MathDB
three sequences

Source: Croatian MO 2004 4th Grade P3

April 9, 2021
algebraSequences

Problem Statement

The sequences (xn),(yn),(zn),n∈N(x_n),(y_n),(z_n),n\in\mathbb N, are defined by the relations xn+1=2xnxn2−1,yn+1=2ynyn2−1,zn+1=2znzn2−1,x_{n+1}=\frac{2x_n}{x_n^2-1},\qquad y_{n+1}=\frac{2y_n}{y_n^2-1},\qquad z_{n+1}=\frac{2z_n}{z_n^2-1},where x1=2x_1=2, y1=4y_1=4, and x1y1z1=x1+y1+z1x_1y_1z_1=x_1+y_1+z_1. (a) Show that xn2≠1x_n^2\ne1, yn2≠1y_n^2\ne1, zn2≠1z_n^2\ne1 for all nn; (b) Does there exist a k∈Nk\in\mathbb N for which xk+yk+zk=0x_k+y_k+z_k=0?
Back to ProblemsView on AoPS