The Olympiad Corner Vi Turkish Mathematical Olympiad Second round 5. given the Angle X O Y , Variable Points M and N Are Considered On

We begin with the problems of the two days of the Turkish Mathematical Olympiad 1998. Thanks go to Ed Barbeau for collecting these problems at the IMO in Romania. 1. On the base of the isosceles triangle ABC (jABj = jACj) we choose a point D such that jB D j : jDCj = 2 : 1 and on AD] we choose a point P such that m(\ B A C) = m(\ B P D). Prove that m(\ DPC) = m(\ B A C)=2. 2. Prove that (a + 3b)(b + 4c)(c + 2a) 60abc for all real numbers 0 a b c. 3. The points of a circle are coloured by three colours. Prove that there exist innnitely many isosceles triangles with vertices on the circle and of the same colour. the arms OX] and OY], respectively, so that jOMj + jONj is constant. Determine the geometric locus of the mid-point of M N ]. 6. Some of the vertices of unit squares of an n n chessboard are coloured so that any k k square formed by these unit squares on the chess board has a coloured point on at least one of its sides. If l(n) stands for the minimum number of coloured points required to satisfy this condition, prove that lim n!1 l(n) n 2 = 2 7 .