Question 34
The ordered pairs $(2011,2),(2010,3),(2009,4), \ldots,(1008,1005),(1007,1006)$ are written from left to right on a blackboard. Every minute, Elizabeth selects a pair of adjacent pairs $\left(x_{i}, y_{i}\right)$ and $\left(x_{j}, y_{j}\right)$, with $\left(x_{i}, y_{i}\right)$ left of $\left(x_{j}, y_{j}\right)$, erases them, and writes $\left(\frac{x_{i} y_{i} x_{j}}{y_{j}}, \frac{x_{i} y_{i} y_{j}}{x_{j}}\right)$ in their place. Elizabeth continues this process until only one ordered pair remains. How many possible ordered pairs $(x, y)$ could appear on the blackboard after the process has come to a conclusion?