2016年欧洲女子数学奥林匹克第一天

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