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Sequence of rational numbers

Source: ITAMO 2016, Problem 5

May 11, 2016
algebraSequence

Problem Statement

Let x0,x1,x2,…x_0,x_1,x_2,\ldots be a sequence of rational numbers defined recursively as follows: x0x_0 can be any rational number and, for n≥0n\ge 0, xn+1={∣xn2−1∣if the numerator of xn is even,∣1xn−1∣if the numerator of xn is odd, x_{n+1}=\begin{cases} \left|\frac{x_n}2-1\right| & \text{if the numerator of }x_n\text{ is even}, \\ \left|\frac1{x_n}-1\right| & \text{if the numerator of }x_n\text{ is odd},\end{cases} where by numerator of a rational number we mean the numerator of the fraction in its lowest terms. Prove that for any value of x0x_0: (a) the sequence contains only finitely many distinct terms; (b) the sequence contains exactly one of the numbers 00 and 2/32/3 (namely, either there exists an index kk such that xk=0x_k=0, or there exists an index mm such that xm=2/3x_m=2/3, but not both).
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