HMMT 二月 2019 · 冲刺赛 · 第 4 题

HMMT February 2019 — Guts Round — Problem 4

[ 3 ] Tessa has a figure created by adding a semicircle of radius 1 on each side of an equilateral triangle with side length 2, with semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2019, February 16, 2019 — GUTS ROUND Organization Team Team ID#

解析

[ 3 ] Tessa has a figure created by adding a semicircle of radius 1 on each side of an equilateral triangle with side length 2, with semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points? Proposed by: Yuan Yao Answer: 3 Note that both points must be in different semicircles to reach the maximum distance. Let these points be M and N , and O and O be the centers of the two semicircles where they lie respectively. Then 1 2 M N ≤ M O + O O + O N. 1 1 2 2 Note that the the right side will always be equal to 3 ( M O = O N = 1 from the radius condition, 1 2 and O O = 1 from being a midline of the equliateral triangle), hence M N can be at most 3. Finally, 1 2 if the four points are collinear (when M and N are defined as the intersection of line O O with the 1 2 two semicircles), then equality will hold. Therefore, the greatest possible distance between M and N is 3.