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Found problems: 1

2020 USA TSTST, 5
Tags: counting , bijective proof , combinatorics
Let $\mathbb{N}^2$ denote the set of ordered pairs of positive integers. A finite subset $S$ of $\mathbb{N}^2$ is [i]stable[/i] if whenever $(x,y)$ is in $S$, then so are all points $(x',y')$ of $\mathbb{N}^2$ with both $x'\leq x$ and $y'\leq y$. Prove that if $S$ is a stable set, then among all stable subsets of $S$ (including the empty set and $S$ itself), at least half of them have an even number of elements. [i]Ashwin Sah and Mehtaab Sawhney[/i]