Language:English
Day:1
Problem1.Showthat
Tuesday,April12,2016
Letnbeanoddpositiveinteger,andletx1,x2,...,xnbenon-negativerealnumbers.
i=1,...,n
2
min(x2i+xi+1)≤max(2xjxj+1),
j=1,...,n
wherexn+1=x1.
Problem2.LetABCDbeacyclicquadrilateral,andletdiagonalsACandBDintersectatX.Let
C1,D1andMbethemidpointsofsegmentsCX,DXandCD,respectively.LinesAD1andBC1intersectatY,andlineMYintersectsdiagonalsACandBDatdi?erentpointsEandF,respectively.ProvethatlineXYistangenttothecirclethroughE,FandX.
Problem3.Letmbeapositiveinteger.Considera4m×4marrayofsquareunitcells.Twodi?erentcellsarerelatedtoeachotheriftheyareineitherthesameroworinthesamecolumn.Nocellisrelatedtoitself.Somecellsarecolouredblue,suchthateverycellisrelatedtoatleasttwobluecells.Determinetheminimumnumberofbluecells.
Language:English
Time:4hoursand30minutesEachproblemisworth7points
2016年欧洲女子数学奥林匹克第一天
