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). -- 錦瑟無端五十絃...一絃一柱思華年... 莊生曉夢迷蝴蝶...望帝春心託杜鵑... 滄海月明珠有淚...藍田日暖玉生煙... 此情可待成追憶...只是當時已惘然...多情者...情場殺手... --

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