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HMMT 二月 2009 · 冲刺赛 · 第 9 题

HMMT February 2009 — Guts Round — Problem 9

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 6 ] An icosidodecahedron is a convex polyhedron with 20 triangular faces and 12 pentagonal faces.
    How many vertices does it have?
    12 HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUNDth

英文原题

[ 6 ] An icosidodecahedron is a convex polyhedron with 20 triangular faces and 12 pentagonal faces.
How many vertices does it have?
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1
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12 th HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUND

解析

英文解析

  1. [ 6 ] An icosidodecahedron is a convex polyhedron with 20 triangular faces and 12 pentagonal faces.
    How many vertices does it have?
    Answer: 30
    Solution: Since every edge is shared by exactly two faces, there are (20 · 3 + 12 · 5) / 2 = 60 edges.
    Using Euler’s formula v − e + f = 2, we see that there are 30 vertices.
    12 HARVARD-MIT MATHEMATICS TOURNAMENT, 21 FEBRUARY 2009 — GUTS ROUNDth