一些比较难的英文数学题(追加20分)
1.Apolynomialp(x)satisfiesp(5−x)=p(5+x)forallrealx.Supposep(x)=0hasfour
distinctrealroots.Findthesumoftheseroots.
2.Apersoncashesacheckatabank.Bymistakethetellerpaysthenumberofcentsas
dollarsanddollarsascents.Thepersonspends$3.50beforenoticing,thenlaterfindsthattheremainingmoneyisexactlydoubletheamountofthecheck.Whatwastheamount?
3.Leta,b,cbeintegerssuchthat2005dividesbothab+9b+81andbc+9c+81.Provethat2005alsodividesca+9a+81.
4.Twothievesstoleanopenchainwith2kwhitebeadsand2mblackbeads.Theywanttosharethelootequally,bycuttingthechaininpiecesinsuchawaythateachonegetskwhitebeadsandmblackbeads.Whatistheminimumnumberofcutsthatissufficient?Describeanalgorithmtomakethecuts.
5.LetABCDbeaunitsquareandmarkoffpointsE,F,G,HsuccessivelyinsidesAB,BC,CD,DA
sothatAE=BF=CG=DH=2011/2012.ProvethattheregionthatistheintersectionoftrianglesAGB,BHC,CED,DFAisasquarewitharea
1/(20112+20122)^2.