12)(x+2)

B)(x–6)(x–4)

C)(x+12)(x–2)

D)(x+6)(x+4)

Answer:B

7.

Ifn<0andp>0,whichinequalityisalwaystrue?

n>p

-n>p

C)11

n p

D)11

n p

Answer:D

8.

Iamthinkingofanumber.WhenIsubtract2fromthenumber,thentake300%oftheresult,thenumberIgetismyoriginalnumber.Whatismyoriginalnumber?

0

2

3

6

Answer:C

9.

x(x(x(x+1)+1)+1)+1=

A)4x+4

B)x4+2x3+3x2+2x+1

C)x4+4

D)x4+x3+x2+x+1

Answer:D

10.

Howmanysolutionsdoestheequation|x+2|=2xhave?

0

1

2

3Answer:B

11.

Howmanydifferentnumberssatisfytheequation(x–2)2(x–1)3(x+1)4(x+2)5=0?

4

5

14

D)120

Answer:A

12.

TrianglesABCandDCBarebothisoscelesrighttriangles,asshown.IfAC=12,whatistheareaoftriangleBCE?

Answer:18

13.

Inacertaincircle,thelengthofaradiusisequaltothereciprocalofthelengthofadiameter.Whatistheareaofthiscircle?

12

π

2π

2

Answer:A

14.

Thesumof20positivenumbersiswhatpercentoftheaverageofthesesame20numbers?

Note:Thisisthewaytoenteryouranswer.If,forexample,youransweris5%,pleaseenter5.

Answer:2000

15.

Whatarethethreeorderedpairsofnon-negativeintegers(a,b)forwhich

ab

ab?

Note:Thisisthewaytoenteryouranswer.Ifyourthreeorderedpairsare(a1,b1),(a2,b2),and(a3,b3),pleaseenterthevalueofa1+b1+a2+b2+a3+b3.

Answer:2

16.

Ihikedintothewoodsat4km/handreturnedbythesamerouteat3km/h.Ifthe

roundtriptook31

2

Answer:6

hours,howmanykmintothewoodsdidIhike?

17.

Howmanytwo-digitwholenumberscontainno0andnomorethanone1?Answer:80

18.

AllbuttwoofmymathbooksaretitledAlgebra,allbuttwoaretitledTrigonometry,andallbuttwoaretitledGeometry.IhaveatleastonemathbooktitledAlgebra.HowmanymathbooksdoIhave?

Answer:3

19.

IntheentirehistoryofOz,eachpresidenthasservedexactlyone4-yearterm.WhatisthemaximumnumberofpresidentsthatcouldhaveheldofficeinOzduringsome10-yearperiod?

Answer:4

20.

Whatareallvaluesofxforwhichxand

17arebothintegers?

x

Note:Thisisthewaytoenteryouranswer.Ifyouranswercontainsthreevaluesofx:x1,x2,andx3,pleaseenterthevalueofx1+x2+x3,similarforothernumbersofvaluesofx.

Answer:0

21.

Whilereviewingtheinventoryofhisbicycleshop,Patnoticedthat,betweenthenewbicyclesandthenewtricycles,therewereatotalof176wheelsand152pedals.HowmanynewtricycleswereinPat’sshop?

Answer:24

22.

Inarighttriangle,thelengthsofthelegsare333and444.Whatisthelengthofthehypotenuse?

Answer:555

23.

AnnandBobtaketurnsinagame.Ataturn,aplayertakesoneortwocoinsfromacupthatatfirstholdssixcoins.Theplayertakingthelastcoinwins.IfAnngoesfirstandbothplayperfectly,whowillwin?[Note:Write“1”iftheanswerisAnn.Write“2”iftheanswerisBob.]

Answer:2

24.

Allrealvaluesofxwhichsatisfyx+|x|=0areintheintervalx≤p.Whatisthevalueofp?

Answer:0

25.

Simultaneously,twoparticlesstartatpointQ,oncircleO,andmoveindifferentdirectionsaroundthecircle.Ifthespeedofoneparticleis4timesthatoftheother,thentheparticleswillnextmeetatpointPonthecircle.If∠POQisacute,whatism∠POQ,indegrees?

Answer:72

26.

Ipayfora$1burgerwith48coinsandgetnochange.WhatisthelargestnumberofnickelsIcoulduse?

Answer:13

27.

Whatistheonlypositivevalueofbforwhich(x+b)2=x2+2bx+b3isanidentityforallrealvaluesofx?

Answer:1

28.

Express,asafractioninlowestterms,thevalueofthefollowingproductof99factors:

111111...1

1...1

1.

2

3

4

n1

100

Answer:

1

100

29.

Figurebelow,thebluecircleistheincircleoftriangleABC.(Theincircleisthecirclewhichliesinsideatriangleandtouchesallsides.)Oisthecentertheincircle.DisthecenterofthecirclewhichtouchesACandtheextensionsofBAandBC.EisthecenterofthecirclewhichtouchesABandtheextensionsofCAandCB.FisthecenterofthecirclewhichtouchesBCandtheextensionsofABandAC.IsittruethatEisalwaystheorthocenteroftriangleODFforanytriangleABC?(Analtitudeofatriangleisastraightlinethroughavertexandperpendiculartotheoppositeside.Thethreealtitudesofanytriangleintersectinasinglepoint,calledtheorthocenterofthetriangle.)

Choices:

Yes

NoAnswer:(a)

30.

Whenall8positiveintegralfactorsof30aremultipliedtogether,theproductis30k.Whatisthevalueofk?

Answer:4

31.

InsquareABCD,drawnbelow,EB=3andFC=4.If AC andEF bisecteachother,whatistheareaofsquareABCD?

Answer:49

32.

Smalltownhasonlytwopeoplewhohavethesame(two-letter)initials.WhatisthelargestpopulationthatSmalltowncouldpossiblyhave?

Note:

Initials:thefirstletterofeachwordinafullname.Forexample,DWistheinitialsofDavidWarren.

Answer:677

33.

Theaverageof10numbersis63.Theaverageof6ofthesenumbersis57.Whatistheaverageoftheother4?

Answer:72

34.

x

If2

x2

2

0,whatisthevalueofx6?

Answer:64

35.

Winkinsaid“We’reallliars,”Blinkinreplied,“Onlyyouarealiar,”andFinkinsaid“You’rebothliars.”Ifeachofthemalwaystellsthetruthoralwayslies,howmanyofthethreeareliars?

Answer:2

36.

Toexpress10asasumofdifferentpowersof2,wewouldwrite10=23+21.Thesumoftheexponentsofthesepowersis3+1,or4.If100wereexpressedasasumofdifferentpowersof2,whatwouldbethesumoftheexponentsofthesepowers?

Answer:13

37.

If,forallvaluesofx,(x–a)2=(x+a)2,whatisthevalueofa?Answer:0

38.

Ifxisreal,andx64=64,whatisthevalueofx32?Answer:8

39.

ThepopulationofthetownofDecrease,now11800,isdecreasingattherateof120peopleeachweek.ThepopulationofthetownofIncrease,now2200,isincreasingattherateof80peopleeachweek.Inhowmanyweekswillthesetwotownshaveequalpopulations?

Answer:48

40.

Whatarebothvaluesofxthatsatisfyinascendingorder.)

Answer:-1,1

x

12

x

4?(Note:Pleaseenteryouranswers

41.

Thenumber4938271603950617283willbeaperfectsquareifoneofitsdigitsisincreasedby1(andalltheotherdigitsareleftunchanged).Whatisthevalueofthisdigitbeforetheincrease?

Answer:3

42.

Whatistheonlyorderedpairofpositiveintegers(x,y)forwhichx2–y2=7?(Note:Pleaseenterthevalueofxfirst.)

Answer:(4,3)

43.

Twoshadedsquaresaredrawninsidearectangle,asshown.Ifeachsquarehasanareaof64,whatistheareaoftherectangle?

Answer:256

44.

AlandPateachtakecoinsfromapiggybanktobuysomegum,butAlis10￠shortandPatis15￠shortofthepriceofthegum.Iftheycombinetheirresources,theystillcan’taffordtobuythegum.Whatisthegreatestpossible(integral)priceofthegum,incents?

Answer:24

45.

InanAlmostPythagoreantriangle,thesquareofthelongestsideis1morethanthesumofthesquaresoftheothertwosides.Ifallofitssideshaveintegrallengths,whatistheleastperimeterofsuchatriangle?

Answer:7

46.

Whatisthesumofallintegersxforwhich

9equalsaninteger?

x

Answer:0

47.

Althoughfourstudentstriedtofindthesumofthefirst21positiveprimes,onlyonegotthecorrectanswer.Patgot709,Leegot711,Sandygot712,andDalegot713.Whogotthecorrectanswer?[Note:Enter“1”ifPatdid;enter“2”ifLeedid;enter“3”ifSandydid;enter“4”ifDaledid.]

Answer:3

48.

Forwhichn>0does(32+42)+(32+42)+(32+42)+(32+42)=n2?

Answer:10

49.

Whatisthelargestpossibleareaofarectanglewithintegersidesandperimeter22?Answer:30

50.

Thesumofthesquaresofthelengthsofthefourlegsoftworighttrianglesis100.Ifthelengthofthehypotenuseofoneofthetrianglesis6,howlongisthehypotenuseoftheothertriangle?

Answer:8

51.

Thevaluesofxwhichsatisfyboth|x–8|<6and|x–3|>5arepreciselythevaluesofxthatsatisfya<x<b.Whatisa+b?

Answer:22

52.

Disregardingorder,wecanwrite4asasumofoneormoreintegers,eachapowerof2,inonlyfourways:4,2+2,2+1+1,and1+1+1+1.Inatmosthowmanydifferentwayscan5bewrittenassuchasum?

Answer:4

53.

Aplaneispartitionedinto2regionsby1lineandinto4regionsby2intersectinglines.Intohowmanydisjointregionsdo5coplanarlinespartitiontheplane,ifno2ofthelinesareparallelandno3ofthemareconcurrent?

Answer:16

54.

Twononcongruentcirclesareexternallytangent.Eachbaseofanisoscelestrapezoidisadiameterofoneofthecircles.Ifthedistancebetweenthecentersofthecirclesis

8,whatistheareaofthetrapezoid?Answer:64

55.

The4thtermofasequenceis4andthe6thtermis6.Everytermafterthe2ndisthesumofthe2precedingterms.Whatisthe8thtermofthissequence?

Answer:14

56.

Onediagonalofasquareservesastheshorterbaseofatrapezoid,andalinethroughoneoftheverticesof