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Sequence and solutions of a quadratic equation

Source: Romanian IMO Team Selection Test TST 1988, problem 13

October 1, 2005
quadraticsalgebra proposedalgebra

Problem Statement

Let aa be a positive integer. The sequence {xn}n≥1\{x_n\}_{n\geq 1} is defined by x1=1x_1=1, x2=ax_2=a and xn+2=axn+1+xnx_{n+2} = ax_{n+1} + x_n for all n≥1n\geq 1. Prove that (y,x)(y,x) is a solution of the equation ∣y2−axy−x2∣=1 |y^2 - axy - x^2 | = 1 if and only if there exists a rank kk such that (y,x)=(xk+1,xk)(y,x)=(x_{k+1},x_k). Serban Buzeteanu
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