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HMMT 二月 2009 · GEN1 赛 · 第 6 题

HMMT February 2009 — GEN1 Round — Problem 6

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 4 ] The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacentto the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placedon a table?
解析

英文解析

  1. [ 4 ] The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacentto the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placedon a table?
    √
    Answer: 2 3 / 3
    √
    Solution: The major diagonal has a length of 3. The volume of the pyramid is 1 / 6, and so its
    √
    √
    1 3 2
    height h satisfies · h · ( 2) = 1 / 6 since the freshly cut face is an equilateral triangle of side length
    3 4
    √ √