Question

一块长方体形状的奶酪尺寸为 $10$ cm、 $13$ cm、 $14$ cm。从这块奶酪上切下十片。每片厚 $1$ cm，且平行于奶酪的某一个面切下。各片不一定彼此平行。切下十片后，剩余奶酪块的体积最大可能是多少立方 cm？

A block of cheese in the shape of a rectangular solid measures $10$ cm by $13$ cm by $14$ cm. Ten slices are cut from the cheese. Each slice has a width of $1$ cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?

答案

答案：729

知识点：算术-几何平均不等式体积最优化

难度评级：1970

解答：

每片厚 $1$ cm 且平行于某个面，所以每次切割后剩余奶酪仍是一个长方体，只是某个维度减少 $1$ 。若十片分别使三个维度减少 $p$ ， $q$ ， $r$ ，其中 $p + q + r = 10$ ，则剩余长方体尺寸为 $(10 - p) \times (13 - q) \times (14 - r)$ ，这些维度之和为 $37 - 10 = 27$ 。

由 AM-GM 不等式，和为 $27$ 的正数乘积在三者都等于 $9$ 时最大；这可以通过从 $10$ cm 维度切 $1$ 片、从 $13$ cm 维度切 $4$ 片、从 $14$ cm 维度切 $5$ 片实现。最大体积为 $9^3 = 729$ 立方 cm。

Every slice is $1$ cm wide and parallel to a face, so after each cut the remaining cheese is still a rectangular block, with one dimension shortened by $1.$ If the ten slices shorten the three dimensions by $p,$ $q,$ and $r$ with $p + q + r = 10,$ the remaining block measures $(10 - p) \times (13 - q) \times (14 - r),$ and these dimensions sum to $37 - 10 = 27.$

By the AM-GM inequality, a product of positive numbers with fixed sum $27$ is greatest when all three are equal to $9,$ which is achieved by taking $1$ slice from the $10$ cm dimension, $4$ from the $13$ cm dimension, and $5$ from the $14$ cm dimension. The maximum volume is $9^3 = 729$ cubic cm.

2008 AIME II 第 3 题

答案：729

难度评级：1970

解答：

其他年份的第 3 题