IMO 2007 Ha Noi, Vietnam | IMO 2007 Ha Noi Vietnam Day 1 - 25 July 2007 T Real numbers 1 tt2 . an are given. For each 3 1 i n define di max ữỹ I 1 j 0 min ữỹ I i j n and let d max I 1 i n . a Prove that for any real numbers X x 2 xn max xi CLi I 1 i n Ệ-. b Show that there are real numbers Xi X2 xn such that the equality holds in . 2 Consider five points A B ơ D and E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. Let f be a line passing through A. Suppose that Ể. intersects the interior of the segment DC at F and intersects line BC at G. Suppose also that EF EG EC. Prove that -Ể is the bisector of angle DAB. -3 In a mathematical competition some competitors are friends. Friendship is always mutual. 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. This file was downloaded from the AoPS MathLinks Math Olympiad Resources Page http Page 1 http MathLinks Everyone IMO 2007 Ha Noi Vietnam Day 2 - 26 July 2007 ỊTỊ In triangle ABC the bisector of angle BCA intersects the circumcircle again at R the perpendicular 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 RPK and RQL have the same area. 5 Let a and b be positive integers. Show that if 4ab 1 divides 4tt2 I 2 then a b. 6 Let n be a positive integer. Consider s x y z I X ý z e 0 1 . n X y z 0 as a set of n I 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 0 0 . This file was downloaded from the AoPS
