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1、USAAMC1020001Intheyear,theUnitedStateswillhosttheInternationalMathematicalOlympiad.Let,,andbedistinctpositiveintegerssuchthattheproduct.What'sthelargestpossiblevalueofthesum?SolutionThesumisthehighestiftwofactorsarethelowest.So,and.2Solution.3Eachday,Jennyateofthejellybeansthatwereinherjaratthebegin
2、ningoftheday.Attheendofthesecondday,remained.Howmanyjellybeanswereinthejaroriginally?Solution-19-4Chandrapaysanonlineserviceproviderafixedmonthlyfeeplusanhourlychargeforconnecttime.HerDecemberbillwas,butinJanuaryherbillwasbecausesheusedtwiceasmuchconnecttimeasinDecember.Whatisthefixxedmonthlyfee?Sol
3、utionLetbethefixedfee,andbetheamountshepaysfortheminutessheusedinthefirstmonth.Wewantthefixedfee,whichis5Pointsandarethemidpointsofsidesandof.Asmovesalongalinethatisparalleltoside,howmanyofthefourquantitieslistedbelowchange?(a)thelengthofthesegment(b)theperimeterof(c)theareaof(d)theareaoftrapezoid-1
4、9-Solution(a)Clearlydoesnotchange,and,sodoesn'tchangeeither.(b)Obviously,theperimeterchanges.(c)Theareaclearlydoesn'tchange,asboththebaseanditscorrespondingheightremainthesame.(d)Thebasesanddonotchange,andneitherdoestheheight,sothetrapezoidremainsthesame.Onlyquantitychanges,sothecorrectansweris.6The
5、FibonacciSequencestartswithtwo1sandeachtermafterwardsisthesumofitspredecessors.WhichoneofthetendigitsisthelasttoappearinthetunitspositionofanumberintheFibonacciSequence?SolutionThepatternoftheunitsdigitsareInorderofappearance:.isthelast.7Inrectangle,,ison,andandtrisect.Whatistheperimeterof?-19-Solut
6、ion.Sinceistrisected,.Thus,.Adding,.8AtOlympicHighSchool,ofthefreshmenandofthesophomorestooktheAMC-10.Giventhatthenumberoffreshmenandsophomorecontestantswasthesame,whichofthefollowingmustbetrue?Therearefivetimesasmanysophomoresasfreshmen.Therearetwiceasmanysophomoresasfreshmen.Thereareasmanyfreshmen
7、assophomores.Therearetwiceasmanyfreshmenassophomores.Therearefivetimesasmanyfreshmenassophomores.Solution-19-Letbethenumberoffreshmanandbethenumberofsophomores.Therearetwiceasmanyfreshmenassophomores.
