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Bất đẳng thức luyện thi mang tính chất tham khảo, giúp ích cho các bạn tự học, ôn thi, với phương pháp giải hay, thú vị, rèn luyện kỹ năng giải đề, nâng cao vốn kiến thức cho các bạn trong các kỳ thi sắp tới. Tác giả hy vọng tài liệu này sẽ giúp ích cho các bạn. | www.batdangthuc.net 5 Inequalities From 2007 Mathematical Competition Over The World Example 1 Iran National Mathematical Olympiad 2007 . Assume that a b c are three different positive real numbers. Prove that a b b c c a ----Ä Ä------1 a b b c c a 1. Example 2 Iran National Mathematical Olympiad 2007 . Find the largest real T such that for each non-negative real numbers a b c d e such that a b c d e then Ja2 b2 c2 d2 e2 T a Vb - c Vd e 2. Example 3 Middle European Mathematical Olympiad 2007 . Let a b c d be positive real numbers with a b c d 4. Prove that a2bc b2cd c2da d2ab 4. Example 4 Middle European Mathematical Olympiad 2007 . Let a b c d be real num- bers which satisfy a b c d 2 and abcd 1. Find the maximum value of 1 a r b b 1 c 1 cd d a Example 5 China Northern Mathematical Olympiad 2007 . Let a b c be side lengths of a triangle and a b c 3. Find the minimum of a2 b2 c2 4 . Example 6 China Northern Mathematical Olympiad 2007 . Let a 3 be acute angles. Find the maximum value of 1 Vtan a tan 3 cot a cot 3 Example 7 China Northern Mathematical Olympiad 2007 . Let a b c be positive real numbers such that abc 1. Prove that ak bk ck 3 a b b c c a 2 for any positive integer k 2. 6 www.batdangthuc.net Example 8 Croatia Team Selection Test 2007 . Let a b c 0 such that a b c 1. Prove that b- C2 3 a2 b c2 . b c a Example 9 Romania Junior Balkan Team Selection Tests 2007 . Let a b c three pos- itive reals such that 1 1 1 1 a b 1 b c 1 c a 1 Show that a b c ab bc ca. Example 10 Romania Junior Balkan Team Selection Tests 2007 . Let x y z 0 be real numbers. Prove that X y Z xyz x - y y - z z - x . 3 Example 11 Yugoslavia National Olympiad 2007 . Let k be a given natural number. Prove that for any positive numbers x y z with the sum 1 the following inequality holds xk 2 yk 2 k 2 Xk i yk Zk 1 yk 1 zk xk zk 1 xk yk 7 Example 12 Cezar Lupu 2 ai bi G R 1 i n such that a2 n Tudorel Lupu Romania TST 2007 . For n G N n 2 E b2 1 n i aibi 0. Prove that n 2 i bj n. Example 13 .
