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HMMT 十一月 2016 · 冲刺赛 · 第 16 题

HMMT November 2016 — Guts Round — Problem 16

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 10 ] Create a cube C with edge length 1. Take the centers of the faces and connect them to form anoctahedron O . Take the centers of the octahedron’s faces and connect them to form a new cube C .1
    1 2
    Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons. 2
解析

英文解析

  1. [ 10 ] Create a cube C with edge length 1. Take the centers of the faces and connect them to form anoctahedron O . Take the centers of the octahedron’s faces and connect them to form a new cube C .1
    1 2
    Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.
    Proposed by: Shyam Narayanan
    √
    54+9 3
    Answer:
    The lengths of the second cube are one-third of the lengths of the first cube, so the surface area 8
    decreases by a factor of one-ninth. Since the first cube has surface area 6 and the first octahedron has
    √
    ( )
    √ √
    1 1 54 + 9 3
    surface area 3, the total area is (6 + 3) · 1 + + + · · · =
    9 9 82 2