Every integer occurs in this sequence exactly once
Source: Iran Third Round MO 1998, Exam 1, P1
October 31, 2010
modular arithmeticinductionnumber theory unsolvednumber theory
Problem Statement
Define the sequence by and for all
x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}
where . Prove that every integer occurs in this sequence exactly once.