AMO-Bench 2025 Problem 42

A point M on an infinite number line moves one unit to the left or to the right at each step. The probability of moving to the right is $p (p > \frac{1}{2})$, and the probability of moving to the left is $1-p$. The point M starts at the origin $x = 0$. Let $x$ be the probability that M is at position $x = 3$ after $n$ moves, where $n \geq 2$, and let $y$ be the probability that M is at position $x = 4$ after $m$ moves, where $m \geq 2$. Find the conditions on $m$ and $n$ such that $x > (1-p)y$.