HMMT 二月 2008 · TEAM1 赛 · 第 9 题

HMMT February 2008 — TEAM1 Round — Problem 9

[ 35 ] Determine, with proof, a simple closed-form expression for f (2 , d ). Incircles [ 180 ] In the following problems, ABC is a triangle with incenter I . Let D, E, F denote the points where the incircle of ABC touches sides BC, CA, AB , respectively. A F E I B D C At the end of this section you can find some terminology and theorems that may be helpful to you. ′

解析

[ 35 ] Determine, with proof, a simple closed-form expression for f (2 , d ). a d Answer: (2 − 1)2 + 1 a a d Solution: From Problem ?? , f (2 , d ) ≥ (2 − 1)2 + 1. We prove by induction on a that a a d d f (2 , d ) ≤ (2 − 1)2 + 1. When a = 1, Problem ?? shows that f (2 , d ) ≤ 2 + 1. Fix a > 1 and suppose that the assertion holds for smaller values of a . Using Problem ?? , ( ) a a − 1 f (2 , d ) ≤ f (2 , d ) + 2 f (2 , d ) − 1 d a − 1 d ≤ 2 + 1 + 2 · (2 − 1)2 a d = (2 − 1)2 + 1 . a a d Thus f (2 , d ) = (2 − 1)2 + 1. Incircles [ 180 ] In the following problems, ABC is a triangle with incenter I . Let D, E, F denote the points where the incircle of ABC touches sides BC, CA, AB , respectively. 3 A F E I B D C At the end of this section you can find some terminology and theorems that may be helpful to you. ′