HMMT 二月 2012 · TEAM2 赛 · 第 2 题

英文解析

2. [ 10 ] You are given two line segments of length 2 for each integer 0 ≤ n ≤ 10. How many distinctnondegenerate triangles can you form with three of the segments? Two triangles are considered distinctif they are not congruent.
   a b c
   Answer: 55 First, observe that if we have three sticks of distinct lengths 2 < 2 < 2 , thena b b +1 c
   2 + 2 < 2 ≤ 2 , so we cannot form a triangle. Thus, we must have (exactly) two of our sticks thea a bsame length, so that our triangle has side lengths 2 , 2 , 2 . This triangle is non-degenerate if and only
   ( )
   a +1 b 11
   if 2 > 2 , and since a 6 = b , this happens if and only if a > b . Clearly, there are = 55 ways tochoose such a, b .2