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X_k will win [Two sequences of positives: some x_k is > y_k]

Source: All-Russian Olympiad 2006 finals, problem 11.5

May 6, 2006
logarithmsalgebra unsolvedalgebra

Problem Statement

Two sequences of positive reals, (xn) \left(x_n\right) and (yn) \left(y_n\right), satisfy the relations x_{n \plus{} 2} \equal{} x_n \plus{} x_{n \plus{} 1}^2 and y_{n \plus{} 2} \equal{} y_n^2 \plus{} y_{n \plus{} 1} for all natural numbers n n. Prove that, if the numbers x1 x_1, x2 x_2, y1 y_1, y2 y_2 are all greater than 1 1, then there exists a natural number k k such that xk>yk x_k > y_k.
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