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The mathscope - All the best from Vietnamese Problem Solving Journals

It’s now not too hard to find problems and solutions on the Internet due to the increasing number of websites devoted to mathematical problem solving. It is our hope that this collection saves you considerable time searching the problems you really want. We intend to give an outline of solutions to the problems in the future. Now enjoy these “cakes” from Vietnam first. | moti V I I c om mathematics 4 teachers n1 students THE MATHSCOPE All the best from Vietnamese Problem Solving Journals February 12 2007 please download for free at our website www.imo.org.yu translated by Phạm Văn Thuận Eckard Specht Vol I Problems in Mathematics Journal for the Youth The Mathscope is a free problem resource selected from mathematical problem solving journals in Vietnam. This freely accessible collection is our effort to introduce elementary mathematics problems to foreign friends for either recreational or professional use. We would like to give you a new taste of Vietnamese mathematical culture. Whatever the purpose we welcome suggestions and comments from you all. More communications can be addressed to Phạm Văn Thuận of Hanoi University at pvthuan@gmail.com It s now not too hard to find problems and solutions on the Internet due to the increasing number of websites devoted to mathematical problem solving. It is our hope that this collection saves you considerable time searching the problems you really want. We intend to give an outline of solutions to the problems in the future. Now enjoy these cakes from Vietnam first. Pham Van Thuan www.MATHVN.com 1 www.MATHVN.com moti V I c_cz m mathematics 4 teachers n students 153.1 Nguyễn Đông Yên Prove that if y y x1 2 x 1 then x2 y 1. Find all pairs of x y such that the first inequality holds while equality in the second one attains. 153.2 Tạ Văn Tự Given natural numbers m n and a real number a 1 prove the inequality 2n n 1 n 1 V am 1 n a m a m . 153.3 Nguyễn Minh Đức Prove that for each 0 e 1 there exists a natural number n0 such that the coefficients of the polynomial x y n x2 2 e xy y2 are all positive for each natural number n n0 . 200.1 Phạm Ngọc Quang In a triangle ABC let BC a CA b AB c I be the incenter of the triangle. Prove that a. IA2 b. IB2 c. IC2 abc. 200.2 Trần Xuân Đáng Let a b c 2 R such that a b c 1 prove that 15 a3 b3 c3 ab bc ca 9abc 7. 200.3 Đặng Hùng Thắng Let a b c be integers such