USAJMO 2025 Problem 3

Let $m$ and $n$ be positive integers, and let $\mathcal R$ be a $2m\times{2n}$ grid of unit squares.

A domino is a $1\times2{}$ or $2\times{1}$ rectangle. A subset $S$ of grid squares in $\mathcal R$ is domino-tileable if dominoes can be placed to cover every square of $S$ exactly once with no domino extending outside of $S$ . Note: The empty set is domino tileable.

An up-right path is a path from the lower-left corner of $\mathcal R$ to the upper-right corner of $\mathcal R$ formed by exactly $2m+2n$ edges of the grid squares.

Determine, with proof, in terms of $m$ and $n$ , the number of up-right paths that divide $\mathcal R$ into two domino-tileable subsets.