TheFifty-petitionSaturday,December2,1995A–1Letbeasetofrealnumberswhichisclosedundermultiplication(thatis,ifandarein,thensois).(notnecessarilydistinct)elementsofisinandthattheproductofanythreeelementsofisin,–2Forwhatpairsofpositiverealnumbersdoestheimproperintegralconverge?A–3Thenumberhasnine(notnecessarilydis-tinct)-digitnumbersformedbyreplac-ingjustoneofthedigitsisbythecorre-spondingdigit():thatis,eachoftheninenumbersformedbyreplac-,foreach,isdivisibleby7.[Forexample,if,thenmaybe2or9,sinceandaremultiplesof7.]A––5Letbedifferentiable(real-valued)func-tionsofasinglevariablewhichsatisfy......,?A–6Supposethateachofpeoplewritesdownthenumbers1,2,3inrandomorderinonecolumnofamatrix,(ifnecessary),–1Forapartitionof,,therearetwodis-tinctnumbersandinsuchthatand.[Apartitionofasetisacollectionofdisjointsubsets(parts)whoseunionis.]B–2Anellipse,whosesemi-axeshavelengthsand,,pletesonerev-olutionwhenittraversesoneperiodofthecurve?B–3Toeachpositiveintegerwithdecimaldigits,weas-sociatethedeterminantofthematrixobtainedbywrit-,for,,asafunctionof,thesumofallthedetermi-nantsassociatedwith-digitint
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