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connected sequences revisited

Source: Bundeswettbewerb Mathematik 1992, round two, problem 4

June 19, 2004
algebra unsolvedalgebra

Problem Statement

For three sequences (xn),(yn),(zn)(x_n),(y_n),(z_n) with positive starting elements x1,y1,z1x_1,y_1,z_1 we have the following formulae: x_{n+1} = y_n + \frac{1}{z_n}   y_{n+1} = z_n + \frac{1}{x_n}   z_{n+1} = x_n + \frac{1}{y_n}   (n = 1,2,3, \ldots) a.) Prove that none of the three sequences is bounded from above. b.) At least one of the numbers x200,y200,z200x_{200},y_{200},z_{200} is greater than 20.
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