- Day 1 - 25 July 2007. - 1 Real numbers a 1 , a 2. - (a) Prove that, for any real numbers x 1 x 2 x n , maxfjx i a i j j 1 i ng d. - (b) Show that there are real numbers x 1 x 2 x n such that the equality holds in. - 2 Consider ve points A, B, C, D and E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. - Let ` be a line passing through A. - Suppose that ` intersects the interior of the segment DC at F and intersects line BC at G. - Prove that ` is the bisector of angle DAB.. - 3 In a mathematical competition some competitors are friends. - Call a group of competitors a clique if each two of them are friends. - (In particular, any group of fewer than two competitiors is a clique.) The number of members of a clique is called its size.. - Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.. - http://www.artofproblemsolving.com/. - This le was downloaded from the AoPS MathLinks Math Olympiad Resources Page Page 1 http://www.mathlinks.ro/. - Day 2 - 26 July 2007. - 4 In triangle ABC the bisector of angle BCA intersects the circumcircle again at R, the per- pendicular bisector of BC at P , and the perpendicular bisector of AC at Q. - The midpoint of BC is K and the midpoint of AC is L. - Prove that the triangles RP K and RQL have the same area.. - 5 Let a and b be positive integers. - Show that if 4ab 1 divides (4a 2 1) 2 , then a = b.. - 6 Let n be a positive integer. - as a set of (n + 1) 3 1 points in the three-dimensional space. - Determine the smallest possible number of planes, the union of which contains S but does not include (0. - This le was downloaded from the AoPS MathLinks Math Olympiad Resources Page Page 2 http://www.mathlinks.ro/