HMMT 十一月 2016 · 冲刺赛 · 第 16 题

HMMT November 2016 — Guts Round — Problem 16

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

解析

[ 10 ] Create a cube C with edge length 1. Take the centers of the faces and connect them to form an 1 octahedron O . Take the centers of the octahedron’s faces and connect them to form a new cube C . 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: 8 The lengths of the second cube are one-third of the lengths of the first cube, so the surface area 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 + + + · · · = 2 9 9 8 2