2022 USAJMO Problems/Problem 2

Problem

Let $a$ and $b$ be positive integers. The cells of an $(a + b + 1)\times (a + b + 1)$ grid are colored amber and bronze such that there are at least $a^2+ab-b$ amber cells and at least $b^2+ab-a$ bronze cells. Prove that it is possible to choose $a$ amber cells and $b$ bronze cells such that no two of the $a+b$ chosen cells lie in the same row or column.

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2022 USAJMO Problems/Problem 2

Problem

Solution 1: Pigeonhole

Solution 2: Induction