剑桥模拟试题3 【剑桥国际高中入学资料】

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1、ModelTestforCambridgeInternationalSchoolSolvethefollowingquestionscomprehensively.1.Threeunitcirclesarearrangedsothateachtouchestheothertwo.Findtheradiiofthetwocircleswhichtouchallthree.2.Findallrealnumbersxsuchthatx+1=

2、x+3

3、-

4、x-1

5、.3.(1)Givenx=(1+1/n)n,y=(1

6、+1/n)n+1,showthatxy=yx.(2)Showthat12-22+32-42+...+(-1)n+1n2=(-1)n+1(1+2+...+n).4.Allcoefficientsofthepolynomialp(x)arenon-negativeandnoneexceedp(0).Ifp(x)hasdegreen,showthatthecoefficientofxn+1inp(x)2isatmostp(1)2/2.5.Whatisthemaximumpossiblevalueforthesum

7、oftheabsolutevaluesofthedifferencesbetweeneachpairofnnon-negativerealnumberswhichdonotexceed1?6.ABisadiameterofacircle.XisapointonthecircleotherthanthemidpointofthearcAB.BXmeetsthetangentatAatP,andAXmeetsthetangentatBatQ.ShowthatthelinePQ,thetangentatXandt

8、helineABareconcurrent.第2页1.Fourpointsonacircledivideitintofourarcs.Thefourmidpointsformaquadrilateral.Showthatitsdiagonalsareperpendicular.2.Findthesmallestpositiveintegerbforwhich7+7b+7b2isafourthpower.3.Showthattherearenopositiveintegersm,nsuchthat4m(m+1

9、)=n(n+1).4.ABCDisaconvexquadrilateralwitharea1.ThelinesAD,BCmeetatX.ThemidpointsofthediagonalsACandBDareYandZ.FindtheareaofthetriangleXYZ.5.Asquarehastensdigit7.Whatistheunitsdigit?6.Findallorderedtriples(x,y,z)ofrealnumberswhichsatisfythefollowingsystemof

10、equations:xy=z-x-yxz=y-x-zyz=x-y-z第2页