国际数学奥林匹克IMO试题(官方版)1963_eng

FifthInternationalOlympiad,19631963/1.Findallrealrootsoftheequation√x2−p+2√x2−1=x,wherepisarealparameter.1963/2.PointAandsegmentBCaregiven.DeterminethelocusofpointsinspacewhichareverticesofrightangleswithonesidepassingthroughA,andtheothersideintersectingthesegmentBC.1963/3.Inann-gonallofwhoseinterioranglesareequal,thelengthsofconsecutivesidessatisfytherelationa1≥a2≥···≥an.Provethata1=a2=···=an.1963/4.Findallsolutionsx1,x2,x3,x4,x5ofthesystemx5+x2=yx1x1+x3=yx2x2+x4=yx3x3+x5=yx4x4+x1=yx5,whereyisaparameter.1963/5.Provethatcospi7−cos2pi7+cos3pi7=12.1963/6.Fivestudents,A,B,C,D,E,tookpartinacontest.OnepredictionwasthatthecontestantswouldfinishintheorderABCDE.Thispredictionwasverypoor.Infactnocontestantfinishedinthepositionpredicted,andnotwocontestantspredictedtofinishconsecutivelyactuallydidso.Asecondpre-dictionhadthecontestantsfinishingintheorderDAECB.Thispredictionwasbetter.Exactlytwoofthecontestantsfinishedintheplacespredicted,andtwodisjointpairsofstudentspredictedtofinishconsecutivelyactuallydidso.Determinetheorderinwhichthecontestantsfinished.