数学.docx

数学.docx

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数学

数学

数学

1018BigNumber

TimeLimit:

2000/1000MS（Java/Others）    MemoryLimit:

65536/32768K（Java/Others）

TotalSubmission（s）:

26720    AcceptedSubmission（s）:

12156

ProblemDescription

Inmanyapplicationsverylargeintegersnumbersarerequired.Someoftheseapplicationsareusingkeysforsecuretransmissionofdata,encryption,etc.Inthisproblemyouaregivenanumber,youhavetodeterminethenumberofdigitsinthefactorialofthenumber.

Input

Inputconsistsofseverallinesofintegernumbers.Thefirstlinecontainsanintegern,whichisthenumberofcasestobetested,followedbynlines,oneinteger1≤n≤107oneachline.

Output

Theoutputcontainsthenumberofdigitsinthefactorialoftheintegersappearingintheinput.

SampleInput

2

10

20

SampleOutput

7

19

1028IgnatiusandthePrincessIII

TimeLimit:

2000/1000MS（Java/Others）    MemoryLimit:

65536/32768K（Java/Others）

TotalSubmission（s）:

13512    AcceptedSubmission（s）:

9558

ProblemDescription

"Well,itseemsthefirstproblemistooeasy.Iwillletyouknowhowfoolishyouarelater."feng5166says.

"Thesecondproblemis,givenanpositiveintegerN,wedefineanequationlikethis:

  N=a[1]+a[2]+a[3]+...+a[m];

  a[i]>0,1<=m<=N;

MyquestionishowmanydifferentequationsyoucanfindforagivenN.

Forexample,assumeNis4,wecanfind:

  4=4;

  4=3+1;

  4=2+2;

  4=2+1+1;

  4=1+1+1+1;

sotheresultis5whenNis4.Notethat"4=3+1"and"4=1+3"isthesameinthisproblem.Now,youdoit!

"

Input

Theinputcontainsseveraltestcases.EachtestcasecontainsapositiveintegerN（1<=N<=120）whichismentionedabove.Theinputisterminatedbytheendoffile.

Output

Foreachtestcase,youhavetooutputalinecontainsanintegerPwhichindicatethedifferentequationsyouhavefound.

SampleInput

4

10

20

SampleOutput

5

42

627

1056HangOver

TimeLimit:

2000/1000MS（Java/Others）    MemoryLimit:

65536/32768K（Java/Others）

TotalSubmission（s）:

9720    AcceptedSubmission（s）:

4096

ProblemDescription

Howfarcanyoumakeastackofcardsoverhangatable?

Ifyouhaveonecard,youcancreateamaximumoverhangofhalfacardlength.（We'reassumingthatthecardsmustbeperpendiculartothetable.）Withtwocardsyoucanmakethetopcardoverhangthebottomonebyhalfacardlength,andthebottomoneoverhangthetablebyathirdofacardlength,foratotalmaximumoverhangof1/2+1/3=5/6cardlengths.Ingeneralyoucanmakencardsoverhangby1/2+1/3+1/4+...+1/（n+1）cardlengths,wherethetopcardoverhangsthesecondby1/2,thesecondoverhangsthathirdby1/3,thethirdoverhangsthefourthby1/4,etc.,andthebottomcardoverhangsthetableby1/（n+1）.Thisisillustratedinthefigurebelow.

Theinputconsistsofoneormoretestcases,followedbyalinecontainingthenumber0.00thatsignalstheendoftheinput.Eachtestcaseisasinglelinecontainingapositivefloating-pointnumbercwhosevalueisatleast0.01andatmost5.20;cwillcontainexactlythreedigits.

Foreachtestcase,outputtheminimumnumberofcardsnecessarytoachieveanoverhangofatleastccardlengths.Usetheexactoutputformatshownintheexamples.

SampleInput

1.00

3.71

0.04

5.19

0.00

SampleOutput

3card（s）

61card（s）

1card（s）

273card（s）

1060LeftmostDigit

TimeLimit:

2000/1000MS（Java/Others）    MemoryLimit:

65536/32768K（Java/Others）

TotalSubmission（s）:

13545    AcceptedSubmission（s）:

5208

ProblemDescription

GivenapositiveintegerN,youshouldoutputtheleftmostdigitofN^N.

Input

Theinputcontainsseveraltestcases.ThefirstlineoftheinputisasingleintegerTwhichisthenumberoftestcases.Ttestcasesfollow.

EachtestcasecontainsasinglepositiveintegerN（1<=N<=1,000,000,000）.

Output

Foreachtestcase,youshouldoutputtheleftmostdigitofN^N.

SampleInput

2

3

4

SampleOutput

2

2

Hint

Inthefirstcase,3*3*3=27,sotheleftmostdigitis2.

Inthesecondcase,4*4*4*4=256,sotheleftmostdigitis2.

1066Lastnon-zeroDigitinN!

TimeLimit:

2000/1000MS（Java/Others）    MemoryLimit:

65536/32768K（Java/Others）

TotalSubmission（s）:

6337    AcceptedSubmission（s）:

1567

ProblemDescription

TheexpressionN!

readas"Nfactorial,"denotestheproductofthefirstNpositiveintegers,whereNisnonnegative.So,forexample,

NN!

01

11

22

36

424

5120

103628800

Forthisproblem,youaretowriteaprogramthatcancomputethelastnon-zerodigitofthefactorialforN.Forexample,ifyourprogramisaskedtocomputethelastnonzerodigitof5!

yourprogramshouldproduce"2"because5!

=120,and2isthelastnonzerodigitof120.

Input

Inputtotheprogramisaseriesofnonnegativeintegers,eachonitsownlinewithnootherletters,digitsorspaces.ForeachintegerN,youshouldreadthevalueandcomputethelastnonzerodigitofN!

.

Output

Foreachintegerinput,theprogramshouldprintexactlyonelineofoutputcontainingthesinglelastnon-zerodigitofN!

.

SampleInput

1

2

26

125

3125

9999

SampleOutput

1

2

4

8

2

8

1071Thearea

TimeLimit:

2000/1000MS（Java/Others）    MemoryLimit:

65536/32768K（Java/Others）

TotalSubmission（s）:

7855    AcceptedSubmission（s）:

5520

ProblemDescription

Ignatiusboughtalandlastweek,buthedidn'tknowtheareaofthelandbecausethelandisenclosedbyaparabolaandastraightline.Thepicturebelowshowsthearea.Nowgivenalltheintersectantpointsshowsinthepicture,canyoutellIgnatiustheareaoftheland?

Note:

ThepointP1inthepictureisthevertexoftheparabola.

Input

Theinputcontainsseveraltestcases.ThefirstlineoftheinputisasingleintegerTwhichisthenumberoftestcases.Ttestcasesfollow.

Eachtestcasecontainsthreeintersectantpointswhichshowsinthepicture,theyaregivenintheorderofP1,P2,P3.Eachpointisdescribedbytwofloating-pointnumbersXandY（0.0<=X,Y<=1000.0）.

Output

Foreachtestcase,youshouldoutputtheareaoftheland,theresultshouldberoundedto2decimalplaces.

SampleInput

2

5.0000005.000000

0.0000000.000000

10.0000000.000000

10.00000010.000000

1.0000001.000000

14.0000008.222222

SampleOutput

33.33

40.69

1099Lottery

TimeLimit:

2000/1000MS（Java/Others）    MemoryLimit:

65536/32768K（Java/Others）

TotalSubmission（s）:

2559    AcceptedSubmission（s）:

1158

ProblemDescription

Eddy'scompanypublishesakindoflottery.Thissetoflotterywhicharenumbered1ton,andasetofoneofeachisrequiredforaprize.Withonenumberperlottery,howmanylotteryonaveragearerequiredtomakeacompletesetofncoupons?

Input

Inputconsistsofasequenceoflineseachcontainingasinglepositiveintegern,1<=n<=22,givingthesizeofthesetofcoupons.

Output

Foreachinputline,outputtheaveragenumberoflotteryrequiredtocollectthecompletesetofncoupons.Iftheanswerisanintegernumber,outputthenumber.Iftheanswerisnotinteger,thenoutputtheintegerpartoftheanswerfollowedbyaspaceandthenbytheproperfractionintheformatshownbelow.Thefractionalpartshouldbeirreducible.Thereshouldbenotrailingspacesinanylineofouput.

SampleInput

2

5

17

SampleOutput

3

5

11--

12

340463

58------

720720

1110EquipmentBox

TimeLimit:

2000/1000MS（Java/Others）    MemoryLimit:

65536/32768K（Java/Others）

TotalSubmission（s）:

1547    AcceptedSubmission（s）:

364

ProblemDescription

ThereisalargeroominthePyramidcalledRoom-of-No-Return.Itsflooriscoveredbyrectangulartilesofequalsize.Thenameoftheroomwaschosenbecauseoftheveryhighnumberoftrapsandmechanismsinit.TheACMgrouphasspentseveralyearsstudyingthesecretplanofthisroom.Ithasmadeacleverplantoavoidallthetraps.AspeciallytrainedmechanicwassenttodeactivatethemostfearedtrapcalledShatteredBones.Afterdeactivatingthetrapthemechanichadtoescapefromtheroom.Itisveryimportanttosteponthecenterofthetilesonly;hemustnottouchtheedges.Onewrongstepandalargerockfallsfromtheceilingsquashingthemechaniclikeapancake.Afterdeactivatingthetrap,herealizedahorriblething:

theACMplandidnottakehisequipmentboxintoconsideration.Theboxmustbelaidontothegroundbecausethemechanicmusthavebothhandsfreetopreventcontactwithothertraps.Butwhentheboxislaidontheground,itcouldtouchthelineseparatingthetiles.Andthisisthemainproblemyouaretosolve.

Input

TheinputconsistsofTtestcases.Thenumberofthem（T）isgivenonthefirstlineoftheinput.Eachtestcaseconsistsofasingleline.Thelinecontainsexactlyfourintegernumbersseparatedbyspaces:

A,B,XandY.AandBindicatethedimensionsofthetiles,XandYarethedimensionsoftheequipmentbox（1<=A,B,X,Y<=50000）.

Output

Yourtaskistodeterminewhetheritispossibletoputtheboxonasingletile--thatis,ifthewholeboxfitsonasingletilewithouttouchingitsborder.Ifso,youaretoprintonelinewiththesentence"Escapeispossible.".Otherwiseprintthesentence"Boxcannotbedropped.".

SampleInput

2

101088

881010

SampleOutput

Escapeispossible.

Boxcannotbedropped.

1210Eddy's洗牌问题

TimeLimit:

2000/1000MS（Java/Others）    MemoryLimit:

65536/32768K（Java/Others）

TotalSubmission（s）:

3389    AcceptedSubmission（s）:

2275

ProblemDescription

Eddy是个ACMer,他不仅喜欢做ACM题,而且对于纸牌也有一定的研究,他在无聊时研究发现,如果他有2N张牌，编号为1,2,3..n,n+1,..2n。

这也是最初的牌的顺序。

通过一次洗牌可以把牌的序列变为n+1,1,n+2,2,n+3,3,n+4,4..2n,n。

那么可以证明，对于任意自然数N，都可以在经过M次洗牌后第一次重新得到初始