Question 195
Consider a rectangular billiards table $a$ meters wide and $b$ meters long with pockets at its corners, where $a$ and $b$ are both positive integers. A ball is placed at the lower left corner of the table and shot at a $45$ degree angle. It travels without friction until it lands in one of the pockets. Every time it hits an edge of the table the angle of reflection is equal to the angle of incidence. How many ordered pairs of integers $(a, b)$ are there with $1 \le b \le a \le 2009$ such that on a $a$ by $b$ table, the ball eventually lands in the upper left pocket?