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ĐỀ THI TOÁN APMO (CHÂU Á THÁI BÌNH DƯƠNG)_ĐỀ 23

Tham khảo tài liệu 'đề thi toán apmo (châu á thái bình dương)_đề 23', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 40th United States of America Mathematical Olympiad Day I 12 30 PM - 5 PM EDT April 27 2011 USAMO 1. Let a b c be positive real numbers such that a2 b2 c2 a b c 2 4. Prove that ab 1 bc 1 ca 1 a b 2 b c 2 c a 2 3. USAMO 2. An integer is assigned to each vertex of a regular pentagon so that the sum of the hve integers is 2011. A turn of a solitaire game consists of subtracting an integer m from each of the integers at two neighboring vertices and adding 2m to the opposite vertex which is not adjacent to either of the hrst two vertices. The amount m and the vertices chosen can vary from turn to turn. The game is won at a certain vertex if after some number of turns that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers there is exactly one vertex at which the game can be won. USAMO 3. In hexagon ABCDEF which is nonconvex but not self-intersecting no pair of opposite sides are parallel. The internal angles satisfy A 3ZD C 3 F and E 3ZB. Furthermore AB DE BC EF and CD FA. Prove that diagonals AD BE and CF are concurrent. Copyright Committee on the American Mathematics Competitions Mathematical Association of .