HMMT 二月 2008 · TEAM2 赛 · 第 9 题
HMMT February 2008 — TEAM2 Round — Problem 9
题目详情
英文原题
- [ 5 ] Show that the converse of the previous statement is false by providing a non-jugglingsequence j (0) j (1) j (2) of length 3 where the average ( j (0) + j (1) + j (2)) is an integer. Show 1
that your example works.3
Incircles [180]2
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.
解析
英文解析
- [ 5 ] Show that the converse of the previous statement is false by providing a non-jugglingsequence j (0) j (1) j (2) of length 3 where the average ( j (0) + j (1) + j (2)) is an integer. Show 1
that your example works.3
Solution: One such example is 210. This is not a juggling sequence since f (0) = f (1) = 2.
Incircles [180]4
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.