2000 USAMO 第 4 题

Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.

找到最小的正整数 $n$，使得如果 $1000 \times 1000$ 棋盘的 $n$ 个方块被着色，则将存在三个彩色方块，其中心形成直角三角形，边平行于棋盘的边缘。