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AIME 2008 II · 第 3 题

AIME 2008 II — Problem 3

专题
Contest Math
难度
L4
来源
AIME

题目详情

Problem

A block of cheese in the shape of a rectangular solid measures 1010 cm by 1313 cm by 1414 cm. Ten slices are cut from the cheese. Each slice has a width of 11 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?

解析

Solution

Let the lengths of the three sides of the rectangular solid after the cutting be a,b,ca,b,c, so that the desired volume is abcabc. Note that each cut reduces one of the dimensions by one, so that after ten cuts, a+b+c=10+13+14−10=27a+b+c = 10 + 13 + 14 - 10 = 27. By AM-GM, a+b+c3=9≥abc3⟹abc≤729\frac{a+b+c}{3} = 9 \ge \sqrt[3]{abc} \Longrightarrow abc \le \boxed{729}. Equality is achieved when a=b=c=9a=b=c=9, which is possible if we make one slice perpendicular to the 1010 cm edge, four slices perpendicular to the 1313 cm edge, and five slices perpendicular to the 1414 cm edge.

Solution 2

A more intuitive way to solve it is by seeing that to keep the volume of the rectangular cheese the greatest, we must slice the cheese off to decrease the greatest length of the cheese (this is easy to check). Here are the ten slices:

10,13,14→10,13,13→10,12,13→10,12,12→10,11,12→10,11,11→10,10,11→10,10,10→9,10,10→9,9,10→9,9,9.{10, 13, 14} \rightarrow {10, 13, 13} \rightarrow {10, 12, 13} \rightarrow {10, 12, 12} \rightarrow {10, 11, 12} \rightarrow {10, 11, 11} \rightarrow {10, 10, 11} \rightarrow {10, 10, 10} \rightarrow {9, 10, 10} \rightarrow {9, 9, 10} \rightarrow {9, 9, 9}.

So the greatest possible volume is thus 9×9×9=7299 \times 9 \times 9 = \boxed{729}