CAT (QA) PYQs

Let $N, x$ and $y$ be positive integers such that $N = x + y,$ $2 < x < 10$ and $14 < y < 23$. If $N > 25$, then how many distinct values are possible for $N?$

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 $m^2 + 105 = n^2$?