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MATHCOUNTS® and Mathlete® are registered trademarks of the MATHCOUNTS Foundation.
2010–2011
School Handbook:Volume II
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 200 creative math problemsthat meet NCTM standards for grades 6-8.
MATHCOUNTS 2010-2011 9
Warm-Up 71. ��������� Melinda chooses a three-digit positive integer, subtracts it from 3000, and triples
the result. What is the largest integer Melinda can get?
9. ��������� TriangleABChasverticeswithcoordinates(1,0),(5,0),and(3,7).Kadimaccidentallyswitchedthex- and y-coordinatesofeverysinglevertexwhencalculatingtheareaofthetriangle.WhatisthepercentofchangebetweentheareaoftheoriginaltriangleABCandtheareaofthenewtriangleKadimmade?
3. ��������� Achevronisinscribedinasquareofside12yardswherepointXisatthe center of the square and the upper corners of the chevron touch the upper corners of the square. What is the area of the chevron?
8. ��������� Theproductofthreedifferentpositiveintegersis2010.Whatisthemaximumpossible sum of the three integers?
9. ��������� Youhaveasupplyofboxesofvolumes1,3,9,27and81cubicmeters.Giventhattheboxesmustbefilledcompletely,whatistheleastnumberofboxesthatwillholdexactly300cubicmeters of sand?
10. �������� If F (n)=3n–5,G(n)=n 2 + 3n−2andH(n)=15−0.3n,whatisthevalueof× −−
240 square centimeters. Another object is a smaller-scale replica of the original object.Ifthesurfaceareaofthesmallerobjectis60squarecentimeters,whatisits volume?
3. ��������� Tricia has a bag of marbles. She gave one-third of them to Barbara and one-fourth oftheremainingmarblestoSam.Iftherearenow24marblesinthebag,howmanymarbles did Tricia give to Barbara?
4. ��������� Acubehasasidelengthof50inches.Whatisthepositivedifferencebetweenthenumerical values of the surface area of the cube in square inches and the volume of the cube in cubic inches?
7. ��������� ThestudentsatValleyMiddleSchoolheldaWalk-a-Thontoraisemoneyforalocalcharity.Thebargraphshowshowmuchmoneywascollectedbyeachgrade. If the data had been graphed in a pie chart instead,howmanydegreeswouldbeinthecentralangle of the sector representing grade 7?
8. ��������� The mean, median and range are all the same for a particular set of four numbers. What is the largest number in the set if the smallest number is 17?
2. ��������� Thegraphshowstheamountoffuelusedbyfivedifferentvehiclesthateachdrove200milesduring a road test. What is the average miles per gallonforthetest?Expressyouranswerasadecimalto the nearest tenth.
3. ��������� Whatisthelengthofthesideoftheonlyequilateraltrianglewhoseperimeterhasthesamenumericalvalueinunitsasitsareainsquareunits?Expressyouranswerasa decimal to the nearest tenth.
4 . What is theresultifPisdividedbyS?Expressyouranswerasacommonfraction.
5. ��������� Theregionshowniscomposedofarighttriangleandthreesemicircles. What is the area of the entire enclosed region if the trianglehaslegsof6and8units?Expressyouranswerintermsofπ.
7. ��������� What is the product of all integer values of xforwhich −2 9x is a prime number?
8. ��������� ConvexoctagonABCDEFGHhascongruentanglesA,B,EandFandcongruentanglesC,D,GandH.If m∠A=(2x)°,m∠B = y°andm∠C=(x+50)°,whatisthe value x + y?
9. ��������� Asquareofwidth100unitscontainsacircletangenttoallfoursidesofthesquare.Twosmallersquares,congruenttoeachotherandeach containing a circle tangent to all four sides of the square, rest atopthelargesquare,withthethreesquarestogetherformingarectangle.Whatisthedifferencebetweentheareaoftheshadedsemicircleandthecombinedareaofthetwosmallershadedcircles?
10. �������� TwosubsetsofthesetS={a, b, c, d, e}aretobechosensothattheirunionisSandtheirintersectioncontainsexactlythreeelements.EachoftheelementsofSisdistinct.Inhowmanywayscanthisbedone,assumingthattheorderinwhichthesubsets are chosen is irrelevant?
6. ��������� Themeanofthesquaresofthreenon-zeronumbersa, b and c is three times the square of their mean. If b = 2a,whatisthevalueof ca ?Expressyouranswerasacommon fraction.
7. ��������� When creating a particular pattern, the designer beginswithalinesegment1unitlong.Eachsuccessivestageiscreatedbyusingcongruentsegments that are half as long as those in the previousstage.IfyoutracethedesignatStage4,howmanyunitslongerisitthanthelength of the design at Stage 1?
8. ��������� Twodifferentunitsquaresarerandomlyselectedfromthe16unitsquares in the 4 ×4gridshown.Whatistheprobabilitythattheydonothaveavertexincommon?Expressyouranswerasacommonfraction.
9. ��������� The graphs of y = x2–8x–35andy = –2x2+16x+3intersectintwopoints.Whatis the sum of the x-coordinatesofthetwopointsofintersection?
10. �������� Inthefigureshown,AC is the diameter of the circle. If the measureofarcBCis60°,AC BE⊥ ,andBE=12units,whatisthe length of AB?
apples,onebananaandfourcarrotscostatotalof$1.89.Threeapples,twobananasandthreecarrotscostatotalof$1.98.Whatisthetotal cost, in cents, of one apple, one banana and one carrot?
6. ��������� Stella sells 30 gismos a month at a price of $20 per gismo. She notices that each time sheraisesthepriceofagismoby$4,shesells2fewergismospermonth.Whatisthemaximumamountofrevenue,indollars,thatStellacanmakeinamonthofsellinggismos?
9. ��������� A circular garden is placed inside a rectangular patio that measures20feetby14feet.Thecirculargardenistangenttothreesidesoftherectangularpatio,asshown.Whatpercentofthepatioisoutsideofthegarden?Expressyouranswertothenearestwholenumber.Use 22
8. ��������� Runningat12milesperhourburnsabout1200caloriesperhour.Runningat5milesperhourburnsabout450caloriesinanhour.Howmanyminuteslongerdoesonehavetoruntoburn300caloriesat5milesperhourthanat 12 miles per hour?
3. ��������� AlinethroughpointP(4,3)isparalleltothelinex – 2y=8.WhatistheequationofthelinethroughpointP?Expressyouranswerinslope-interceptformwheretheslope is a common fraction.
5. ��������� Eightsophomoresareonastudentcouncilcommittee.Apartyplanning subcommittee must be selected from these eight student councilcommitteemembers.Howmanydifferentpartyplanningsubcommittees are possible if it must have more than 1 but less than5members?
7. ��������� What is the integer closest to +150 75 ?
8. ��������� Arectanglehasbeenpartitionedintoninesquares,asshown.Ifthe area of each of the three medium squares is 100 square units, whatistheareaofthelargestsquare?
4. ��������� The largest angle of a quadrilateral is four times its smallest angle. Another angle is10degreesmorethantwicethesmallest.Thefourthangleis60degreeslessthan 3 times the smallest. What is the measure of the largest angle?
5. ��������� AtMelvilleMiddleSchool73studentsareintheband,65studentsareinthechorusand114studentsplaysports.Thirty-twoarebothinthebandandplaysports,12playsportsandsinginthechorus,butnooneisinboththebandandthechorus.Fivehundredfifty-eightstudentsdonotparticipateinanyoftheseactivities.Howmanystudentsarethere at Melville Middle School?
7. ��������� InisoscelestriangleABC,AB=AC=13ft,andBC=10ft.SquareRSTUisinscribedinthetriangle,withRandSonBC,withTonACandwithUonAB. What isthelengthofasideofthesquare?Expressyouranswerasacommonfraction.
8. ��������� Alinewithslopeequalto1andalinewithslopeequalto2intersectatthepointP(4,6),asshown.PointsQandRareboth on the x-axis.WhatistheareaoftrianglePQR?
4. ��������� Thetablelistspoints(x, y)thatsatisfythelinearrelationy = mx + b. What is the product of m and b ?Expressyouranswerasacommonfraction.x 1 5 9 13y 2 5 8 11
5. ��������� SquareABCDhassidesoflength4units.MisthemidpointofAB ,andKisonCDsothatDK:KC=3:1.DM and AK intersect at J. MC and BK intersect at L. What is the area of quadrilateral JKLM?Expressyouranswerasacommonfraction.
6. ��������� Giventhatn>1,whatisthesmallestpositiveintegernwhosedivisorshaveaproduct of n 8?
4. ��������� Themean,medianandrangeareall6foracollectionoffivepositiveintegers.Whatis the sum of all possible distinct values that could be the greatest number in all the possible collections?
5. ��������� What is the sum of the reciprocals of the roots of 2x 2 + 3x–8=0?Expressyouranswerasacommonfraction.
6. ��������� ThisSudoku-likefigureisa4×4gridtobefilledsothateachofthedigits1,2,3and4appearsineachrowandineachcolumn.The 4 × 4 grid is divided into four 2 × 2 squares. Each of these 2 × 2 squares also is to contain each of the digits 1, 2, 3 and 4. What digit replaces P?
7. ��������� A small square is inscribed in a circle that has another, larger square circumscribed about it. What is the ratio of the area of the small square to the area of the large square?Expressyouranswerasacommonfraction.
8. ��������� Three rectangles have the same area. Their dimensions are mbyn,(m+2)by(n–2)and(m-2)by(n+10).Whatistheareaofeachrectangle?
Workout 91. ��������� Each of the integers 1 through 9 is to be placed in one of the nine
given circles so that the sum, S, of the four numbers along each sideofthetriangleisthesame.Twoofthenumbershavebeenentered. What is the value of S?
2. ��������� IntrapezoidABCD,basesABandCDhavelengths10and6,respectively.LinesegmentPQisdrawnparalleltothebases,withPonAD and Q on BC so that the areaofABQPisthree-fourthsthatofABCD.WhatisthelengthofPQ?Expressyouranswerinsimplestradicalform.
8. ��������� Ifapositivetwo-digitintegerisc times the sum of its digits, the number formed byinterchangingthedigitsisthesumofthedigitsmultipliedbywhatexpressionthat involves c?
10. �������� There are several natural numbers, n, such that B = n 2+2008n and B has a units digit of 4. What is the largest possible digit that can be in the tens place of B?
6
9
units
ways
mm3
numbers
cm2
miles
ordered pairs
MATHCOUNTS 2010-2011 27
Solids Stretch IfyourevolverectangleABCDaboutsideAB,youwillgenerateacylinder.LineABistheaxisofrevolutionforthissolidofrevolution. Forquestions1and2,describethesolidofrevolutiongeneratedbyrevolvingthegivenfigure360°aboutsegment AB.
6. ��������� Considerthisshadedrectanglewithb>a.LetVa denote the volume of the cylinderthatiscreatedwhentheshadedrectangleisrevolvedaroundsidea, and let Vbdenotethevolumeofthecylinderthatiscreatedwhentheshaded rectangle is revolved around side b. What is the value of the ratio
7. ��������� Let Sadenotethetotalsurfaceareaofthecylinderthatiscreatedwhentheshadedrectangleusedinquestion6isrevolvedaroundsidea,andletSb denote the total surface areaofthecylinderthatiscreatedwhentheshadedrectangleisrevolvedaroundsideb.What is the value of the ratio b
a
SS ?Expressyouranswerasacommonfractionintermsof
a and b.
8. ��������� Theseconescanbeformedbyrevolvingarighttriangleaboutitstwolegs.Whatistheratioofthe volume of cone A to the volume of cone B? Expressyouranswerasacommonfraction.
2. ��������� A quadratic equation of the form x2 + kx + m = 0 has solutions 3 2 2x = + and 3 2 2x = − . What is the value of k + m ?
3. ��������� What is the sum of the reciprocals of the solutions of 4x2 − 13x+3=0?Expressyouranswerasacommonfraction.
4. ��������� If r and s are the solutions of 2x2 + 9x+3=0,whatisthevalueofr 2 + s 2? Expressyouranswerasacommonfraction.
5. ��������� If r and s are the solutions of x2+6x −2=0,whatisthevalueofr 3 + s 3?
6. ��������� The solutions of x2 + bx + c=0areeach5morethanthesolutionsof x2 + 7x + 3 = 0. What are the values of b and c ?Expressyouranswerasanorderedpair(b, c ).
7. ��������� A cubic equation of the form x3 + bx2 + cx + d = 0 has solutions x = 3, x = 4 and x=5.Whatarethevaluesofb, c and d ?Expressyouranswerasanorderedtriple(b, c, d ).
8. ��������� What is the sum of the reciprocals of the solutions of x3 − 3x2 − 13x+15=0?Expressyouranswerasacommonfraction.
9. ��������� What is the sum of the squares of the solutions of x3 −15x2+66x −80=0?
10. �������� The solutions of x3 −63x2 + cx −1728=0formageometricsequence.Whatisthevalue of c ?