... integers n such that1Anis an integer.7. Prove that a =1 . . . 120125 . . . 5 2011 6 is a perfect square.8. Determine the greatest number m such that the systemx2+ y2= ... Sn= xn1+ xn2+ xn3, for n = 1, 2, . . . ,. Prove that S2012is an integer.Copyrightc 2011 HEXAGON1Hanoi Open Mathematical Olympiad • • • www.hexagon.edu.vn12. Let M be a point ... BD.A DCBBy the Thales theorem we getOEBC+OFAD=ODBD+OCAC=ODBD+OBBD= 1. From this,1OE=1BC+1AD. Likewise,1OF=1BC+1AD. Hence, OE = OF . That is,2EF=1OE=1BC+1AD=13+16=12....