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AM, GM algebraic computations

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March 5, 2005

Problem Statement

We denote with mam_a, mgm_g the arithmetic mean and geometrical mean, respectively, of the positive numbers x,yx,y. a) If ma+mg=yxm_a+m_g=y-x, determine the value of xy\dfrac xy; b) Prove that there exists exactly one pair of different positive integers (x,y)(x,y) for which ma+mg=40m_a+m_g=40.