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HMMT 十一月 2016 · 冲刺赛 · 第 33 题

HMMT November 2016 — Guts Round — Problem 33

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 17 ] Camille the snail lives on the surface of a regular dodecahedron. Right now he is on vertex Pof the face with vertices P , P , P , P , P . This face has a perimeter of 5. Camille wants to get to 1
    1 2 3 4 5
    the point on the dodecahedron farthest away from P . To do so, he must travel along the surface adistance at least L . What is L ?21
    HMMT November 2016, November 12, 2016 — GUTS ROUND
    Organization Team Team ID#
解析

英文解析

  1. [ 17 ] Camille the snail lives on the surface of a regular dodecahedron. Right now he is on vertex Pof the face with vertices P , P , P , P , P . This face has a perimeter of 5. Camille wants to get to 1
    1 2 3 4 5
    the point on the dodecahedron farthest away from P . To do so, he must travel along the surface adistance at least L . What is L ?21
    Proposed by: Saranesh Prembabu
    √
    17+7 5
    Answer:
    Consider the net of the dodecahedron. It suffices to look at three pentagons ABCDE , EDF GH , and 2
    GF IJK , where AJ = L . This can be found by the law of cosines on triangle AEJ . We have AE = 1,
    √
    2 17+7 5
    ° ° 2 ° ° °
    EJ = tan 72 , and ∠ AEJ = 162 . Thus L = 1 + tan 72 + 2 · tan 72 · cos 18 = 2