HMMT 二月 2014 · 冲刺赛 · 第 9 题

HMMT February 2014 — Guts Round — Problem 9

[ 6 ] Compute the side length of the largest cube contained in the region 2 2 2 { ( x, y, z ) : x + y + z  25 and x 0 } of three-dimensional space.

解析

[ 6 ] Compute the side length of the largest cube contained in the region 2 2 2 { ( x, y, z ) : x + y + z ≤ 25 and x ≥ 0 } of three-dimensional space. √ 5 6 Answer: The given region is a hemisphere, so the largest cube that can fit inside it has one 3 face centered at the origin and the four vertices of the opposite face on the spherical surface. Let the side length of this cube be s . Then, the radius of the circle is the hypotenuse of a triangle with side √ √ 2 6 lengths s and s . So, by the Pythagorean Theorem, the radius equals s . Since the radius of the 2 2 √ 5 6 hemisphere is 5, the side length of the cube is . 3