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HMMT 十一月 2014 · 冲刺赛 · 第 10 题

HMMT November 2014 — Guts Round — Problem 10

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 8 ] Let ABC be a triangle with CA = CB = 5 and AB = 8. A circle ω is drawn such that the interior of triangle ABC is completely contained in the interior of ω . Find the smallest possible area of ω .
解析
  1. [ 8 ] Let ABC be a triangle with CA = CB = 5 and AB = 8. A circle ω is drawn such that the interior of triangle ABC is completely contained in the interior of ω . Find the smallest possible area of ω . Answer: 16 π We need to contain the interior of AB , so the diameter is at least 8. This bound is sharp because the circle with diameter AB contains all of ABC . Hence the minimal area is 16 π .