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2011 – 2012 Log1 Contest Round 3 Theta Individual Name: __________________ 4 points each 1 Findthemedianandmeanforthefollowingsetofnumbers: 8,10,16,3,8,6,12,4,20,13,11,14,5 2 Fred,inNewYorkcheckedhisthermometerandhesawthatitwas45° Fahrenheit. Hecallshisfriend,Sergei,inRussiaandtellshimthatitwas45°.Sergeithoughtwe meant45°Celsius.IndegreesFahrenheit,whatis45°Celsius.Thereislinear 32 and 100 212 .Unitis relationshipbetweenFahrenheitandCelsius:0 optional. 3 Findthesumofthenumberoffaces,vertices,andedgesofapyramidwhose baseisintheshapeofaregularpentagon 4 Theseventhelementofanarithmeticsequenceis22andtheeleventh elementis36.Whatisthefirstelement? 5 Whatisthesum1 4 9 16 ⋯ 100? Median: _____ Mean: _____ 5 points each 6 Solveforx:512 7 Ifadigitalclockstartsatmidnight,12:00,andlosesone secondeveryhour,what timewilltheclockreadinthreeyears?Assumethereare365daysinayear. 8 Simplifytheexpression. 9 10 Ifa@bmeans 8 ,evaluate6@2,where means“a itemschooseb”. Howmanycombinationsof13lightbulbslinedupinastraightrowwilltherebeif youhave4red,4green,3yellowand2blue? 6 points each 11 Findthecoefficientofthex6terminthebinomialexpansionof5x 3x‐2 7. 12 Abagcontains2blue,3redand5greenballs.Billwantstoknowwhatthe probabilitywillbeofpickingablueballonhisseconddraw.Previouslydrawnballs arenotreturnedtothebag. 13 9pointsarearrangedina3rowby3columnpattern.Whatistheprobabilityof randomlyselectingthreedistinctpointsthatformatriangle? 14 3 1 0 4 2 5 , find the value of A22, the entry in the 2 9 6 8 4 7 second row and second column of A. 15 Howmanypositivefactorsdoestheexpression 20 11 have? 2011 – 2012 Log1 Contest Round 3 Alpha Individual Name: __________________ 4 points each 1 Findthemedianandmeanforthefollowingsetofnumbers: 8,10,16,3,8,6,12,4,20,13,11,14,5 2 Fred,inNewYorkcheckedhisthermometerandhesawthatitwas45° Fahrenheit. Hecallshisfriend,Sergei,inRussiaandtellshimthatitwas45°.Sergeithoughtwe meant45°Celsius.IndegreesFahrenheit,whatis45°Celsius.Thereislinear relationshipbetweenFahrenheitandCelsius:0 32 and 100 212 .Unitis optional. 3 Theseventhelementofanarithmeticsequenceis22andtheeleventh elementis36.Whatisthefirstelement? 4 Solveforx:512 5 Ifadigitalclockstartsatmidnight,12:00,andlosesone secondeveryhour,what timewilltheclockreadinthreeyears?Assumethereare365daysinayear. 8 Median: _____ Mean: _____ 5 points each 6 Simplifytheexpression. 7 Thesymbol meansthenumberofwaysonecanchoosey itemsfromx 2 3 4 6 5 distinguishableitems.Whatis ? 0 4 1 2 3 Howmanycombinationsof13lightbulbslinedupinastraightrowwilltherebeif youhave4red,4green,3yellowand2blue? 9 Abagcontains2blue,3redand5greenballs.Billwantstoknowwhatthe probabilitywillbeofpickingablueballonhisseconddraw.Previouslydrawnballs arenotreturnedtothebag. 10 DaveandMollyareplayingagamewheretheytaketurnsspinningaspinner.The spinnerhasaone‐thirdchanceofcomingup“WIN”andtwo‐thirdschanceof“PASS”. Davegoesfirstandtheytaketurnsuntiloneofthemspins“WIN”.Whatisthe probabilitythatDavewins? 8 6 points each 11 3 1 0 4 2 5 , find the value of A22, the entry in the 2 9 6 8 4 7 second row and second column of A. 12 Howmanypositivefactorsdoestheexpression 20 11 have? 13 Findthenumberofcommonprimefactorsof2002and1729. 14 Convertthisequationfrompolartorectangularform. 15 Evaluatecos 75° sin 30° 4 sin 2011 – 2012 Log1 Contest Round 3 Mu Individual Name: __________________ 4 points each 1 Fred,inNewYorkcheckedhisthermometerandhesawthatitwas45° Fahrenheit. Hecallshisfriend,Sergei,inRussiaandtellshimthatitwas45°.Sergeithoughtwe meant45°Celsius.IndegreesFahrenheit,whatis45°Celsius.Thereislinear relationshipbetweenFahrenheitandCelsius:0 32 and 100 212 .Unitis optional. 2 Theseventhelementofanarithmeticsequenceis22andtheeleventh elementis36.Whatisthefirstelement? 3 Solveforx:512 4 Ifadigitalclockstartsatmidnight,12:00,andlosesone secondeveryhour,what timewilltheclockreadinthreeyears?Assumethereare365daysinayear. 5 Simplifytheexpression. 8 5 points each 6 Howmanycombinationsof13lightbulbslinedupinastraightrowwilltherebeif youhave4red,4green,3yellowand2blue? 7 Findthecoefficientofthex6terminthebinomialexpansionof5x 3x‐2 7. 8 9pointsarearrangedina3rowby3columnpattern.Whatistheprobabilityof randomlyselectingthreedistinctpointsthatformatriangle? 9 Thereare5coinsinabox.2havea40%chanceoflandingonheadsandtheother3 arefaircoins.Ifyoupicktwoatrandomandflipthem,whatistheprobabilitythat theywillbothbetails?Expressasapercentage. 10 Howmanypositivefactorsdoestheexpression 20 11 have? 6 points each 11 Whatisthesmallestpositivethree‐digitnumberthathasaremainderof2when dividedby3,aremainderof4whendividedby5andaremainderof6whendivided by7? 12 Convertthisequationfrompolartorectangularform. 4 sin 13 Evaluatecos 75° 14 Giventhefunction7 5. 15 Calculatethevolumeofanobjectthatisproducedwhenthefunction 3 2onthedomainx 0,4 isrevolvedaroundthex‐axis. sin 30° 2,determinethevalueof when 2011 – 2012 Log1 Contest Round 3 Theta Individual Name: __________________ 4 points each 1 Findthemedianandmeanforthefollowingsetofnumbers: 8,10,16,3,8,6,12,4,20,13,11,14,5 2 Fred,inNewYorkcheckedhisthermometerandhesawthatitwas45° Fahrenheit. Hecallshisfriend,Sergei,inRussiaandtellshimthatitwas45°.Sergeithoughtwe meant45°Celsius.IndegreesFahrenheit,whatis45°Celsius.Thereislinear relationshipbetweenFahrenheitandCelsius:0 32 and 100 212 .Unitis optional. 113 3 Findthesumofthenumberoffaces,vertices,andedgesofapyramidwhose baseisintheshapeofaregularpentagon 22 4 Theseventhelementofanarithmeticsequenceis22andtheeleventh elementis36.Whatisthefirstelement? 1 5 Whatisthesum1 4 9 16 ⋯ Median: __10_ Mean: __10_ 100? 385 5 points each 6,‐3 6 Solveforx:512 7 Ifadigitalclockstartsatmidnight,12:00,andlosesone secondeveryhour,what timewilltheclockreadinthreeyears?Assumethereare365daysinayear. 8 Simplifytheexpression. 9 10 Ifa@bmeans 8 4:42 5 34 7 18 ,evaluate6@2. Howmanycombinationsof13lightbulbslinedupinastraightrowwilltherebeif youhave4red,4green,3yellowand2blue? 900,900 6 points each 11 Findthecoefficientofthex6terminthebinomialexpansionof5x 12 Abagcontains2blue,3redand5greenballs.Billwantstoknowwhatthe probabilitywillbeofpickingablueballonhisseconddraw.Previouslydrawnballs arenotreturnedtothebag. 1 5 13 9pointsarearrangedina3rowby3columnpattern.Whatistheprobabilityof randomlyselectingthreedistinctpointsthatformatriangle? 19 21 14 3 1 0 4 2 5 , find the value of A22, the entry in the 2 9 6 8 4 7 second row and second column of A. A22 ‐87 15 Howmanypositivefactorsdoestheexpression 20 11 have? 3x‐2 7. 102060 45 2011 – 2012 Log1 Contest Round 3 Alpha Individual Name: __________________ 4 points each 1 Findthemedianandmeanforthefollowingsetofnumbers: 8,10,16,3,8,6,12,4,20,13,11,14,5 2 Fred,inNewYorkcheckedhisthermometerandhesawthatitwas45° Fahrenheit. Hecallshisfriend,Sergei,inRussiaandtellshimthatitwas45°.Sergeithoughtwe meant45°Celsius.IndegreesFahrenheit,whatis45°Celsius.Thereislinear relationshipbetweenFahrenheitandCelsius:0 32 and 100 212 .Unitis optional. 3 Theseventhelementofanarithmeticsequenceis22andtheeleventh elementis36.Whatisthefirstelement? 4 Solveforx:512 5 Ifadigitalclockstartsatmidnight,12:00,andlosesone secondeveryhour,what timewilltheclockreadinthreeyears?Assumethereare365daysinayear. 8 Median: __10_ Mean: __10_ 113 1 6,‐3 4:42 5 points each 6 Simplifytheexpression. 7 Thesymbol 8 meansthenumberofwaysonecanchoosey itemsfromx 2 3 4 6 5 distinguishableitems.Whatis ? 0 2 4 1 3 Howmanycombinationsof13lightbulbslinedupinastraightrowwilltherebeif youhave4red,4green,3yellowand2blue? 5 34 35 900,900 9 Abagcontains2blue,3redand5greenballs.Billwantstoknowwhatthe probabilitywillbeofpickingablueballonhisseconddraw.Previouslydrawnballs arenotreturnedtothebag. 1 5 10 DaveandMollyareplayingagamewheretheytaketurnsspinningaspinner.The spinnerhasaone‐thirdchanceofcomingup“WIN”andtwo‐thirdschanceof“PASS”. Davegoesfirstandtheytaketurnsuntiloneofthemspins“WIN”.Whatisthe probabilitythatDavewins? 3 5 6 points each 11 3 1 0 4 2 5 , find the value of A22, the entry in the 2 9 6 8 4 7 second row and second column of A. 12 Howmanypositivefactorsdoestheexpression 20 11 have? 45 13 Findthenumberofcommonprimefactorsof2002and1729. 2 14 Convertthisequationfrompolartorectangularform. 15 Evaluatecos 75° sin 30° 4 sin A22 ‐87 4 orequivalent √ or equivalent 2011 – 2012 Log1 Contest Round 3 Mu Individual Name: __________________ 4 points each 1 Fred,inNewYorkcheckedhisthermometerandhesawthatitwas45° Fahrenheit. Hecallshisfriend,Sergei,inRussiaandtellshimthatitwas45°.Sergeithoughtwe meant45°Celsius.IndegreesFahrenheit,whatis45°Celsius.Thereislinear relationshipbetweenFahrenheitandCelsius:0 32 and 100 212 .Unitis optional. 2 Theseventhelementofanarithmeticsequenceis22andtheeleventh elementis36.Whatisthefirstelement? 3 Solveforx:512 4 Ifadigitalclockstartsatmidnight,12:00,andlosesone secondeveryhour,what timewilltheclockreadinthreeyears?Assumethereare365daysinayear. 5 Simplifytheexpression. 8 113 1 6,‐3 4:42 5 34 5 points each 6 Howmanycombinationsof13lightbulbslinedupinastraightrowwilltherebeif youhave4red,4green,3yellowand2blue? 900,900 7 Findthecoefficientofthex6terminthebinomialexpansionof5x 3x‐2 7. 102060 8 9pointsarearrangedina3rowby3columnpattern.Whatistheprobabilityof randomlyselectingthreedistinctpointsthatformatriangle? 9 Thereare5coinsinabox.2havea40%chanceoflandingonheadsandtheother3 arefaircoins.Ifyoupicktwoatrandomandflipthem,whatistheprobabilitythat theywillbothbetails?Expressasapercentage. 10 Howmanypositivefactorsdoestheexpression 20 11 have? 19/21 29.1% 45 6 points each 11 Whatisthesmallestpositivethree‐digitnumberthathasaremainderof2when dividedby3,aremainderof4whendividedby5andaremainderof6whendivided by7? 12 Convertthisequationfrompolartorectangularform. 4 sin 13 Evaluatecos 75° 104 4 or equivalent √ sin 30° orequivalent 14 Giventhefunction7 15 Calculatethevolumeofanobjectthatisproducedwhenthefunction 3 2onthedomainx 0,4 isrevolvedaroundthex‐axis. 2,determinethevalueof when 5. 1 10 14672 15 2011 – 2012 Log1 Contest Round 3 Individual Solutions Mu Al 1 Th 1 Solution Median:10 Mean:130/13 10 3,4,5,6,8,8,10,11,12,13,14,16,20 1 2 2 ConvertCelsiustoFahrenheit 45 32 32 3 Thispyramidhas6vertices.Eachedgeonthebasepentagoncorrespondstoexactly onelateralfaceonthepyramid.Thusthereare6facesonthispyramid includingthe base .Eachvertexonthebasepentagoncorrespondstoexactlyonelateraledgeon thepyramid.Thusthereare5lateraledgesonthepyramidand5edgesonthebase pentagon.Thesumis22. 45 32 113 2 3 4 Thedifferencebetweenthe11th and7th elementsis4timesthecommondifference. The7thand1stelementsdifferby6timesthecommondifference.The1stelementis then22– 6/4 36‐22 22‐21 1 5 Theformulaisn n 1 2n 1 /6 10 11 21/6 385. 6 8 2 512 9 5 3 3 3 5 3 3 15 3 0 3 18 0 6 3 6, 3 7 Thereare3*365*24 26280hoursinexactlythreeyear.Thus,26280secondsare lost.Thisisequivalentto438minutesor7hours,18minutes.Exactlythreeyears later,insteadofreadingmidnight,theclockwillbeslowbythisamount,equivalentto 4:42. 3 4 4 5 2 Knowingthat 1 1 1 1 1 1 1 1 1 ⋯ 2 5 3 5 8 3 14 17 3 Factoringoutthecommon1/3,onlythefirstandlasttermsremain.Allthemiddle termscancelout. Therefore, 5 6 8 9 Evaluate. Onecanevaluateeachcombinationseparatelybutthereisaformulasometimes 7 referredtoas“hockey‐stick”whichhasthissumequal 35.Inotherwords, 4 1 ∑ 7 6 8 10 13!/ 4!4!3!2! 900,900 7 11 9 12 8 13 9 10 7 243 4 21 5 20412.Thisnumberisthenmultipliedby5toattainthefinalresultof102060 Forthefactor 3x‐2 7,thecoefficientofthex5 termis 3 1 2 1 2 2 1 18 1 10 9 90 5 9 Thereare 84waystochoose3pointsoutofa3x3lattice.Ofthese84points,the 3 onlypossiblecombinationsthatCANNOTformatrianglearethosethatarecollinear. Thesewouldbethe3columns,3rowsand2diagonalsetsofpoints.Thusthereare76 possiblecombinationsthatcanformatriangle.Theprobabilityis 8 10 1 Createachoicetable: H 50 50 50 40 40 T 50 50 50 60 60 Definethefollowingprobabilities: P TT probabilityofthrowingtwotailsregardlessofthetypeofcoin. P T50T50 probabilityofdrawing2regularcoinsANDthrowingtwotails. 3 2 1 1 3 5 4 2 2 40 P T60T60 probabilityofdrawing2biasedcoinsANDthrowingtwotails. 2 1 3 3 9 5 4 5 5 250 P T50T60 probabilityofdrawingabiasedandaregularcoinANDthrowingtwotails. Thereare6outof10possiblewaystodrawabiasedandaregular coin. P TT P T50T50 P T60T60 P T50T60 P TT 11 14 2 9 Inthefirstturn,Davehasa1/3chanceofwinning.InorderforMollytowin,Dave mustpassandthenMolly“win”withprobability 2/3 1/3 2/9.Thereisa5/9 chanceofsomeonewinninginthefirstroundand4/9chancethatitgoestoround two.OnecantreatthisasaninfinitesequenceornotethatDave’sprobabilityof ,sothedesiredchanceis3/5. winningis 2 3 1 0 4 2 9 6 8 3 0 1 6 3 4 2 0 9 6 2 4 6 20 54 80 1 8 9 8 6 20 8 15 2 5 Evaluating 54 80 50 87 4 7 Forthisproblem,onereallyonlyneeds: 7 2 4 Intermsofprimefactors 20 11 4 1 2 1 2 1 45 9 8 87 2 5 11 .Therefore,thetotalnumberoffactorsis 10 12 15 13 OnecanfindthegreatestcommondivisorusingtheEulermethod.2002 1729 1 273,1729 273 6 91and273 91 3 .ThismeanstheGCF 91withprimefactorsof 7and13. 11 If1isaddedtothenumber,itisdivisibleby3,5and7or3*5*7 105.Thereforethe numberis105‐1 104. 12 14 Since and 4 sin 4 . 4 Thus, Otherpossibilities 2 4or sin .Theequation 4 4 sin ,becomes 0.Itisacircleofradius2centeredat 0,2 . Theanswerwilldependonwhetherasumformulaorhalfangleformulaisusedto evaluatecos 75 . I.cos 75 cos 45 √ √ 30 √ √ √ sotheansweris 13 15 II.LetA cos 75 A sin 15 √3 2 4 1 2 7 14 7 cos 30 2 2 √3 8 7 2 0 7 14 7 14 7 14 5 1 10 Usingthediscmethod, 15 3 2 6 13 12 4 1 3 13 4 6 4 0 5 2 3 1024 768 832 192 32 2 3 2 2 5 6144 11520 8320 2880 480 30 29344 30 14672 15 √ , 1 sin 30 Implicitlydifferentiating, 14 √ √ sin 30 Therefore, 7 √ 1 2 1 2 √3 2 2 1 1 2 √3 2