Problem 1. Let ABC be a triangle with ∠BAC≠90°. Let O be the circumcenter of the triangle ABC and let Γ be the circumcircle of the triangle BOC. Suppose that Γ inter sects the line segment AB at P different from B, and the line segment AC at Q different from C. Let ON be a diameter of the circle Γ. Prove that the quadrilateral APNQ is a parallelogram. Problem 2. For a positive integer k, call an integer a pure k-th power if it can be represented as m^k for some integer m. Show that for every positive integer n there exist n distinct positive integers such that their sum is a pure 2009-th power, and their product is a pure 2010-th power. Problem 3. Let n be a positive integer. n people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants？ Problem 4. Let ABC be an acute triangle satisfying the condition AB > BC and AC > BC. Denote by O and H the circumcenter and the orthocenter, respectively, of the triangle ABC. Suppose that the circumcircle of the triangle AHC intersects the line AB at M different from A, and that the circumcircle of the triangle AHB intersects the line AC at N different from A. Prove that the circumcenter of the triangle MNH lies on the line OH. Problem 5. Find all functions f from the set R of real numbers into R which satisfy for all x,y,z belonging to R the identity f(f(x) + f(y) + f(z)) = f(f(x) - f(y)) + f(2xy + f(z)) + 2f(xz-yz). -- 锦瑟无端五十弦...一弦一柱思华年... 庄生晓梦迷蝴蝶...望帝春心托杜鹃... 沧海月明珠有泪...蓝田日暖玉生烟... 此情可待成追忆...只是当时已惘然...多情者...情场杀手... --

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