HMMT 十一月 2016 · 冲刺赛 · 第 33 题

HMMT November 2016 — Guts Round — Problem 33

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

解析

[ 17 ] Camille the snail lives on the surface of a regular dodecahedron. Right now he is on vertex P 1 of the face with vertices P , P , P , P , P . This face has a perimeter of 5. Camille wants to get to 1 2 3 4 5 the point on the dodecahedron farthest away from P . To do so, he must travel along the surface a 1 2 distance at least L . What is L ? Proposed by: Saranesh Prembabu √ 17+7 5 Answer: 2 Consider the net of the dodecahedron. It suffices to look at three pentagons ABCDE , EDF GH , and 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