1972 AMC 12 第 28 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
28.
将一个直径为 的圆盘放在宽度为 的 棋盘上，使二者中心重合。被圆盘完全覆盖的棋盘小方格数为：
A circular disc with diameter is placed on an checkerboard with width so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is:
小提示：
与棋盘外边界相接的小方格都不可能被完全覆盖
No square touching the outside border can be completely covered
大提示：
在内部的 个小方格中，单独检查四个角上的方格
Among the interior squares, test the four corner squares separately
解答：
边界上的 个小方格没有被完全覆盖。考虑剩余的 内部网格，并以小方格边长为距离单位，则圆盘半径为 。该内部网格的四个外角到中心的距离为 所以四个角方格未被完全覆盖。其他每个内部方格都在圆盘内：其最远角点到中心的距离至多为 因此共有 个方格被完全覆盖。
因此，正确答案是 E。
The border squares are not fully covered. Consider the remaining interior grid and measure distances in square side lengths, so the disc has radius The four outer corners of this interior grid are at distance from the center, so those four corner squares are not fully covered. Every other interior square lies within the disc: its farthest possible corner is at distance at most Hence squares are completely covered.
Therefore, the correct answer is E.
其他年份的第 28 题
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