Junior Balkan MO 1997 Belgrad, Yugoslavia 1 Show that given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than 1. 8 Bulgaria 2 Let x2+y2 + x2−y2 = k. Compute the following expression in terms of k: x2−y2 x2+y2 x8+y8 x8−y8 E(x,y) = − . x8−y8 x8+y8 Ciprus 3 Let ABC be a triangle and let I be the incenter. Let N, M be the midpoints of the sides AB and CA respectively. The lines BI and CI meet MN at K and L respectively. Prove that AI +BI +CI > BC +KL. Greece √ 4 Determine the triangle with sides a,b,c and circumradius R for which R(b+c) = a bc. Romania 5 Let n , n , ..., n be positive integers such that 1 2 1998 n2+n2+···+n2 = n2 . 1 2 1997 1998 Show that at least two of the numbers are even. This file was downloaded from the AoPS Math Olympiad Resources Page Page 1 http://www.artofproblemsolving.com/