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Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 26', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | THE 1990 ASIAN PACIFIC MATHEMATICAL OLYMPIAD Time allowed 4 hours NO calculators are to be used. Each question is worth seven points. Question 1 Given triagnle ABC let D E F be the midpoints of BC AC AB respectively and let G be the centroid of the triangle. For each value of ABAC how many non-similar triangles are there in which AEGF is a cyclic quadrilateral Question 2 Let a-1 a2 . an be positive real numbers and let Sk be the sum of the products of a1 a2 . an taken k at a time. Show that Sk Sn-k Q a1a2 an for k 1 2 . n 1. Question 3 Consider all the triangles ABC which have a fixed base AB and whose altitude from C is a constant h. For which of these triangles is the product of its altitudes a maximum Question 4 A set of 1990 persons is divided into non-intersecting subsets in such a way that 1. No one in a subset knows all the others in the subset 2. Among any three persons in a subset there are always at least two who do not know each other and 3. For any two persons in a subset who do not know each other there is exactly one person in the same subset knowing both of them. a Prove that within each subset every person has the same number of acquaintances. b Determine the maximum possible number of subsets. Note It is understood that if a person A knows person B then person B will know person A an acquaintance is someone who is known. Every person is assumed to know one s self. Question 5 Show that for every integer n 6 there exists a convex hexagon which can be dissected into exactly n congruent .