2025 USAJMO Problems/Problem 3

Problem

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 $2\times1$ 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.

Solution

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2025 USAJMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
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2025 USAJMO Problems/Problem 3

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