HMMT 二月 2001 · 冲刺赛 · 第 5 题
HMMT February 2001 — Guts Round — Problem 5
题目详情
- [ ± 6] Let ABC be a triangle with incenter I and circumcenter O . Let the circumradius be R . What is the least upper bound of all possible values of IO ?
解析
- [ ± 6] Let ABC be a triangle with incenter I and circumcenter O . Let the circumradius be R . What is the least upper bound of all possible values of IO ? Solution: I always lies inside the convex hull of ABC , which in turn always lies in the circumcircle of ABC , so IO < R . On the other hand, if we first draw the circle Ω of radius R about O and then pick A , B , and C very close together on it, we can force the convex hull of ABC to lie outside the circle of radius R − about O for any . Thus the answer is R .