HMMT 二月 2006 · COMB 赛 · 第 4 题
HMMT February 2006 — COMB Round — Problem 4
题目详情
英文原题
- A dot is marked at each vertex of a triangle ABC . Then, 2, 3, and 7 more dots are marked on thesides AB , BC , and CA , respectively. How many triangles have their vertices at these dots?
解析
英文解析
- A dot is marked at each vertex of a triangle ABC . Then, 2, 3, and 7 more dots aremarked on the sides AB , BC , and CA , respectively. How many triangles have theirvertices at these dots?
Answer: 357
( )
Solution: Altogether there are 3 + 2 + 3 + 7 = 15 dots, and thus = 45515
( ) ( ) ( )3
2+2 2+3 2+7
combinations of 3 dots. Of these combinations, + + = 4 + 10 + 84 = 98
3 3 3
do not give triangles because they are collinear (the rest do give triangles). Thus
455 − 98 = 357 different triangles can be formed.