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Sequence with Number Theory

Source: Romania 2017 IMO TST 1, problem 3

March 18, 2018
Sequencenumber theory

Problem Statement

Consider the sequence of rational numbers defined by x1=43x_1=\frac{4}{3}, and xn+1=xn2xn2−xn+1x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}. Show that the nu,erator of the lowest term expression of each sum x1+x2+...+xkx_1+x_2+...+x_k is a perfect square.
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