HMMT 二月 2005 · 几何 · 第 1 题

英文解析

1. The volume of a cube (in cubic inches) plus three times the total length of its edges
   (in inches) is equal to twice its surface area (in square inches). How many inches longis its long diagonal?
   √
   Solution: 6 3
   3 2
   If the side length of the cube is s inches, then the condition implies s + 3 · 12 s = 2 · 6 s ,
   2 2
   or s ( s − 12 s + 36) = s ( s − 6) = 0. Therefore s = 6, and the long diagonal has length
   √ √
   s 3 = 6 3.