CAT (QA) PYQs

How many different pairs $(a, b)$ of positive integers are there such that $a ≤ b$ and $\frac{1}{a} + \frac{1}{b} = \frac{1}{9}$

The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x > y \geq 3$ and $x + y < 14$ is

How many pairs $(m, n)$ of positive integers satisfy the equation $m^2 + 105 = n^2$?