Download Find the sum of the number of faces, vertices, and edges of a

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