上交ACM04-05赛季笔试

Direction:Youwillhave20problemsandyouareexpectedtofillintheblanksonANSWERSHEET.

(1)What’sthesmallestpositiveintegerinthefollowingset?

󰀃󰀁󰀂󰀃

126x+336y+450z󰀃x,y,z∈Z(2)Findthelastdigitofthenumber20042003

2002

.

(3)LetABCbeatrianglewithsidesBC,CA,ABoflengthsa,b,crespec-tively.LetD,EbethemidpointsofthesidesAC,AB,respectively.If

2+c2

themedianBDisperpendiculartoCE,what’sthevalueofba2?(4)Howmanypositiveintegerslessthanorequalto2004,cannotbedividedby2norby5?(5)LetSbeasubsetof{1,2,...,9},suchthatthesumsformedbyadding

eachunorderedpairofdistinctnumbersfromSarealldifferent.Forexample,thesubset{1,2,3}hasthisproperty,but{1,2,3,4}doesnot,sincethepairs{1,4}and{2,3}havethesamesum,namely5.

WhatisthemaximumnumberofelementsthatScancontain?(6)ThepointPliesonthesideBCoftriangle∆ABCsothatPC=2BP,

ABC=45◦andAPC=60◦.DetermineACB.(7)Howmanyfive-digitpositiveintegershavethepropertythattheprod-uctoftheirdigitsis2000?(8)Letn!=n·(n−1)·...·2·1,howmanyconsecutivezerosattheend

of2004!?(9)Ifthenumbersfrom1to6arewritteninrandomorder,anumber

consistingof6digitsisobtained.Whatistheprobabilitythatthisnumberisdivisibleby6?1

iswritteninitsdecimalform,whatisthe2004thdigitafter(10)If

7000

thedecimalpoint?(11)Lookatthepicturebelow.Supposethesidelengthofthesquareis

1.What’stheareaoftheblackpart?(intersectionoffourcircleswithradius1)Hint:0.30Page1

SJTUProgrammingContest2004—Paper-basedTest

(12)Fiveofthesixedgesofatetrahedronareknowntobeatmost2004

unitslong.What’sthemaximumpossiblevolumeofthetetrahedron?(13)Findthesmallestpositiveintegernsuchthatforeveryintegermwith

0mkm+1

<<.2004n2005

(14)Apermutationoftheintegers1901,1902,···,2003,2004isasequence

a1,a2,···,a104inwhicheachofthoseintegersappearsexactlyonce.Givensuchapermutation,weformthesequenceofpartialsums

s1=a1,

s2=a1+a2,

···

s104=a1+···+a104

Howmanyofthesepermutationswillhavenotermsofthesequences1,···,s104divisiblebythree?(15)Ifyouknowthat

1111π21+2+2+2+2+···=

23456

thendetermine

1+

1111

++++···32527292

(16)Howmanynonnegativeintegersolutionsfortheequation

x1+x2+x3+···+x10=2004

where

xi≥0andxi∈Zfori=1,2,3,···,10

(17)Inaboxtherearefourkindsofmarbles:20redones,12yellowones,

8blueonesand6greenones.Whatisthesmallestnumberofmarblesonehastotakeoutoftheboxtobesurethat10marblesareofthesamecolor?(18)Letf(x)=5x13+13x5+9ax.Findthesmallestpositiveintegerasuch

that65dividesf(x)foreveryintegerx.(19)Asmallclasshasfivestudents.Theteachercollectsatestandimme-diatelyhandsitoutagainsuchthatthestudentscancorrectthetestthemselves.Inhowmanywayscanthetestbehandedoutwithoutanystudentreceivinghis/herowntest?(20)Pisapointinsideanequilateraltrianglesuchthatthedistancesfrom

Ptothethreeverticesare3,4and5,respectively.Findtheareaofthetriangle.

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