HMMT 二月 2008 · TEAM2 赛 · 第 9 题

HMMT February 2008 — TEAM2 Round — Problem 9

[ 5 ] Show that the converse of the previous statement is false by providing a non-juggling 1 sequence j (0) j (1) j (2) of length 3 where the average ( j (0) + j (1) + j (2)) is an integer. Show 3 that your example works. 2 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

[ 5 ] Show that the converse of the previous statement is false by providing a non-juggling 1 sequence j (0) j (1) j (2) of length 3 where the average ( j (0) + j (1) + j (2)) is an integer. Show 3 that your example works. Solution: One such example is 210. This is not a juggling sequence since f (0) = f (1) = 2. 4 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.