HMMT 二月 2014 · 冲刺赛 · 第 9 题

英文解析

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.
   √
   5 6
   Answer: The given region is a hemisphere, so the largest cube that can fit inside it has oneface centered at the origin and the four vertices of the opposite face on the spherical surface. Let the 3
   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