AIME II #3 2008

[stupidityismygam]

A block of cheese in the shape of a rectangular solid measures [unparseable or potentially dangerous latex formula] cm by [unparseable or potentially dangerous latex formula] cm by [unparseable or potentially dangerous latex formula] cm. Ten slices are cut from the cheese. Each slice has a width of [unparseable or potentially dangerous latex formula] 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?

[Michael T]

You start with a 10x13x14. The smallest area you can cut off is the 10x13 face, leaving a 10x13x13 prism.
Repeat cutting off the smallest area-d face
10x13x13 (1)
10x13x12 (2)
10x12x12 (3)
10x12x11 (4)
10x11x11 (5)
10x10x11 (6)
10x10x10 (7)
10x10x9 (8)
9x10x9 (9)
9x9x9 (10)

So after ten slices, we have a 9x9x9 cube, which has volume of 729 cubic units.

Whoo.

[stupidityismygam]

More rigorous time.
With each cut, one of the edge lengths decreases by 1. So, after 10 cuts the "total number of lengths" is [unparseable or potentially dangerous latex formula] So, we will form an [unparseable or potentially dangerous latex formula] by [unparseable or potentially dangerous latex formula] by [unparseable or potentially dangerous latex formula] rectangular prism such that [unparseable or potentially dangerous latex formula]

By AM-GM, [unparseable or potentially dangerous latex formula] So, [unparseable or potentially dangerous latex formula] Thus the maximum volume is [unparseable or potentially dangerous latex formula]