HMMT 二月 2008 · TEAM1 赛 · 第 9 题
HMMT February 2008 — TEAM1 Round — Problem 9
题目详情
英文原题
- [ 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 wherethe incircle of ABC touches sides BC, CA, AB , respectively.
IEFA
B D C
At the end of this section you can find some terminology and theorems that may be helpful to you.
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解析
英文解析
- [ 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 thata a d df (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 wherethe incircle of ABC touches sides BC, CA, AB , respectively.
A3
B D CIEF
At the end of this section you can find some terminology and theorems that may be helpful to you.
′