2022 USAMO 第 1 题

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.

设$a$和$b$为正整数。 $(a+b+1)\times (a+b+1)$ 网格的单元格颜色为琥珀色和青铜色，因此至少有 $a^2+ab-b$ 琥珀色单元格和至少 $b^2+ab-a$ 青铜色单元格。证明可以选择 $a$ 琥珀色单元格和 $b$ 青铜色单元格，使得 $a+b$ 所选单元格中没有两个位于同一行或同一列。