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2005_eng - BC = DA and BC not parallel with DA Let two...

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46rd IMO 2005 Problem 1 . Six points are chosen on the sides of an equilateral triangle ABC : A 1 , A 2 on BC , B 1 , B 2 on CA and C 1 , C 2 on AB , such that they are the vertices of a convex hexagon A 1 A 2 B 1 B 2 C 1 C 2 with equal side lengths. Prove that the lines A 1 B 2 , B 1 C 2 and C 1 A 2 are concurrent. Problem 2 . Let a 1 , a 2 , . . . be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer n the numbers a 1 , a 2 , . . . , a n leave n different remainders upon division by n . Prove that every integer occurs exactly once in the sequence a 1 , a 2 , . . . . Problem 3 . Let x, y, z be three positive reals such that xyz 1. Prove that x 5 - x 2 x 5 + y 2 + z 2 + y 5 - y 2 x 2 + y 5 + z 2 + z 5 - z 2 x 2 + y 2 + z 5 0 . Problem 4 . Determine all positive integers relatively prime to all the terms of the infinite sequence a n = 2 n + 3 n + 6 n - 1 , n 1 . Problem 5 . Let ABCD be a fixed convex quadrilateral with
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Unformatted text preview: BC = DA and BC not parallel with DA . Let two variable points E and F lie of the sides BC and DA , respectively and satisfy BE = DF . The lines AC and BD meet at P , the lines BD and EF meet at Q , the lines EF and AC meet at R . Prove that the circumcircles of the triangles PQR , as E and F vary, have a common point other than P . Problem 6 . In a mathematical competition, in which 6 problems were posed to the participants, every two of these problems were solved by more than 2 5 of the contestants. Moreover, no contestant solved all the 6 problems. Show that there are at least 2 contestants who solved exactly 5 problems each. 1...
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